Quantum Information, Bell
Inequalities and the
No-Signalling Principle
Damia´n Pitalu´a-Garc´ıa
Gonville and Caius College
University of Cambridge
A thesis submitted for the degree of
Doctor of Philosophy
September 2013

Declaration
This dissertation is the result of my own work and includes nothing
which is the outcome of work done in collaboration except where
specifically indicated in the text.
The main results presented in chapter 2 were obtained in collaboration
with my PhD supervisor, Adrian Kent.

A mis padres, a mi hermana, y a mis abuelos.

Acknowledgements
I am extremely grateful to Adrian Kent for supervising my PhD. His
creative, rigorous and passionate approach to physics has influenced
my attitude towards research. He always encouraged me to think
freely and to pursue my own ideas, and offered me advice and support.
I also thank Adrian for inviting me to collaborate in the interesting
research project whose results correspond to chapter 2 of this thesis.
I enjoyed very much our collaborative work and many interesting and
very helpful conversations we had about quantum physics and my
research.
I would like to thank all the members of the Centre for Quantum
Information and Foundations in the Department of Applied Mathe-
matics and Theoretical Physics for their support during my PhD. I
am particularly grateful to Tony Short, Boris Groisman and Nilan-
jana Datta for several very helpful discussions about my research. I
am also grateful to Sabri Al-Safi, Sergii Strelchuk and Min-Hsiu Hsieh
for helpful conversations.
I would also like to thank Boris Bukh for helpful discussions, my
College tutor Jonathan Evans for his support, and my examiners Boris
Groisman and Jonathan Barrett for their careful reading of this thesis
and helpful feedback.
I thank all my friends in Cambridge for very enjoyable times, and my
friends in Mexico and my family for their encouragement.
I am very grateful to CONACYT Me´xico for sponsoring my PhD, to
Gobierno de Veracruz for partial support, and to Gonville and Caius
College for travel grants.

Abstract
This PhD thesis contains a general introduction and three main chap-
ters. Chapter 2 investigates Bell inequalities that generalize the CHSH
and Braunstein-Caves inequalities. Chapter 3 shows a derivation of
an upper bound on the success probability of a class of quantum tele-
portation protocols, denoted as port-based teleportation, from the
no-cloning theorem and the no-signalling principle. Chapter 4 intro-
duces the principle of quantum information causality.
Chapter 2 considers the predictions of quantum theory and local hid-
den variable theories (LHVT) for the correlations obtained by mea-
suring a pair of qubits by projections defined by randomly chosen axes
separated by a given angle θ. The predictions of LHVT correspond
to binary colourings of the Bloch sphere with antipodal points oppo-
sitely coloured. We show a Bell inequality for all θ, which generalizes
the CHSH and the Braunstein-Caves inequalities in the sense that the
measurement choices are not restricted to be in a finite set, but are
constrained only by the angle θ. We motivate and explore the hypoth-
esis that for a continuous range of θ > 0, the maximum correlation
(anticorrelation) is obtained by assigning to one qubit the colouring
with one hemisphere black and the other white, and assigning the
same (reverse) colouring to the other qubit. We describe numerical
tests that are consistent with this hypothesis and bound the range of
θ.
Chapter 3 shows a derivation of an upper bound on the success prob-
ability of port-based teleportation from the no-cloning theorem and
the no-signalling principle.
Chapter 4 introduces the principle of quantum information causality,
a quantum version of the information causality principle. The quan-
tum information causality principle states the maximum amount of
quantum information that a transmitted quantum system can commu-
nicate as a function of its dimension, independently of any quantum
physical resources previously shared by the communicating parties.
These principles reduce to the no-signalling principle if no systems are
transmitted. We present a new quantum information task, the quan-
tum information causality game, whose success probability is upper
bounded by the new principle, and show that an optimal strategy to
perform it combines the quantum teleportation and superdense coding
protocols with a task that has classical inputs.
Contents
List of Figures xv
1 Introduction 1
1.1 History of Quantum Information . . . . . . . . . . . . . . . . . . 1
1.2 Quantum Information . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Quantum Operations . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 The Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.3.1 Classical Entropy . . . . . . . . . . . . . . . . . . 10
1.2.3.2 Quantum Entropy . . . . . . . . . . . . . . . . . 12
1.2.4 Fundamental Principles . . . . . . . . . . . . . . . . . . . . 14
1.2.4.1 The No-Cloning Theorem . . . . . . . . . . . . . 15
1.2.4.2 The No-Signalling Principle . . . . . . . . . . . . 16
1.2.5 Fundamental Protocols . . . . . . . . . . . . . . . . . . . . 16
1.2.5.1 Superdense Coding . . . . . . . . . . . . . . . . . 16
1.2.5.2 Quantum Teleportation . . . . . . . . . . . . . . 18
1.3 Bell Inequalities and the No-Signalling Principle . . . . . . . . . . 21
1.3.1 Bell Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.1.1 The EPR Argument . . . . . . . . . . . . . . . . 23
1.3.1.2 Bell’s Theorem . . . . . . . . . . . . . . . . . . . 25
1.3.1.3 The CHSH Inequality . . . . . . . . . . . . . . . 26
1.3.1.4 The Braunstein-Caves Inequality . . . . . . . . . 28
1.3.1.5 Bell Experiments and Loopholes . . . . . . . . . 30
1.3.2 The No-Signalling Principle . . . . . . . . . . . . . . . . . 33
xi
Contents
1.3.2.1 No-Signalling and the CHSH Inequality . . . . . 38
1.3.2.2 No-Signalling and Quantum Information . . . . . 41
2 Bloch Sphere Colourings and Bell Inequalities 45
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2 Bloch Sphere Colourings and Correlation Functions . . . . . . . . 47
2.3 The Hemispherical Colouring Maximality Hypothesis . . . . . . . 52
2.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.5 Related Questions for Exploration . . . . . . . . . . . . . . . . . . 64
2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3 Bound on the Success Probability of Port-Based Teleportation
from No-Cloning and No-Signalling 77
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.1.1 Port-Based Teleportation . . . . . . . . . . . . . . . . . . . 78
3.2 The Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3 Summary of the Proof . . . . . . . . . . . . . . . . . . . . . . . . 80
3.4 A More General No-Cloning Theorem . . . . . . . . . . . . . . . . 84
3.5 Conditions on the Port States . . . . . . . . . . . . . . . . . . . . 90
3.6 Implications from Superdense Coding . . . . . . . . . . . . . . . . 92
3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4 Quantum Information Causality 99
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.1.1 The Holevo Bound . . . . . . . . . . . . . . . . . . . . . . 100
4.1.2 Information Causality . . . . . . . . . . . . . . . . . . . . 101
4.2 Quantum Information Causality . . . . . . . . . . . . . . . . . . . 107
4.2.1 Achievability of the Bound . . . . . . . . . . . . . . . . . . 109
4.2.2 The Case of Information Causality . . . . . . . . . . . . . 111
4.3 The Quantum Information Causality Game . . . . . . . . . . . . . 111
4.4 Upper Bound on the Success Probability in the QIC Game . . . . 115
4.4.1 Equivalence of the Two Versions of the Game . . . . . . . 117
4.4.2 Reduction to a Covariant Strategy . . . . . . . . . . . . . 120
4.4.3 A Useful Bound . . . . . . . . . . . . . . . . . . . . . . . . 122
xii
Contents
4.5 Strategies in the QIC Game . . . . . . . . . . . . . . . . . . . . . 123
4.5.1 Teleportation Strategies . . . . . . . . . . . . . . . . . . . 124
4.5.2 An Optimal Strategy . . . . . . . . . . . . . . . . . . . . . 128
4.5.3 Nonlocal Strategies . . . . . . . . . . . . . . . . . . . . . . 131
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5 Conclusions 135
A Details of Numerical Work 139
B Code for the Computer Program 147
C Details of the Primed PBT Protocol 155
D Bound for Nonlocal Strategies in the IC-2 Game 159
References 165
xiii

List of Figures
1.1 Spacetime diagram showing space-like and time-like separations . 22
1.2 Spacetime diagram of the EPR-Bohm experiment . . . . . . . . . 24
1.3 Spacetime diagram showing that superluminal signalling leads to
violation of relativistic causality . . . . . . . . . . . . . . . . . . . 34
1.4 Schematic of an instantaneous nonlocal quantum computation . . 43
2.1 Alice’s and Bob’s measurement axes separated by an angle θ . . . 49
2.2 Diagram of the measurements performed by Alice and Bob that
are used in the proof of Lemma 2.2 . . . . . . . . . . . . . . . . . 54
2.3 Diagram of the measurements performed by Alice and Bob that
are used in the proof of Theorem 2.1 . . . . . . . . . . . . . . . . 56
2.4 Some antipodal colouring functions on the sphere . . . . . . . . . 61
2.5 Plots of the correlations for the antipodal colouring functions shown
schematically in Figure 2.4, the singlet state quantum correlation,
and the bounds given by Theorem 2.1 . . . . . . . . . . . . . . . . 62
2.6 Plots of correlations for colouring 3δ . . . . . . . . . . . . . . . . . 63
2.7 Plots of correlations for colouring 2∆ . . . . . . . . . . . . . . . . 63
2.8 Colourings Aν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.9 Colourings Dν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.10 Set of measurements on the sphere . . . . . . . . . . . . . . . . . 74
3.1 Probabilistic port-based teleportation . . . . . . . . . . . . . . . . 82
3.2 A superdense coding protocol without communication . . . . . . . 93
4.1 The information causality game . . . . . . . . . . . . . . . . . . . 102
xv
List of Figures
4.2 Setting for quantum information causality . . . . . . . . . . . . . 108
4.3 Version I of the QIC game . . . . . . . . . . . . . . . . . . . . . . 113
4.4 The IC-2 game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.5 Teleportation strategies in the QIC game . . . . . . . . . . . . . . 126
4.6 Plots of the success probabilities in the QIC game for m = 1
achieved with different strategies and the upper bound obtained
from quantum information causality . . . . . . . . . . . . . . . . . 127
4.7 Superdense coding strategies in the IC-2 game . . . . . . . . . . . 130
B.1 Code for the computer program that defines the integral function
and the correlation function C2(θ) . . . . . . . . . . . . . . . . . . 148
B.2 Code for the computer program that defines the correlation func-
tion C3(θ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
B.3 Code for the computer program that defines the correlation func-
tion C4(θ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
B.4 Code for the computer program that defines the correlation func-
tion C2∆(θ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
B.5 Code for the computer program that defines the correlation func-
tion C3δ(θ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
B.6 Code for the computer program that outputs the values for the
correlation functions plotted in Figures 2.5, 2.6 and 2.7 . . . . . . 153
xvi
Chapter 1
Introduction
“I think I can safely say that nobody understands quantum mechan-
ics.” – Richard Feynman
1.1 History of Quantum Information
Quantum physics originated at the end of the nineteenth century. At that time,
there were physical phenomena that could not be explained with the existing
physical theories, which now we call classical physics. The problem of the black
body radiation was a cornerstone for the development of quantum mechanics, also
called quantum theory. The energy spectrum that classical physics predicted for
the radiation emitted by a black body, a perfect absorber and emitter of radiation,
in a thermal bath at a constant temperature was different to what was observed
experimentally. In 1900, Max Planck discovered that if the black body emitted
and absorbed radiation in discrete packets of energy proportional to the radiation
frequency then the energy spectrum observed experimentally would be justified
theoretically. In 1905, Albert Einstein proposed that not only the electromagnetic
radiation was interchanged discretely, but also that it was discrete itself. These
ideas developed later into the concept of the photon, a particle of light. The
emerging theory took its name from the Latin word ‘quantus’, which means ‘how
much’, to refer to the discreteness of energy discovered by Planck and Einstein.
Nonrelativistic quantum mechanics was developed in the first decades of the
1
Chapter 1. Introduction
twentieth century mainly by Max Planck, Albert Einstein, Niels Bohr, Werner
Heisenberg, Max Born, Louis de Broglie, Erwin Schro¨dinger, Wolfang Pauli, Paul
Dirac and John von Neumann. In 1926, Erwin Schro¨dinger obtained an equa-
tion that described the time evolution of the quantum state. The linearity of
Schro¨dinger’s equation implies that quantum systems can be in a linear superpo-
sition of different quantum states. When applied to composite systems, quantum
superposition leads to the property of quantum entanglement. Two systems that
are entangled present correlations that cannot be explained by classical physics.
In 1935, Einstein, Podolsky and Rosen found an apparent paradox arisen from
quantum entanglement [1]. The EPR argument considers a thought experiment
in which a pair of particles created in an entangled state are sent to different
laboratories that are arbitrarily far-apart. EPR proposed a criterion for the
existence of an element of physical reality associated to a physical quantity and
assumed local causality : the elements of physical reality associated to one of
the particles cannot be instantaneously altered by an experiment performed on
the other distant particle. They concluded that there are elements of reality
associated to two physical quantities corresponding to one of the particles that
quantum mechanics does not describe simultaneously, and thus that quantum
mechanics does not provide a complete description of physical reality.
In 1964, John Bell gave a mathematical description for the criterion of physical
reality and the assumption of local causality made by EPR [2]. The hypothetical
physical theories satisfying these conditions are denoted as local hidden variable
theories (LHVT). Bell proved that there are statistical prediction of quantum
mechanics that cannot be explained by LHVT. Bell’s model for LHVT allows
the derivation of some inequalities, the Bell inequalities, for the correlations be-
tween measurement outcomes obtained on distant physical systems. The Bell
inequalities, satisfied by LHVT, can be violated by entangled quantum systems.
The violation of the Bell inequalities, commonly associated with the term of
nonlocality, has been verified experimentally. Nevertheless, the Bell experiments
performed so far present experimental deficiencies called loopholes, which do not
allow us to make a definite claim about whether nature violates the EPR criterion
of reality and local causality.
The phenomena of quantum superposition, quantum entanglement and quan-
2
1.1. History of Quantum Information
tum nonlocality have important applications to the processing of information.
The idea that quantum systems can be used to encode and process information
evolved in the last decades of the twentieth century. The mathematical theory
of classical information began with a paper by Claude Shannon in 1948 [3], in
which the classical entropy for random variables was defined. An extension of
Shannon’s entropy for quantum systems is the quantum entropy, originally de-
fined by John von Neumann in 1932 [4]. In 1969, Stephen Wiesner developed
two protocols for coding information using quantum mechanics, which were not
accepted for publication at that time [5], but which were published in 1983 with
the name of conjugate coding [6]. The first protocol provides a way to transfer two
messages that are encoded in the polarization of light, but only one of them can
be received. The second protocol presents the concept of quantum money : money
encoded in quantum systems that is impossible to counterfeit due to the laws of
quantum mechanics. In 1973, Alexander Holevo proved a bound on the amount
of classical information that can be communicated by transmitting a quantum
system [7]. In 1982, the impossibility of copying unknown quantum states, the
no-cloning theorem, was proven by William Wootters and Wojciech Zurek [8],
and independently by Dennis Dieks [9]. The stronger no-cloning theorem was
proven by Richard Jozsa [10, 11]. In 1992, Charles Bennett and Stephen Wies-
ner presented the superdense coding protocol, in which a two bit message can be
communicated by transmitting only a spin-12 particle if the communicating par-
ties share quantum entanglement [12]. The quantum teleportation protocol was
published in 1993 by Charles Bennett, Gilles Brassard, Claude Cre´peau, Richard
Jozsa, Asher Peres and William Wootters [13]. In quantum teleportation, quan-
tum entanglement shared by two distant parties allows one party to transfer an
unknown quantum state at her location to the other party, by only communi-
cating classical information. In 1995, Benjamin Schumacher presented the idea
of storing information in quantum states and compressing the quantum informa-
tion. Schumacher’s theorem states that the minimum rate at which a quantum
information source can be compressed is given by its quantum entropy. Moreover,
Schumacher named in his paper for the first time the elementary unit of quantum
information as the quantum bit, or qubit [14].
Modern computer science began with a paper published by Alan Turing in
3
Chapter 1. Introduction
1936 [15]. Turing proposed a model of computation, named a Turing machine in
his honour. The concept of a quantum computer, a computer operating with the
laws of quantum mechanics, was first raised by Paul Benioff [16–18] and Richard
Feynman [19,20] in the early 1980s. The first formal model of quantum computa-
tion was introduced by David Deutsch in 1985 [21]. In 1992, David Deutsch and
Richard Jozsa presented an algorithm for quantum computation that is exponen-
tially faster than any algorithm performed on a classical computer [22]. In 1994,
Peter Shor obtained a quantum algorithm that finds the prime factors of a com-
posite number in a polynomial time [23], which cannot be achieved by any known
classical factoring algorithm. Since the security of many cryptographic systems
used today are based on the mathematical difficulty of finding the prime factors of
a large composite number, a quantum computer running Shor’s algorithm would
be able to decrypt such systems.
On the one hand, quantum computers would make the currently used cryp-
tography insecure. On the other hand, quantum systems can be used in new
models of cryptography, whose security is guaranteed by the laws of quantum
mechanics. Quantum cryptography has its roots in the 1960s ideas of Wiesner,
which were published until 1983. The term ‘quantum cryptography’ was used
for the first time in 1982, in a work by Charles Bennett, Gilles Brassard, Seth
Breidbart and Stephen Wiesner [24]. It was consolidated by Bennett and Bras-
sard in 1984, when they presented the first quantum key distribution protocol,
the BB84 protocol [25]. Quantum key distribution allows two parties to generate
a random secret string of bits, a key, which is used to encode secret messages.
In the BB84 protocol, the key is encoded in a series of quantum systems that
are prepared from a set of quantum states that are not mutually orthogonal.
The impossibility of perfectly distinguishing non-orthogonal quantum states im-
plies the security of the protocol. A different quantum key distribution protocol
was proposed by Artur Ekert in 1991 [26]. Ekert’s protocol requires that the
communicating parties share entangled particles. Its security is guaranteed if a
Bell inequality is violated. Jonathan Barrett, Lucien Hardy and Adrian Kent
showed in 2005 that quantum key distribution is secure even against eavesdrop-
pers not restricted by the laws of quantum mechanics, as long as the impossibility
of sending messages faster than the speed of light is satisfied [27], leading to the
4
1.2. Quantum Information
development of device-independent quantum key distribution.
1.2 Quantum Information
Quantum information science studies how information can fundamentally be en-
coded, processed and communicated using quantum systems [5]. Quantum sys-
tems are described by quantum states. The quantum state allows us to compute
the outcome probabilities for the measurement of the physical properties of the
described system. Quantum states can be pure or mixed. Mathematically, a pure
state is a vector |ψ〉 in a Hilbert space H. A mixed state is a density opera-
tor, also called a density matrix, which is a positive linear operator ρ ∈ D(H)
of unit trace, where we define D(H) to be the set of density operators acting
on the Hilbert space H. The dimension of the Hilbert space equals the number
of possible distinguishable outcomes in a measurement of the described system.
For example, if the described physical system is the polarization of a photon or
the spin of an electron, whose measurement gives one of two possible values, the
dimension of the associated Hilbert space is two. An infinite dimensional Hilbert
space is associated, for example, to the spatial position of a particle or to a field
in relativistic quantum theory. In this thesis, we only consider nonrelativistic
quantum theory and finite dimensional Hilbert spaces. We are interested in the
properties of the quantum state, but not in the particular physical system that
is described. For example, if we talk about a quantum state with Hilbert space
of dimension two, we do not consider whether it describes the polarization of a
photon, the spin of an electron or any other physical system. That is, we only
discuss the properties of the quantum information.
1.2.1 Quantum Operations
A general physical operation allowed by quantum theory is called a quantum
operation. There are three elementary quantum operations, from which a general
quantum operation can be implemented [5].
Unitary evolution The time evolution of a quantum state |ψ(t)〉 is given by
5
Chapter 1. Introduction
the Schro¨dinger equation:
i~
∂
∂t
|ψ(t)〉 = H(t)|ψ(t)〉, (1.1)
where H(t) is the corresponding Hamiltonian and h = 2pi~ is Planck’s
constant. If the described quantum system is in a quantum state |ψ(t0)〉 at
the time t0, its quantum state at the time tf > t0 is
|ψ(tf )〉 = exp
(
−
i
~
∫ tf
t0
dtH(t)
)
|ψ(t0)〉. (1.2)
Since the Hamiltonian is a Hermitian operator, that is H(t) = H†(t), the
operator U ≡ exp
(
− i~
∫ tf
t0
dtH(t)
)
is unitary : UU † = U †U = I.
Adding or discarding a system If the original system A is in a state ρA ∈
D(HA) and an ancilla E in a state ρE ∈ D(HE) is added, the combined
system AE is in the state ρA ⊗ ρE ∈ D(HA ⊗HE), where ⊗ denotes the
tensor product. If an original composite system AB is in a state ρAB ∈
D(HA ⊗HB) and the system B is discarded, the state ρA of the system A
is obtained by taking the partial trace over HB: ρA = TrB(ρAB) ∈ D(HA).
Projective measurements A projective measurement consists of a set of pro-
jectors {Πj}
n−1
j=0 , where n is the number of possible measurement outcomes.
The projectors are linear operators that are Hermitian and satisfy ΠjΠk =
δj,kΠj and
∑n−1
j=0 Πj = I. If the system subject to the projective measure-
ment is in the state ρ before the measurement then the outcome k is ob-
tained with probability P (k) = Tr(Πkρ). After the outcome k is obtained,
the state transforms into ΠkρΠkTr(Πkρ) .
An arbitrary quantum operation on a system A can be implemented by adding
an ancillaE of sufficiently big dimension, applying a unitary operation on the joint
system AE, possibly performing a projective measurement, and finally discarding
the ancilla. For example, a generalized measurement on a system A can be
implemented by adding an ancilla E, applying a unitary operation on AE, and
then performing a projective measurement on AE.
6
1.2. Quantum Information
A generalized measurement of n possible outcomes consists of a set of measure-
ment operators {Mj}
n−1
j=0 , which are linear operators satisfying the completeness
equation:
n−1∑
j=0
M †jMj = I. (1.3)
If the measured system is in the state ρ before the measurement then the outcome
k is obtained with probability P (k) = Tr(MkρM
†
k). After the outcome k is
obtained, the state transforms into MkρM
†
k
Tr(MkρM
†
k)
.
It is often useful to analyze the outcome probabilities of a quantum mea-
surement in terms of a Positive Operator-Valued Measure (POVM ). A POVM
corresponding to a measurement with n possible outcomes consists of a set of n
POVM elements {Fj}
n−1
j=0 , which are positive operators satisfying
∑n−1
j=0 Fj = I.
The probability that a measurement outcome k is obtained when a state ρ is
subject to the measurement is P (k) = Tr(Fkρ). The POVM elements are related
to the measurement operators by Fj = M
†
jMj.
A useful mathematical description of a quantum operation is given by the
operator-sum representation. The operator sum representation of a quantum
operation E is the following:
E(ρ) =
∑
j
EjρE
†
j , (1.4)
where {Ej} is a set of linear operators, the Kraus operators, that satisfy
∑
j
E†jEj ≤ I. (1.5)
The more general quantum operations are non-trace-preserving, for which the
previous equation is satisfied. If
∑
j E
†
jEj = I, the quantum operation is trace-
preserving. A non-trace-preserving quantum operation corresponds to a process
in which information is obtained due to a quantum measurement.
7
Chapter 1. Introduction
An important quantum operation is the depolarizing map. It transforms a
state ρ ∈ D(Cd) as follows:
E(ρ) = p
I
d
+ (1− p)ρ, (1.6)
where p is the probability that ρ is replaced by the completely mixed state Id . The
depolarizing map is covariant. A covariant map Ecov is such that
Ecov(UρU †) = UEcov(ρ)U †, (1.7)
for any quantum state ρ ∈ D(Cd) and unitary operation U ∈ SU(d), where SU(d)
is the special unitary group of degree d.
1.2.2 The Qubit
The elementary unit of quantum information is the qubit, or quantum bit. The
qubit is defined as a quantum system with Hilbert space of dimension two. The
qubit is the quantum generalization of a bit. A bit can be in one of two possible
states: ‘0’ or ‘1’. A probabilistic bit is described by its probability of being in the
state 0, which can be described geometrically by a point on a line of unit length.
The mathematical structure of a qubit is much richer than that of a bit. Its
quantum state can be visualized geometrically by a point in a sphere, the Bloch
sphere [5].
The quantum state ρ of a qubit is related to the Bloch sphere through its
Bloch vector ~r ∈ R3, with ‖~r‖ ≤ 1. The relation is
ρ =
1
2
(I + ~r · ~σ), (1.8)
where ~σ ≡ (σx, σy, σz), I is the identity on C2 and
σx ≡ σ1 ≡
(
0 1
1 0
)
, σy ≡ σ2 ≡
(
0 −i
i 0
)
, σz ≡ σ3 ≡
(
1 0
0 −1
)
,
are the Pauli matrices. Pure states define the surface of the sphere: ‖~r‖ =
1. Mixed states are associated with the interior of the sphere: ‖~r‖ < 1. The
8
1.2. Quantum Information
completely mixed state I2 has Bloch vector zero and corresponds to the centre of
the sphere. The north and south poles correspond to the eigenstates of σz with
eigenvalues 1 and −1, which are denoted as |0〉 and |1〉, respectively. Antipodal
points on the Bloch sphere define an orthonormal basis. The basis {|0〉, |1〉} is
called the computational basis. The expansion of a pure state |ψ〉 of a qubit in
this basis is
|ψ〉 = cos
(θ
2
)
|0〉+ eiφ sin
(θ
2
)
|1〉, (1.9)
where θ ∈ [0, pi] and φ ∈ [0, 2pi] are the polar and azimuthal angles of the Bloch
vector, respectively.
An arbitrary unitary operation on a qubit state can be expressed as follows:
U = eiαe−i
β
2 nˆ·~σ, (1.10)
where α, β ∈ R and nˆ is a unit vector in R3. Up to the global phase eiα, the
unitary operation U has the effect of rotating the Bloch vector by an angle β
along the axis nˆ in the Bloch sphere.
The depolarizing map applied to a qubit state ρ is
E(ρ) = p
I
2
+ (1− p)ρ, (1.11)
where p is the probability that ρ is replaced by the completely mixed state. It
has the effect of contracting uniformly the Bloch sphere as a function of p. The
depolarizing map is the only covariant map acting on a qubit.
A useful identity that is satisfied for any qubit density matrix ρ is the following:
I
2
=
1
4
(ρ+ σxρσx + σyρσy + σzρσz). (1.12)
If we substitute I2 , as given by the previous identity, into (1.11), we obtain that the
operator-sum representation of the qubit depolarizing map has Kraus operators
E0 =
√
1− 3p4 I and Ei =
√
p
2 σi, for i = 1, 2, 3.
A quantum system of dimension d is called a qudit, where d is an integer
bigger than two. In this thesis, we usually consider sets of n qubits, which form
qudits of dimension 2n. In general, these qubits can be entangled. A bipartite
9
Chapter 1. Introduction
quantum state ρ ∈ D(H), with H = HA⊗HB, is said to be entangled if it cannot
be expressed as a convex combination of product states, that is, if it cannot be
expressed in the form
ρ =
N∑
j=1
pjηj ⊗ γj, (1.13)
for some probability distribution {pj}Nj=1, and states {ηj}
N
j=1 ∈ D(HA) and
{γj}Nj=1 ∈ D(HB). On the other hand, a state ρ that can be expressed in the
form (1.13) is called separable.
1.2.3 Entropy
In this section we briefly discuss a few properties of the entropy for classical
random variables and for quantum systems [5].
1.2.3.1 Classical Entropy
Consider a classical random variable X that takes the value x ∈ {0, 1, . . . , d− 1}
with probability Px. Shannon [3] defined the entropy of X by
H(X) ≡ −
d−1∑
x=0
Px log2 Px, (1.14)
where 0 log2 0 ≡ 0. Although there can be other definitions of entropy for classical
variables, we only discuss the Shannon entropy in this thesis, and we refer to it
as the classical entropy.
The classical entropy is a measure of how much a classical information source
can be compressed. A classical information source is defined by a set of ran-
dom variables X1, X2, . . . , XN , whose values x1, x2, . . . , xN are the outputs of the
source. An independent and identically distributed (i.i.d) information source is
one for which Xj = X for j = 1, 2, . . . , N and whose outputs are independent
and identically distributed. Shannon’s noiseless channel coding theorem [3] states
that the minimum number of bits needed to compress reliably the output of an
i.i.d information source X per use of the source in the limit N →∞ is given by
the classical entropy H(X).
10
1.2. Quantum Information
An important property of the classical entropy is that
0 ≤ H(X) ≤ log2 d. (1.15)
The entropy is zero if X is deterministic, that is, if Px = 1 for some x. It is
maximum if X is totally random, that is, if Px = 1d for x = 0, 1, . . . , d − 1. If X
is a variable of n bits, that is, if d = 2n then H(X) ≤ n.
A very useful property of the classical entropy is that it is a concave function.
Consider a set of random variables {Xj}
l−1
j=0 and a probability distribution {qj}
l−1
j=0.
Let Xj take the value x ∈ {0, 1, . . . , d − 1} with probability P
(j)
x . Consider a
random variable X ′ that takes the value x ∈ {0, 1, . . . , d − 1} with probability
P ′x ≡
∑l−1
j=0 qjP
(j)
x . The concavity property of the classical entropy states that
l−1∑
j=0
qjH(Xj) ≤ H(X
′). (1.16)
We can assign a joint entropy to a pair of random variables X and Y . Let X
and Y take the values x ∈ {0, 1, . . . , d−1} and y ∈ {0, 1, . . . , d′−1}, respectively.
Let Px,y be the probability that X = x and Y = y. The joint entropy of X and
Y is defined as
H(XY ) ≡ −
d−1∑
x=0
d′−1∑
y=0
Px,y log2 Px,y. (1.17)
An important inequality satisfied by the classical entropy is subadditivity :
H(XY ) ≤ H(X) +H(Y ), (1.18)
where the probability distributions for X and Y are obtained from the proba-
bilities Px,y as follows: Px =
∑d′−1
y=0 Px,y and Py =
∑d−1
x=0 Px,y. The equality is
achieved if and only if X and Y are independent variables, that is, if Px,y = PxPy
for all x ∈ {0, 1, . . . , d − 1} and y ∈ {0, 1, . . . , d′ − 1}. Another useful inequality
is that
H(Y ) ≤ H(XY ), (1.19)
where the equality is achieved if and only if X is a function of Y . A similar
11
Chapter 1. Introduction
inequality is obtained if X and Y are interchanged in (1.19).
The classical mutual information of X and Y is defined as
H(X : Y ) ≡ H(X) +H(Y )−H(XY ). (1.20)
It is a measure of the information shared by X and Y . From inequalities (1.18)
and (1.19), it follows that the classical mutual information satisfies
0 ≤ H(X : Y ) ≤ H(X). (1.21)
The value ofH(X : Y ) is zero ifX and Y do not share any information. It achieves
H(X : Y ) = H(X) if all the information about X is contained in Y . Formally,
H(X : Y ) = 0 if and only if X and Y are independent, and H(X : Y ) = H(X)
if and only if X is a function of Y . Similar results are obtained if X and Y are
interchanged.
1.2.3.2 Quantum Entropy
The entropy of a quantum system provides a measure of the uncertainty about
its state. The entropy of a quantum state ρ was defined by von Neumann [4], up
to a factor of ln 2, as
S(ρ) ≡ −Tr(ρ log2 ρ). (1.22)
There can be different definitions of entropy for quantum states. In this thesis
we only consider the von Neumann entropy and we refer to it as the quantum
entropy.
The quantum entropy is a measure of how much a quantum information source
can be compressed. A quantum information source is defined by a Hilbert space
H and a density matrix ρ ∈ D(H), which corresponds to an ensemble of signal
states ρi occurring with probability pi, ρ ≡
∑
i piρi. The source is independent
and identically distributed (i.i.d) if N uses of the source produce the ensemble
state ρ⊗N , that is, the outputs of different uses of the channel are in a product
state. Schumacher’s quantum noiseless channel coding theorem [14] states that
the minimum number of qubits needed to compress reliably the output of an i.i.d
quantum information source per use of the source in the limit N → ∞ is given
12
1.2. Quantum Information
by S(ρ).
The quantum entropy of a state ρ ∈ D(Cd) equals
S(ρ) = −
d−1∑
j=0
λj log2 λj, (1.23)
where {λj}
d−1
j=0 is the set of eigenvalues of ρ and we define 0 log2 0 ≡ 0. From this
expression and (1.14), we see that the quantum entropy of ρ equals the classical
entropy of the probability distribution corresponding to its eigenvalues. From
(1.15), it follows that
0 ≤ S(ρ) ≤ log2 d. (1.24)
The entropy of ρ is zero if it only has one nonzero eigenvalue, which equals unity,
that is, if ρ is pure. It achieves its maximum value log2 d if all its eigenvalues are
equal, that is, if ρ is the completely mixed state Id . From (1.24), we see that if ρ
is a state of n qubits, which means that d = 2n, we have that S(ρ) ≤ n.
Similar to the classical entropy, the quantum entropy is concave. Consider a
quantum state ρ′ ≡
∑l−1
j=0 qjρj for a probability distribution {qj}
l−1
j=0 and quantum
states {ρj}
l−1
j=0. The concavity of the quantum entropy states that
l−1∑
j=0
qjS(ρj) ≤ S(ρ
′). (1.25)
Consider two quantum systems A and B that are in a joint quantum state
ρAB. Let ρA ≡ TrB(ρAB) and ρB ≡ TrA(ρAB) be the quantum states of A
and B, respectively. In the rest of this thesis, we adopt the notation S(A) ≡
S(ρA), S(B) ≡ S(ρB) and S(AB) ≡ S(ρAB). Two important inequalities for
the quantum entropy are subadditivity and the triangle inequality, also called the
Araki-Lieb inequality. Subadditivity [28] states that
S(AB) ≤ S(A) + S(B). (1.26)
The equality in (1.26) is achieved if and only if A and B are not correlated, that
13
Chapter 1. Introduction
is, if ρAB = ρA ⊗ ρB. The triangle inequality [29] states that
|S(B)− S(A)| ≤ S(AB). (1.27)
Similar to the classical mutual information, the quantum mutual information
is defined by
I(A : B) ≡ S(A) + S(B)− S(AB). (1.28)
It is a measure of the total correlations between the quantum systems A and
B [30–32]. From (1.26) – (1.28), it follows that
0 ≤ I(A : B) ≤ 2S(A). (1.29)
This inequality is satisfied too if we exchange A and B. Notice the factor of 2
that appears in this inequality compared to the analogous inequality (1.21) for the
classical mutual information. This factor appears due to the triangle inequality
(1.27) for the quantum entropy, which cannot be saturated by classical random
variables X and Y , as follows from (1.19), unless one of these is deterministic,
say X, in which case the upper bound in (1.29) is achieved trivially: H(X : Y ) =
2H(X) = 0.
An important property of the quantum mutual information is that it cannot
increase by quantum operations that act locally on one of the systems. Consider
a quantum operation that acts locally on the system A. Let AB and A′B denote
the composite quantum systems before and after a quantum operation is applied
on A, respectively. The data-processing inequality states that
I(A′ : B) ≤ I(A : B). (1.30)
The properties of the quantum entropy discussed in this section are very useful
in chapter 4, where we introduce a new principle of quantum information.
1.2.4 Fundamental Principles
Two fundamental principles of quantum information are the no-cloning theorem
and the no-signalling principle.
14
1.2. Quantum Information
1.2.4.1 The No-Cloning Theorem
The no-cloning theorem states that an unknown quantum state cannot be copied
perfectly [8, 9]. This theorem holds even probabilistically [33]. That is, it is
impossible to produce a perfect copy of an unknown quantum state with any
nonzero success probability. We introduce a more general theorem in section 3.4,
from which these results are obtained. The proof follows easily from the unitary
evolution and the linearity of quantum theory.
A theorem in the spirit of the no-cloning theorem is the stronger no-cloning
theorem, which considers the question of how much information about a state
|ψ〉 is necessary to produce a copy of it. The stronger no-cloning theorem states
that for a set of pure states {|ψj〉}j, in which no pair of states are orthogonal,
and a set of (possibly mixed) states {ρj}j of an ancilla, the physical operation
|ψj〉 ⊗ ρj → |ψj〉 ⊗ |ψj〉 is possible if and only if the physical operation ρj → |ψj〉
is possible, that is, the ancilla must have complete information about the state
|ψj〉 [10, 11].
Although perfect quantum cloning is impossible, imperfect cloning is not. The
first discovered quantum cloning machine takes as input an unknown pure state
|ψ〉 of a qubit and gives as output two qubits with equal reduced density matrices
ρ such that the fidelity f ≡ 〈ψ|ρ|ψ〉 equals 56 [34], which is its maximum possible
value [35–37]. There exist different classes of quantum cloning machines. The op-
timal symmetric universal quantum cloning machines have been studied in great
detail. A quantum cloning machine is universal if all states are cloned equally
well. It is symmetric if it achieves the same fidelity for all the obtained clones.
It is optimal if the clones have the maximum fidelities allowed by quantum the-
ory [38]. The fidelities of the clones achieved by an optimal symmetric universal
quantum cloning machine that takes N pure qudits as inputs and outputs M
qudit clones are given by [39,40]:
f =
N
M
+
(M −N)(N + 1)
M(N + d)
. (1.31)
We see that the fidelity reduces to f = 56 for the simplest case d = 2, N = 1 and
M = 2.
15
Chapter 1. Introduction
1.2.4.2 The No-Signalling Principle
The no-signalling principle states that communication between two distant par-
ties cannot be performed without the transmission of any physical systems, de-
spite any physical resources that the parties may share. We present a mathemati-
cal expression for this principle and discuss it in detail in section 1.3.2. We discuss
in section 1.3.2 that quantum theory is consistent with relativistic causality by
satisfaction of the no-signalling principle.
1.2.5 Fundamental Protocols
We discuss two fundamental protocols of quantum information theory: superdense
coding and quantum teleportation. These protocols are performed by two distant
parties, Alice and Bob. Superdense coding, originally called 4 way coding, is a
protocol in which Alice communicates Bob two bits of classical information by
transmitting a single qubit [12]. On the other hand, quantum teleportation allows
Alice to transfer an unknown qubit state at her location to Bob’s location, by
transmitting two bits of classical information without the need to transmit any
quantum systems [13]. In the general case, these protocols consider a qudit and
a classical message of d2 possible values.
1.2.5.1 Superdense Coding
Superdense coding [12] is a protocol in which Alice communicates Bob 2 log2 d
bits of classical information, or more precisely, a message of d2 possible values, by
sending him a qudit. We first describe the protocol in the case of a qubit, d = 2.
Alice and Bob need to share a pair of qubits a, at Alice’s location, and b, at
Bob’s location, in a maximally entangled state. Up to local unitary operations,
the maximally entangled state is the singlet state: |Ψ−〉ab = 1√2(|0〉a|1〉b−|1〉a|0〉b).
Alice has a two bit message (x0, x1) that she wants to communicate Bob. Thus,
Alice applies the unitary operation σx0,x1 on her qubit a, where σ0,0 ≡ I is the
identity acting on C2 and σ0,1 ≡ σ1, σ1,0 ≡ σ2 and σ1,1 ≡ σ3 are the Pauli
matrices. The shared entangled state transforms into (σx0,x1 ⊗ I)|Ψ
−〉ab. Alice’s
16
1.2. Quantum Information
operation generates one of four mutually orthogonal states, the Bell states :
|Ψ±〉 =
1
√
2
(|0〉|1〉 ± |1〉|0〉),
|Φ±〉 =
1
√
2
(|0〉|0〉 ± |1〉|1〉). (1.32)
The transformations are obtained in the following way:
(σ0,0 ⊗ I)|Ψ
−〉ab = |Ψ
−〉ab,
(σ0,1 ⊗ I)|Ψ
−〉ab = −|Φ
−〉ab,
(σ1,0 ⊗ I)|Ψ
−〉ab = i|Φ
+〉ab,
(σ1,1 ⊗ I)|Ψ
−〉ab = |Ψ
+〉ab. (1.33)
Then, Alice sends Bob the qubit a. Bob measures the two qubit system ab in the
Bell basis. Bob’s measurement outcome indicates Alice’s message (x0, x1).
Consider now the case in which Alice and Bob have qudits a and b, respec-
tively, in a maximally entangled state |Ψ〉ab. In the schmidt basis {|l〉}
d−1
l=0 , the
expression for |Ψ〉ab is |Ψ〉ab = 1√d
∑d−1
l=0 |l〉a|l〉b. Alice wants to communicate Bob
a classical message (j, k) of d2 possible values, with j, k ∈ {0, 1, . . . , d− 1}. Thus,
Alice applies the following unitary operation on her system a:
Uj,k =
d−1∑
l=0
e
i2pilj
d |l〉〈(l + k) mod d|. (1.34)
The state |Ψ〉ab is transformed into (Uj,k ⊗ I)|Ψ〉ab = |φj,k〉ab. The transformed
state is
|φj,k〉ab =
1
√
d
d−1∑
l=0
e
i2pilj
d |l〉a|(l + k) mod d〉b. (1.35)
Alice sends Bob her system a. Since the states {|φj,k〉}
d−1
j,k=0 form an orthonormal
basis of Cd ⊗ Cd, by measuring the joint system ab in this basis, Bob obtains
Alice’s message (j, k) perfectly.
17
Chapter 1. Introduction
1.2.5.2 Quantum Teleportation
Quantum teleportation [13] provides a way for Alice to transfer an unknown
quantum state at her location to Bob’s location, which can be arbitrarily far and
possibly unknown to her, without the need to transmit any quantum systems.
We first discuss the quantum teleportation protocol for a qubit state.
Alice has a qubit a in an unknown quantum state that she wants to teleport
to Bob’s location. For convenience, we consider that a is in a pure state |ψ〉a.
Due to the linearity of quantum theory, quantum teleportation works too if a is in
a mixed state, possibly in an entangled state with another system. The physical
resource that makes quantum teleportation possible is quantum entanglement.
Alice and Bob need to share a pair of qubits A, at Alice’s location, and B, at
Bob’s location, in a maximally entangled state in order to complete quantum
teleportation faithfully. Thus, consider that Alice and Bob share a singlet state
|Ψ−〉AB = 1√2(|0〉A|1〉B − |1〉A|0〉B). Alice measures their qubits in the Bell basis
{|Ψ±〉, |Φ±〉}. The initial state of the three qubits system can be expressed by
|ψ〉a|Ψ
−〉AB = −
1
2
|Ψ−〉aA ⊗ |ψ〉B −
1
2
|Ψ+〉aA ⊗ σz|ψ〉B +
1
2
|Φ−〉aA ⊗ σx|ψ〉B
−
i
2
|Φ+〉aA ⊗ σy|ψ〉B, (1.36)
for any state |ψ〉 ∈ C2. We see that for any measurement outcome obtained
by Alice, Bob’s qubit projects into the state |ψ〉, up to a possible Pauli error
σx, σy or σz. Bob obtains the teleported state |ψ〉 perfectly after applying the
corresponding Pauli unitary operation that corrects the error. To do so, Bob
needs to receive a two bit message that indicates Alice’s measurement result.
Quantum teleportation is consistent with the no-cloning theorem. From Equa-
tion (1.36), we see that the state of Alice’s qubits projects into one of the four
Bell states. Thus, no copies of |ψ〉 are produced during quantum teleportation.
In principle, it is not necessary that Alice knows Bob’s location, because she
can broadcast her measurement result to all possible regions of space where Bob
might be. On the other hand, sending the system a directly does not allow Alice
to transfer its state to an unknown location of Bob, because according to the no-
cloning theorem, the unknown quantum state |ψ〉 cannot be copied, hence, Alice
18
1.2. Quantum Information
can only transmit a single copy of it and thus is forced to know Bob’s location.
Quantum teleportation is consistent with the no-signalling principle. From
Equation (1.36), we see that Alice’s measurement outcome is totally random. It
follows that, before receiving any message from Alice, Bob’s qubit is in the mixed
state
ρB =
1
4
(|ψ〉〈ψ|+ σx|ψ〉〈ψ|σx + σy|ψ〉〈ψ|σy + σz|ψ〉〈ψ|σz), (1.37)
which is the completely mixed state I2 , as seen from the identity (1.12). Thus,
Bob cannot obtain any information about the state |ψ〉 before receiving a message
from Alice, as stated by the no-signalling principle. Moreover, the no-signalling
principle implies that faithful teleportation of an unknown qubit state cannot be
accomplished if Alice sends Bob less than two bits of classical information, as
shown below [13].
Suppose that a message of c bits is sufficient to complete faithful teleportation
of an unknown qubit state, with c possibly smaller than two. Using the superdense
coding protocol [12], we show that the no-signalling principle implies that c = 2.
Let Alice and Bob share a singlet |Ψ−〉ab that they use to perform superdense
coding and some resource state |ξ〉AB that they use for teleportation. Alice has
the system aA and Bob has the system bB. We do not impose any conditions on
the state |ξ〉AB. Alice is given a random two bit message (x0, x1) that she encodes
in her qubit a by applying the unitary operation σx0,x1 on it, where σ0,0 ≡ I,
σ0,1 ≡ σ1, σ1,0 ≡ σ2 and σ1,1 ≡ σ3. In the superdense coding protocol, Alice then
sends Bob her qubit a and Bob learns the message (x0, x1) after measuring the
pair ab in the Bell basis. Consider instead, that Alice does not send Bob her qubit
a, but that she teleports its state to Bob’s qubit B. However, Alice does not send
Bob the c-bits message y that would allow Bob to complete the teleportation.
Bob guesses the value of y with probability 2−c, in which case, after performing
the teleportation correction operation and the Bell measurement, Bob learns the
two bit message (x0, x1). Hence, Bob learns the value of (x0, x1) with probability
2−c. Since there is not communication in this protocol, the no-signalling principle
implies that Bob can only obtain the two bit random message (x0, x1) with the
probability of making a random guess. Thus, it must be that 2−c = 14 , which
means that c = 2.
19
Chapter 1. Introduction
Now we discuss the teleportation of a quantum state of dimension bigger
than two. Teleportation of a state of n qubits can be performed by applying the
teleportation protocol described above for each of the n qubits. Teleportation
of an arbitrary qudit state |ψ〉a can be performed as follows. Alice and Bob
share a maximally entangled state |Ψ〉AB = 1√d
∑d−1
l=0 |l〉A|l〉B, where {|l〉}
d−1
l=0 is
the Schmidt basis. Alice measures her pair of qudits aA in the basis defined by
the set of orthonormal states
|φj,k〉 =
1
√
d
d−1∑
l=0
e
i2pilj
d |l〉|(l + k) mod d〉. (1.38)
Alice obtains the measurement outcome (j, k) and sends it to Bob. Then, Bob
applies the following unitary operation on B:
Uj,k =
d−1∑
l=0
e
i2pilj
d |l〉〈(l + k) mod d|. (1.39)
After Bob’s operation, the state of his system B transforms into the state |ψ〉.
Similar to the qubit case, the superdense coding protocol and the no-signalling
principle can be used to show that faithful teleportation of an unknown quantum
state of dimension d requires a message of 2 log2 d bits from Alice to Bob [13].
From Equations (1.34), (1.35), (1.38) and (1.39), we notice that the measure-
ment applied by Alice and the unitary correction operations applied by Bob in
the quantum teleportation protocol are the same as the decoding measurement
applied by Bob and the encoding unitary operations applied by Alice in the su-
perdense coding protocol, respectively. In this sense, quantum teleportation can
be considered as the inverse of superdense coding. In section 4.5, we present two
novel quantum information protocols that combine quantum teleportation and
superdense coding, and show that they satisfy a similar inverse relation.
In chapter 3, we discuss a different type of teleportation protocol, denoted
as port-based teleportation. We use the original quantum teleportation protocol
described in this section, together with the superdense coding protocol, the no-
cloning theorem and the no-signalling principle to show an upper bound on the
success probability of port-based teleportation.
20
1.3. Bell Inequalities and the No-Signalling Principle
1.3 Bell Inequalities and the No-Signalling Prin-
ciple
Causality is a fundamental physical principle, which can be understood as stating
that if P is an event occurring before another event F , P can be a cause of
F , but it cannot be a consequence of F . In other words, events in the future
are consequences of events in the past or in the present, but events in the past
cannot be consequences of events in the future. The previous statement seems
obvious according to our human experience of time. Nevertheless, giving a precise
definition of causality is a subtle problem. Although we do not attempt to define
causality in a precise way, we discuss briefly the question of causality in the
frameworks of special relativity and quantum mechanics.
According to relativity, space and time are not separate entities, but a single
physical concept called spacetime. Physical events occurring in different reference
frames in spacetime are observed differently. That is, physical quantities like
distance, time length, speed and energy are relative to observers in difference
reference frames. However, there is a physical quantity that remains constant
in all reference frames, the speed of light in the vacuum, which has a value of
2.998 x 108 meters per second, and is the maximum speed that any physical
system can have in any reference frame.
The speed of light allows us to make a precise statement of which events are
in the future and which are in the past of a given event. Consider the spacetime
diagram in Figure 1.1. The physical events A, B, C and D are observed in two
different inertial reference frames, which displace from each other at a constant
speed. The spatial and time coordinates are x and t for the unprimed frame, and
x′ and t′ for the primed frame. The speed of light in the vacuum is denoted by
c. The dashed lines represent two light beams in the vacuum that reach A and
continue their way in opposite directions. These lines define a two-dimensional
slice of two four-dimensional cones, the light cones of A. Events inside the upper
cone are in the future of A and events in the lower cone are in the past of A.
This is independent of the reference frame, because the speed of light is the same
in any reference frame. We see that the distance and time lengths between the
events observed in the unprimed frame are different in the primed frame, but in
21
Chapter 1. Introduction
both frames C and D are in the past and in the future of A, respectively. On the
other hand, B is neither in the past nor in the future of A. We say that C, A
and D are time-like separated, while A and B are space-like separated. Time-like
separated events are causally connected, but space-like separated events are not.
Thus, a relativistic notion of causality is that a physical event A can be a cause
of a physical event D, only if A and D are time-like separated and D is in the
future of A. This means in particular that if A and B are space-like separated
events then B cannot be caused by A, and vice versa. In this section we describe
how quantum mechanics shakes this notion of causality through the violation of
the Bell inequalities, but still remains consistent with it by satisfaction of the
no-signalling principle.
Figure 1.1: Spacetime diagram for physical events A, B, C and D, which are
observed in two different inertial reference frames that displace away from each
other at a constant speed. The spatial and time coordinates are x and t in one
frame and x′ and t′ in the other frame, respectively. The speed of light in the
vacuum is denoted by c. The dashed lines define a two-dimensional slice of the
four-dimensional light cones of A. The events C, A and D are time-like separated,
while A and B are space-like separated. The events C and D are in the past and
in the future of A, respectively.
22
1.3. Bell Inequalities and the No-Signalling Principle
1.3.1 Bell Inequalities
1.3.1.1 The EPR Argument
In their seminal paper [1], Einstein, Podolsky and Rosen claimed that quantum
mechanics is not a complete theory. Their argument is based on an implicit phys-
ical assumption based on relativistic causality and a defined criterion of physical
reality. These assumptions are the following.
Local causality. A physical event A cannot have any influence on another phys-
ical event B, if A and B are space-like separated.
The EPR criterion of reality. If the value of a physical quantity can be pre-
dicted with certainty without disturbing the system then there exists an element
of physical reality associated to this physical quantity.
We consider Bohm’s [41] version of the EPR thought experiment, whose space-
time diagram is given in Figure 1.2. A pair of qubits is prepared in the singlet
state |Ψ−〉 = 1√
2
(
|0〉|1〉− |1〉|0〉
)
.1 The qubits are sent to two distant laboratories,
one controlled by Alice and the other one controlled by Bob. Alice randomly
chooses a measurement A ∈ {0, 1} from a set of two elements. Similarly, Bob
randomly chooses a measurement B ∈ {0, 1}. Alice and Bob perform a projective
measurements on their qubits in a basis corresponding to the Bloch vectors ~aA
and ~bB, respectively. Alice obtains an outcome a and Bob obtains an outcome b.
The measurement outcomes are assigned numerical values a = ±1 and b = ±1, if
the qubits project into the states with Bloch vectors ±~aA and ±~bB, respectively.
A crucial property of the experiment is that the spacetime region in which Alice
chooses her measurement A is space-like separated from the spacetime region in
which Bob obtains his outcome b, and vice versa.
A property of the singlet state is that if both qubits are measured in the
same basis, the outcomes are opposite. This means that if Alice measures her
qubit in the basis corresponding to the Bloch vector ~aA and obtains outcome a
then, if Bob measures his particle in the same basis, ~bB = ~aA, Bob’s outcome
1The physical quantities considered in the EPR original argument are the momentum and
position of a particle, whereas the quantity considered in Bohm’s version is the spin of a spin- 12
particle, which is a particular type of qubit system.
23
Chapter 1. Introduction
Figure 1.2: Spacetime diagram of the EPR-Bohm experiment. The boxes rep-
resent spacetime regions in which a pair of qubits is created in the singlet state
(S) and sent to Alice’s and Bob’s distant laboratories, Alice randomly chooses
her measurement (A), Alice applies the measurement on her qubit and obtains
a measurement outcome (a), Bob randomly chooses his measurement (B), and
Bob applies the measurement on his qubit and obtains a measurement outcome
(b). The dashed lines represent the two-dimensional slices of the light cones for
the regions corresponding to the measurement choices. The regions A and b are
space-like separated. Similarly, B and a are space-like separated.
is b = −a with probability equal to unity. Since Alice’s and Bob’s experiments
are space-like separated, the assumption of local causality implies that Alice’s
experiment on her qubit cannot in any way disturb the qubit at Bob’s location,
and vice versa. Thus, local causality and the EPR criterion of reality imply the
existence of an element of physical reality associated to Bob’s qubit measurement
~bB = ~aA. This argument holds for any measurement basis ~aA chosen by Alice,
and ~bB = ~aA by Bob. Thus, EPR argued that there are elements of physical
reality associated to any measurement ~bB that Bob can perform on his qubit.
However, quantum mechanics does not describe all possible qubit measurement
values simultaneously. For example, if Bob’s measurement corresponds to~bB = zˆ,
the state of his qubit projects into one of the orthogonal states |0〉 or |1〉. Hence,
Bob obtains complete knowledge about his qubit state in this basis, but his qubit
is then in an equal superposition of the orthogonal states |+〉 = 1√
2
(
|0〉+ |1〉
)
and
24
1.3. Bell Inequalities and the No-Signalling Principle
|−〉 = 1√
2
(
|0〉 − |1〉
)
, which means that Bob does not have any knowledge about
his qubit state in the basis corresponding to ~bB = xˆ. Therefore, EPR concluded
that quantum mechanics cannot be a complete theory, because there are physical
quantities with elements of physical reality that quantum mechanics does not
describe simultaneously.
The EPR assumptions of local causality and physical reality seem as sensible
conditions for a physical theory. Nevertheless, these conditions are not necessar-
ily satisfied by nature. In the following sections we describe how hypothetical
theories based on the EPR criterion of reality and local causality can be tested
experimentally.
1.3.1.2 Bell’s Theorem
Bell [2] provided a mathematical description for local hidden variable theories
(LHVT), which are hypothetical physical theories based on the EPR assumptions
of local causality and physical reality. Bell’s formalism allows us to test these
conditions experimentally.
Consider the EPR-Bohm experiment described in the previous section. Ac-
cording to deterministic LHVT, the outcomes a and b are determined respectively
by the measurement choices A and B and by hidden variables λ shared by both
qubits, in the following way:
a = a(A, λ) ∈ {1,−1}, b = b(B, λ) ∈ {1,−1}. (1.40)
We can also consider probabilistic LHVT, which determine the outcomes a and
b with some probability of the form
P (a, b|A,B, λ) = P (a|A, λ)P (b|B, λ). (1.41)
An LHVT also assigns a probability distribution ρ(λ), independent of A and B,
to the hidden variables, satisfying positivity and normalization:
ρ(λ) ≥ 0,
∫
Λ
dλρ(λ) = 1, (1.42)
25
Chapter 1. Introduction
where Λ is the set of hidden variables.
The LHVT defined above aim to include general theories satisfying local
causality and the EPR criterion of reality. The outcome at one laboratory is
independent of the measurement choice made at the other laboratory, if the ex-
periments are performed at space-like separations, as required by local causality.
The measured properties have elements of physical reality, hence, the outcomes
are predetermined, at least probabilistically, by the values of the hidden variables
λ.
We define the correlation C(A,B) as the average value of the product of
Alice’s and Bob’s outcomes in experiments in which the measurements A and B
are chosen. For a deterministic LHVT we have
C(A,B) =
∫
Λ
dλρ(λ)a(A, λ)b(B, λ). (1.43)
A probabilistic LHVT predicts
C(A,B) =
∑
a,b∈{1,−1}
ab
∫
Λ
dλρ(λ)P (a|A, λ)P (b|B, λ). (1.44)
There are quantum correlations obtained in the EPR-Bohm experiment that
cannot be described by the general form (1.44) predicted by LHVT. Thus, we
can state Bell’s theorem as follows.
Bell’s Theorem. There exist predictions of quantum mechanics that are incon-
sistent with the predictions of local hidden variable theories.
Bell’s theorem is most easily shown by the quantum violation of some in-
equalities, the Bell inequalities, which are satisfied by LHVT. An important Bell
inequality is the CHSH inequality.
1.3.1.3 The CHSH Inequality
The correlations described by LHVT in the EPR-Bohm experiment satisfy the
CHSH inequality [42]:
|I2| ≡
∣
∣C(0, 0) + C(1, 1) + C(1, 0)− C(0, 1)
∣
∣ ≤ 2 . (1.45)
26
1.3. Bell Inequalities and the No-Signalling Principle
We first show this inequality for deterministic LHVT. We have that
|I2| =
∣
∣
∣
∣
∫
Λ
dλρ(λ)
[
a(0, λ)b(0, λ) + a(1, λ)b(1, λ) + a(1, λ)b(0, λ)− a(0, λ)b(1, λ)
]
∣
∣
∣
∣
≤
∫
Λ
dλρ(λ)
∣
∣
∣a(0, λ)
[
b(0, λ)− b(1, λ)
]
+ a(1, λ)
[
b(1, λ) + b(0, λ)
]∣∣
∣
≤
∫
Λ
dλρ(λ)
[∣
∣a(0, λ)
∣
∣
∣
∣b(0, λ)− b(1, λ)
∣
∣+
∣
∣a(1, λ)
∣
∣
∣
∣b(1, λ) + b(0, λ)
∣
∣
]
=
∫
Λ
dλρ(λ)
[∣
∣b(0, λ)− b(1, λ)
∣
∣+
∣
∣b(1, λ) + b(0, λ)
∣
∣
]
≤ 2, (1.46)
where in the first line we used (1.43) and (1.45); in the second and third lines we
arranged terms and used the properties of the modulus function; in the fourth
line we used that |a(0, λ)| = |a(1, λ)| = 1, as follows from (1.40); and in the last
line we used that the value of one of the terms is zero, while the other one is 2,
as obtained from (1.40).
Now we show the CHSH inequality for probabilistic LHVT [43]. From (1.44),
it follows straightforwardly that
C(A,B) =
∫
Λ
dλρ(λ)a¯(A, λ)b¯(B, λ), (1.47)
where a¯(A, λ) and b¯(B, λ) are averaged values defined by
a¯(A, λ) ≡ P (1|A, λ)−P (−1|A, λ), b¯(B, λ) ≡ P (1|B, λ)−P (−1|B, λ). (1.48)
Thus, replacing a(A, λ) and b(B, λ) in (1.46) by their averaged values a¯(A, λ)
and b¯(B, λ), we obtain with the same procedure the corresponding first four lines
of (1.46), with the equality sign replaced by ≤ in the fourth line due to the
inequalities
|a¯(A, λ)| ≤ 1, |b¯(B, λ)| ≤ 1. (1.49)
Thus, we have
|I2| ≤
∫
Λ
dλρ(λ)
[∣
∣b¯(0, λ)− b¯(1, λ)
∣
∣+
∣
∣b¯(1, λ) + b¯(0, λ)
∣
∣
]
. (1.50)
27
Chapter 1. Introduction
It is easy to see from (1.49) and (1.50) that
|I2| ≤
∫
Λ
dλρ(λ)2. (1.51)
Finally, it follows from (1.42) that |I2| ≤ 2, which is the CHSH inequality (1.45).
There exist quantum states and quantum measurements for which the quan-
tum correlations violate the CHSH inequality, up to the value 2
√
2, as given by
the Cirel’son [44] bound
∣
∣IQ2
∣
∣ ≤ 2
√
2, (1.52)
where the label Q indicates that the correlations in (1.45) are quantum. Consider
the singlet state quantum correlation
Q(θ) = − cos θ, (1.53)
where cos θ = ~aA · ~bB. Let ~aA = cos
(
Api
2
)
zˆ + sin
(
Api
2
)
xˆ and ~bB = cos
(
Bpi
2 +
pi
4
)
zˆ + sin
(
Bpi
2 +
pi
4
)
xˆ, for A,B ∈ {0, 1}. We see that cos θ = 1√
2
for the pairs
(A = 0, B = 0), (A = 1, B = 1) and (A = 1, B = 0), and cos θ = − 1√
2
for
(A = 0, B = 1). Thus, from (1.53), each term in (1.45) contributes with a value
of − 1√
2
. Hence, the Cirel’son bound (1.52) is achieved by this set of measurements
on the singlet state.
1.3.1.4 The Braunstein-Caves Inequality
Consider now a version of the EPR-Bohm experiment in which Alice’s and Bob’s
measurement choices belong to a set of N possible elements: A,B ∈ {0, 1, . . . , N−
1}. In this case, the correlations predicted by LHVT satisfy the Braunstein-Caves
inequality [45]:
|IN | ≡
∣
∣
∣
∣
∣
N−1∑
k=0
C(k, k) +
N−2∑
k=0
C(k + 1, k)− C(0, N − 1)
∣
∣
∣
∣
∣
≤ 2N − 2. (1.54)
28
1.3. Bell Inequalities and the No-Signalling Principle
This inequality is valid for probabilistic and deterministic LHVT. We present a
proof similar to (1.46) for deterministic LHVT:
|IN | =
∣
∣
∣
∣
∣
∫
Λ
dλρ(λ)
[
N−1∑
k=0
a(k, λ)b(k, λ)+
N−2∑
k=0
a(k+1, λ)b(k, λ)−a(0, λ)b(N−1, λ)
]∣
∣
∣
∣
∣
≤
∫
Λ
dλρ(λ)
∣
∣
∣
∣
∣
a(0, λ)
[
b(0, λ)−b(N−1, λ)
]
+
N−1∑
k=1
a(k, λ)
[
b(k, λ)+b(k−1, λ)
]
∣
∣
∣
∣
∣
≤
∫
Λ
dλρ(λ)
[
∣
∣a(0, λ)
∣
∣
∣
∣b(0, λ)−b(N−1, λ)
∣
∣+
N−1∑
k=1
∣
∣a(k, λ)
∣
∣
∣
∣b(k, λ)+b(k−1, λ)
∣
∣
]
=
∫
Λ
dλρ(λ)
[
∣
∣b(0, λ)−b(N−1, λ)
∣
∣+
N−1∑
k=1
∣
∣b(k, λ)+b(k−1, λ)
∣
∣
]
≤ 2N−2, (1.55)
where in the first line we used (1.43) and (1.54); in the second and third lines we
arranged terms and used the properties of the modulus function; in the fourth
line we used that |a(A, λ)| = 1, as follows from (1.40); and in the last line we
used that at least one of the terms in the fourth line is zero, while the other N−1
are not bigger than 2, as follows from (1.40), for example, all the right hand side
terms contribute with a value of 2 if and only if all the terms b(k, λ) are equal for
k = 0, 1, . . . , N − 1, which implies that b(0, λ)− b(N − 1, λ) = 0.
We see that the CHSH inequality (1.45) is a special case of the Braunstein-
Caves inequality (1.54) with N = 2. There exist quantum states and measure-
ments for which the quantum correlations violate the Braunstein-Caves inequality,
up to the bound [46]:
∣
∣IQN
∣
∣ ≤ 2N cos
( pi
2N
)
, (1.56)
where the label Q indicates that the correlations in (1.54) are quantum. For
example, if Alice’s and Bob’s qubits are in the singlet state and their mea-
surements are given by the Bloch vectors ~aA = cos
(
Api
N
)
zˆ + sin
(
Api
N
)
xˆ and ~bB =
cos
(
Bpi
N +
pi
2N
)
zˆ+sin
(
Bpi
N +
pi
2N
)
xˆ, for A,B ∈ {0, 1, . . . , N−1}, it is straightforward
to obtain from (1.53) and (1.54) that the equality is achieved in (1.56).
In chapter 2, we introduce a Bell inequality that generalizes the CHSH and
Braunstein-Caves inequalities in the following sense. Instead of restricting Alice’s
29
Chapter 1. Introduction
and Bob’s measurement choices to a finite set, we allow them to choose any qubit
projective measurements defined by Bloch vectors ~a and~b. However, we constrain
these vectors to be separated by a fixed angle θ, hence, cos θ = ~a ·~b.
1.3.1.5 Bell Experiments and Loopholes
The prediction of quantum mechanics that the Bell inequalities are violated has
been verified experimentally. Particularly, the violation of the CHSH inequal-
ity has been observed in several experiments [47–55].1 Ideally, this would rule
out any possible description of the experimental results in terms of local hidden
variable models. However, the Bell experiments performed so far have had defi-
ciencies called loopholes, which allow us to describe the experiments in terms of
local hidden variable models that exploit such loopholes. There are three main
loopholes: the locality loophole [58, 59], the detector efficiency loophole [60] and
the collapse locality loophole [61].
In the locality loophole, the event in which a measurement choice is made at
Alice’s laboratory and the event in which an outcome is obtained at Bob’s lab-
oratory are not space-like separated. Thus, if the locality loophole is not closed,
the experimental results can be described by a local hidden variable model in
which a signal travelling not faster than light from Alice’s to Bob’s laboratory
that indicates the measurement choice at Alice’s location influences the outcome
obtained at Bob’s location, and vice versa. The locality loophole remained open
during the first experimental violations of the Bell inequalities, because the mea-
surements were kept fixed during a whole run of the experiment [47, 48]. In a
further experiment, the measurements were chosen by a pseudo-random genera-
1To be more precise, a different version of the CHSH inequality (1.45) deduced in the
same paper by Clauser, Horne, Shimony and Holt [42], and investigated further by Clauser
and Horne [56], was tested in experiments [47, 49]. Such a version of the CHSH inequality is
particularly suitable for an experimental set up with photons and polarizers in which only the
outcomes a = 1 and b = 1 can be recorded, these corresponding to photons passing through
polarizers at Alice’s and Bob’s laboratories, respectively. The outcomes a = −1 and b = −1
cannot be detected because these correspond to photons being blocked by the polarizers. This
difficulty was removed in other experiments [48,50–54] in which both possible outcomes a = ±1
and b = ±1 were measured, for example by using two-channel polarizers instead of ordinary
polarizers in experiments with photons. Thus, these later experiments tested version (1.45) of
the CHSH inequality. The experiment [55] tested a different version [57] of the CH inequality
[56].
30
1.3. Bell Inequalities and the No-Signalling Principle
tor [49]. Although this was an improvement, the locality loophole was not closed
completely, because the generator worked at a fixed sinusoidal frequency and thus
was not truly random. In two later experiments, the measurements were selected
according to the outcome of a quantum measurement, using a beam splitter to
measure the polarization of a photon. In one experiment, the photons were sepa-
rated by 400 m across the Innsbruck University science campus [51]. In the other
experiment, the photons were sent from Geneva to the villages of Bellevue and
Bernex, which are separated by 10.9 km [50,62]. It is claimed that these experi-
ments closed the locality loophole, because according to quantum mechanics, the
outcome of a measurement of the polarization of a photon, which was used to
choose what measurements to perform in the Bell experiment, is a totally random
event,1 which cannot be determined before the measurement is performed. Nev-
ertheless, we must say that there is still the logical possibility that the outcomes
of a quantum measurement are predetermined by a hidden variables theory, in
which case the outcomes of a quantum measurement are not truly random events
and hence the locality loophole is still open in this case. Moreover, it is logically
possible that any apparently random event, given for example by a quantum
measurement outcome or a human “free” choice, is predetermined by a hidden
variables theory [58]. According to this possibility, closing the locality loophole
is an impossible task, because any measurement choice made at Alice’s labo-
ratory is predetermined, and thus a signal travelling not faster than light from
Alice’s to Bob’s location, indicating this predetermined choice, can influence the
measurement outcome at Bob’s laboratory.
In the detector efficiency loophole, also called the detection loophole, the de-
tector efficiencies are small enough that LHV models can describe the observed
correlations. The experimental result of a projective measurement on a qubit
consists in a detection of the qubit system after it has passed through a measure-
ment process. For example, if the qubit consists in the polarization of a photon,
the photon goes through a beam splitter, whose orientation defines the performed
measurement, which divides the photon trajectory into two possible paths end-
ing each at a photon detector. The measurement outcome is recorded according
1We are considering here that the measured photon is initially prepared in an equal super-
position of the orthogonal polarizations being measured.
31
Chapter 1. Introduction
to which detector is activated. Ideally, all pairs of entangled photons that are
sent to the corresponding laboratories should activate a detection. However, the
detection efficiency achieved in the experiments is not perfect. Therefore, it is
a common practice to assume that the statistics of the detected particles are
the same as the statistics of all the created particles; this is the fair sampling
assumption. However, if the detector efficiencies are small enough, the experi-
mentally observed correlations can be reproduced by LHV models in which the
detection statistics are affected by the hidden variables [60]. A minimum detec-
tion efficiency of 2(
√
2 − 1) = 0.828 [63] is required to close this loophole in the
EPR-Bohm experiment, in which the pair of measured qubits are in a singlet
state. The detection loophole can be closed with a smaller detection efficiency,
with a minimum value of 23 , if the quantum state is optimized, interestingly to a
non-maximally entangled state [57]. Some experiments have closed the detector
efficiency loophole [52, 53, 55], but have left the locality loophole open. On the
other hand, the Innsbruck and the Geneva experiments, which closed the locality
loophole, left the detection loophole open, because their detector efficiencies were
not big enough; the detector efficiency was only 0.05 in the Innsbruck experi-
ment. In fact, no Bell experiment performed so far has been able to close both
the detector efficiency and the locality loopholes. Given that the locality loophole
has only been closed with photons and that recent experiments [55] with photons
achieved to close the detection loophole, it is reasonable to expect that both the
detection and the locality loopholes will be closed in the near future in a Bell
experiment with photons.
The collapse locality loophole is based on the idea that the collapse of the
quantum state, also called state vector reduction or collapse of the wave function,
in a quantum measurement is a well defined physical process. If a measurement
event is not instantaneous, but takes a finite time due to the physical collapse of
the quantum state, then there can be a signal, travelling not faster than light,
departing from Alice’s laboratory indicating her measurement choice that arrives
to Bob’s laboratory after a particle has entered his measurement apparatus, but
before its quantum state collapse is completed, and vice versa. In this case,
the measurements could seem to be space-like separated if state vector reduction
were not considered, but would in fact be time-like separated if the collapse of the
32
1.3. Bell Inequalities and the No-Signalling Principle
quantum state were a physical process that takes a finite time to be completed.
In order to close the collapse locality loophole, it is necessary to have a theory for
the state vector reduction. One suggestion is that gravity induces the collapse
of the quantum state [64–66]. According to gravity induced collapse models, the
sate vector reduction is completed when a superposition state of significantly
different configurations of massive objects is achieved. The violation of a Bell
inequality was observed in an experiment [54,67] that closed the collapse locality
loophole, assuming these collapse models. The collapse locality loophole remains
open for other models of state vector reduction. Independently of the collapse
model under consideration, it is argued that a collapse should not last longer than
the time that takes for a human brain to register a measurement result, which
is of the order of 0.1 seconds [61]. Thus, a Bell experiment in which Alice’s and
Bob’s laboratories are separated by 0.1 light seconds would be able to close the
collapse locality loophole completely. This distance is four orders of magnitude
bigger than the biggest separation between the laboratories achieved by the Bell
experiments performed so far, which is 18 km [54].
1.3.2 The No-Signalling Principle
In the previous sections we have seen that according to relativity, local causality
is a sensible assumption for a physical theory. However, local hidden variable
theories, which aim to include all hypothetical physical theories satisfying EPR’s
criterion of reality and local causality predict the satisfaction of Bell inequalities,
which can be violated by quantum mechanics, and whose violation has been
verified experimentally. In this section we show that quantum mechanics is still
consistent with relativity by satisfaction of the no-signalling principle, which is a
stronger version of the following principle:
No-Superluminal Signalling (NSLS). Information cannot be communicated
at a speed higher than the speed of light in the vacuum.
According to relativity theory, no-superluminal signalling is a necessary condi-
tion for satisfaction of causality. If superluminal signalling were possible, causality
would be violated, as illustrated in Figure 1.3 [68].
33
Chapter 1. Introduction
Figure 1.3: Spacetime diagram showing that superluminal signalling leads to
violation of relativistic causality. Consider that Alice and Bob are at distant
locations and that they have devices that allow them to send messages to each
other at a speed w > c, where c is the speed of light in the vacuum. Alice is
moving at a constant speed v away from Bob, with 0 < v < c. Alice (Bob) is
at rest in the unprimed (primed) reference frame. The events A1 = (x1, ct1),
B = (0, 0) and A2 = (x2, ct2) correspond to Alice sending Bob a message at
speed w, Bob receiving Alice’s message and sending Alice a message at speed w,
in his rest frame, and Alice receiving Bob’s message, respectively. Applying the
corresponding Lorentz transformations, it is straightforward to show that for v
big enough, but still satisfying v < c, we have that t2 < t1, that is, Alice receives
Bob’s message before she has sent hers to Bob. This is a violation of causality,
because the effect, Alice receiving a reply from Bob, precedes the cause, Alice
sending a message to Bob. Logical contradictions can arise. For example, if
Alice’s message indicates what is happening in her present and Bob’s message is
just a copy of Alice’s message, Alice learns from Bob’s message what will happen
in her future; then, Alice can take appropriate actions to change her future, which
means that the message she receives from Bob should be different.
34
1.3. Bell Inequalities and the No-Signalling Principle
Quantum mechanics satisfies the following principle, which is a stronger ver-
sion of no-superluminal signalling.
The No-Signalling Principle (NS). A party, Alice, cannot communicate any
information to another distant party, Bob, if Alice does not transmit any physical
systems to Bob.
An interesting question is, if Alice does send Bob a physical system, how
much information can the transmitted system fundamentally communicate? This
question is discussed in chapter 4, in which an extension of the no-signalling
principle, quantum information causality, is presented.
Satisfaction of NS implies satisfaction of NSLS. This is seen as follows. Ac-
cording to NS, Alice can communicate information to a distant party, Bob, only
if she sends him a physical system. Therefore, since according to relativity, no
physical system can travel faster than light, satisfaction of NS implies that Alice
cannot communicate any information to Bob at a speed higher than the speed of
light, which is NSLS. However, satisfaction of NSLS does not necessarily imply
satisfaction of NS. Suppose that Alice and Bob have devices that allow them to
communicate at a speed not higher than the speed of light, but these devices
do not require the transmission of any physical systems. In this case NSLS is
satisfied but NS is violated.
The no-signalling principle can be stated mathematically in terms of some
conditions on probability distributions. Consider the general situation in which
Alice and Bob try to communicate by using some devices without the transmission
of any physical systems. Alice has a message A that she wants to communicate
Bob. Similarly, Bob has a message B that she wants to communicate Alice. Alice
and Bob choose their messages from sets A and B, respectively. Alice’s and Bob’s
devices output respective values r ∈ R and s ∈ S. These devices are usually
called correlation boxes. The correlation boxes are defined by their outcome
probabilities P (r, s|A,B). Alice and Bob can use their boxes several times; or
they can use them only once, but they have access to several pairs of identical
boxes. The boxes satisfy the no-signalling principle if Alice cannot infer Bob’s
message from her outcome probabilities, and similarly for Bob. This means that
Alice’s (Bob’s) outcome probabilities are independent of Bob’s (Alice’s) inputs.
35
Chapter 1. Introduction
These no-signalling conditions are
∑
s∈S
P (r, s|A,B) =
∑
s∈S
P (r, s|A,B′) ≡ P (r|A), ∀B,B′ ∈ B, r ∈ R, A ∈ A,
∑
r∈R
P (r, s|A,B) =
∑
r∈R
P (r, s|A′, B) ≡ P (s|B), ∀A,A′ ∈ A, s ∈ S, B ∈ B.
(1.57)
A quantum box corresponds to the probability distribution P (r, s|A,B) of
a quantum measurement. In this case, the inputs correspond to measurements
implemented on a quantum state and the outputs correspond to the measurement
outcomes. Quantum mechanics satisfies the no-signalling principle, as we show
below [69].
Consider a quantum system composed of two subsystems, ‘1’ and ‘2’, at dif-
ferent locations in a quantum state with density matrix ρ12 ∈ D(H1⊗H2). Alice
has system 1 and Bob has system 2. Alice and Bob apply measurements labelled
by A ∈ A and B ∈ B, respectively. Alice’s (Bob’s) measurement A (B) consists
of a set of measurement operators MAr (M
B
s ) acting on H1 (H2), where r ∈ R
(s ∈ S) is Alice’s (Bob’s) outcome. The outcome probabilities are given by
P (r, s|A,B) = Tr
((
MAr ⊗M
B
s
)
ρ12
(
MAr ⊗M
B
s
)†
)
. (1.58)
These satisfy the no-signalling conditions (1.57), as we show:
∑
s∈S
P (r, s|A,B) =
∑
s∈S
Tr
((
MAr ⊗M
B
s
)
ρ12
(
MAr ⊗M
B
s
)†
)
=
∑
s∈S
Tr
(((
MAr
†
MAr
)
⊗
(
MBs
†
MBs
))
ρ12
)
= Tr
(((
MAr
†
MAr
)
⊗
∑
s∈S
(
MBs
†
MBs
))
ρ12
)
= Tr
(((
MAr
†
MAr
)
⊗ I2
)
ρ12
)
, (1.59)
where in the second line we used the cyclicity of the trace, in the third line we used
the linearity of the trace, in the fourth line we used the completeness equation
36
1.3. Bell Inequalities and the No-Signalling Principle
(1.3), and I2 denotes the identity acting on H2. Thus, we see that the right hand
side of the previous expression is independent of B, as required by (1.57). In a
similar way we obtain that
∑
r∈R
P (r, s|A,B) = Tr
((
I1 ⊗
(
MBs
†
MBs
))
ρ12
)
, (1.60)
where I1 denotes the identity acting on H1. Thus, the right hand side of the
previous expression is independent of A, as required. The expressions (1.59) and
(1.60) can be stated in the simpler form
P (r|A) = Tr
(
MAr ρ1M
A
r
†)
,
P (s|B) = Tr
(
MBs ρ2M
B
s
†)
, (1.61)
where we have used the cyclicity of the trace and the definitions (1.57), ρ1 ≡
Tr2(ρ12), and ρ2 ≡ Tr1(ρ12).
An application of the no-signalling principle that is useful in chapters 3 and
4 considers the following situation. Alice and Bob are at different locations and
share an arbitrary quantum state. Alice is given a random message from a set
of N elements. Alice and Bob perform an arbitrary quantum strategy using
their quantum state, but without the transmission of any physical systems, with
the goal that Bob guesses the message given to Alice. According to the no-
signalling principle, since Alice does not send Bob any physical systems, Bob
cannot obtain any information about Alice’s message. Since the message given
to Alice is chosen randomly from a set of N elements, that is, with a probability
1
N , the no-signalling principle implies that Bob can only guess it with probability
1
N . This can be shown using the no-signalling conditions (1.57), as follows.
A general quantum strategy performed by Alice and Bob in which no physical
systems are transmitted consists of quantum measurements A ∈ A and B ∈ B
performed by Alice and Bob, respectively, on a shared entangled state. Alice and
Bob obtain respective outcomes r ∈ R and s ∈ S. Alice’s measurement choice
A encodes the received message, hence, |A| = N . Bob’s outcome s corresponds
to his guess of Alice’s message A, hence, the elements of S are in one to one
correspondence with the elements of A. Therefore, without loss of generality, we
37
Chapter 1. Introduction
assume that A = S. Thus, the probability that Bob outputs the correct message
is
Pguess =
∑
x∈A
P (s = x|B,A = x)P (A = x). (1.62)
The no-signalling conditions (1.57) state that Bob’s outcome probabilities are
independent of Alice’s measurement. Therefore, P (s = x|B,A = x) = P (s =
x|B) for all x ∈ A and B ∈ B, which from (1.62) implies that
Pguess =
∑
x∈A
P (s = x|B)P (A = x)
=
1
N
∑
x∈A
P (s = x|B)
=
1
N
, (1.63)
where in the second line we used that the message given to Alice is random and
in the third line we used the normalization of probabilities.
In the following sections we discuss some implications of the no-signalling
principle for the violation of the CHSH inequality (section 1.3.2.1) and for some
quantum information tasks (section 1.3.2.2).
1.3.2.1 No-Signalling and the CHSH Inequality
An interesting question to ask is, why does not quantum mechanics violate the
CHSH inequality up to the maximum possible algebraic value of 4? An interest-
ing hypothesis is that the no-signalling principle restricts the maximum violation
of the CHSH inequality up to the Cirel’son bound. This was shown to be false
because there exist theoretical correlation systems, the PR boxes, which violate
the CHSH inequality up to the value of 4 and still satisfy the no-signalling prin-
ciple [70]. It was shown later that an extension of the no-signalling principle, the
information causality principle [71], does imply the Cirel’son bound, as discussed
in chapter 4. A different approach to this question is given in [72].
In order to present the PR boxes, it is convenient to translate the EPR-Bohm
experiment into an informational task, usually called the CHSH game. Alice and
Bob are at different locations and share a pair of correlated boxes defined by their
38
1.3. Bell Inequalities and the No-Signalling Principle
joint outcome probabilities P (r, s|A,B). Alice (Bob) randomly chooses a value
A ∈ A (B ∈ B) and inputs this into her (his) box, which then outputs a value
r ∈ R (s ∈ S), with A = B = R = S = {0, 1}. The game’s goal is that their
outputs satisfy r⊕ s = AB, where ⊕ denotes sum modulo 2. Thus, we define the
success probability in the CHSH game by
PCHSH ≡ P (r ⊕ s = AB). (1.64)
We show below that the CHSH inequality and the Cirel’son bound impose
bounds on PCHSH when the boxes are described by LHVT and by quantum me-
chanics, respectively. To do so, we associate the EPR-Bohm experiment and the
CHSH game according to the following relations:
r ≡
1− a
2
, s ≡
1− b
2
. (1.65)
We show at the end of this section that the CHSH quantity I2 and the success
probability PCHSH satisfy the relation
PCHSH =
1
2
+
I2
8
. (1.66)
Using the relation (1.66), the CHSH inequality (1.45) and the Cirel’son bound
(1.52), it follows that the success probabilities P LCHSH and P
Q
CHSH achieved by LHV
correlations and quantum correlations in the CHSH game, respectively, satisfy the
following inequalities:
1
4
≤ P LCHSH ≤
3
4
, (1.67)
1
2
(
1−
1
√
2
)
≤ PQCHSH ≤
1
2
(
1 +
1
√
2
)
. (1.68)
The PR box [70] is defined by the following outcome probabilities P (r, s|A,B):
P (0, 0|0, 0) = P (1, 1|0, 0) = P (0, 0|0, 1) = P (1, 1|0, 1) = P (0, 0|1, 0)
= P (1, 1|1, 0) = P (0, 1|1, 1) = P (1, 0|1, 1) =
1
2
, (1.69)
39
Chapter 1. Introduction
with all other outcome probabilities equal to zero. Is is easy to see that the PR
box achieves PCHSH = 1. The PR box satisfies the no-signalling conditions (1.57):
1∑
s=0
P (r, s|A, 0) =
1∑
s=0
P (r, s|A, 1) ≡ P (r|A),
1∑
r=0
P (r, s|0, B) =
1∑
r=0
P (r, s|1, B) ≡ P (s|B), (1.70)
for r, s, A,B ∈ {0, 1}.
Similarly, if we relabel r → r ⊕ 1, we obtain a non-signalling probability
distribution for which PCHSH = 0. Thus, the no-signalling principle does not
impose any restrictions on the success probability in the CHSH game or in the
maximum violation of the CHSH inequality.
We complete this section by showing (1.66). First, a general average correla-
tion in terms of the probabilities of the outcomes a, b ∈ {1,−1}, when measure-
ments A and B are chosen, is
C(A,B) ≡
∑
a,b∈{1,−1}
abP (a, b|A,B). (1.71)
Second, using the change of variables (1.65), the correlation is
C(A,B) ≡
∑
r,s∈{0,1}
(−1)r⊕sP (r, s|A,B). (1.72)
Third, after relabelling A→ A⊕ 1, the CHSH quantity (1.45) is
I2 ≡
∑
A,B∈{0,1}
(−1)ABC(A,B). (1.73)
We notice that the CHSH inequality (1.45) is still satisfied because we have only
changed the measurement labels. From (1.72) and (1.73), the CHSH quantity is
I2 ≡
∑
A,B,r,s∈{0,1}
(−1)r⊕s⊕ABP (r, s|A,B). (1.74)
40
1.3. Bell Inequalities and the No-Signalling Principle
Finally, noting that by definition of the CHSH game, the measurements are chosen
randomly, P (A,B) = 14 , we have
I2
4
=
∑
A,B,r,s∈{0,1}
(−1)r⊕s⊕ABP (r, s|A,B)P (A,B).
= P (r ⊕ s = AB)− P (r ⊕ s 6= AB)
= 2PCHSH − 1, (1.75)
where in the third line we used the normalization of probabilities and the defini-
tion (1.64). Thus, the relation (1.66) follows.
1.3.2.2 No-Signalling and Quantum Information
The no-signalling principle has important implications for quantum information
processing tasks. The no-cloning theorem was first noticed after a publication
claiming a procedure for superluminal communication between distant parties
with access to a hypothetical machine that could produce perfect copies of an
unknown quantum state [38]. Since superluminal communication is not possible
by any quantum process, the assumption of a perfect quantum cloning machine
is false. The no-cloning theorem was shown after such a claim for superluminal
communication. In fact, the maximum fidelity f = 56 achieved by a cloning ma-
chine of qubit states, as mentioned in section 1.2.4.1, can be deduced from the
no-signalling principle [37]. There are other important implications of the no-
signalling principle for quantum information tasks. The security of quantum key
distribution for eavesdroppers not restricted by quantum mechanics is obtained
from the violation of a Bell inequality and satisfaction of the no-signalling prin-
ciple [27]. The maximum guessing probability in quantum state discrimination
can be derived from the no-signalling principle [73]. We discuss below a few other
quantum information tasks that are constrained by the no-signalling principle.
An interesting extension of the no-cloning theorem to relativistic quantum
mechanics is the no-summoning theorem [74], which guarantees the security of a
quantum relativistic protocol for bit commitment [75] and has application to other
quantum information tasks [76]. Consider the following task called summoning.
Alice gives Bob a quantum state ρ at a point P in space-time. The state ρ is known
41
Chapter 1. Introduction
by Alice but unknown by Bob. Alice and Bob agree in advance that Alice will ask
the state back from Bob at a space-time point Q that is time-like separated from
P and that Alice will choose from some set Q with some probability distribution
pQ. Bob succeeds in this task if he gives Alice a copy of ρ at the spacetime point Q.
Consider the example in one spatial dimension in which P = (0, 0), Q = {Q0, Q1},
Qi = (xi, ct) and pQi =
1
2 for i = 0, 1, with x0 = −ct, x1 = ct and t > 0. The
no-summoning theorem states that Bob cannot succeed with probability equal to
unity. The proof follows straightforwardly from the no-signalling principle and
the no-cloning theorem. A general quantum strategy performed by Bob includes
a quantum measurement on the received state ρ together with some ancillary
system, whose outcome is sent at the speed of light to space-time points Qi and
is used to obtain a quantum state ρi at point Qi, for i = 0, 1. If Alice announces
that she wants the state ρ back at the spacetime point Qi, Bob succeeds in the
task if ρi = ρ. From the no-signalling principle, the state ρ0 is independent of
whether or not Alice chooses to ask the state back at Q1, and similarly for ρ1.
Thus, the only way for Bob to succeed for both of Alice’s possible asked points
Q0 and Q1 is that ρ0 = ρ1 = ρ, but this is impossible according to the no-cloning
theorem.
Other important quantum information tasks restricted by the no-signalling
principle are instantaneous nonlocal quantum computation (INLQC) and instan-
taneous nonlocal measurements (INLM) [77–85]. In an INLQC, a distributed
input quantum state |ψ〉 ∈ H is transformed into U |ψ〉 up to local errors that
are corrected after a single round of communication by the parties sharing the
state |ψ〉, where U is a given nonlocal unitary acting on H (see Figure 1.4). The
no-signalling principle requires that this task is completed with at least one round
of communication. The word instantaneous means that, in principle, Alice’s and
Bob’s local operations on |ψ〉 can be performed in an arbitrarily short time.
An INLM is the measurement of a nonlocal variable O on a distributed quan-
tum state |ψ〉 ∈ H among distant parties that is completed after a single round of
communication among the parties sharing the state |ψ〉. Consider the bipartite
case in which H = HA ⊗HB. One party, Alice, has the system A and the other
party, Bob, has the system B. A nonlocal variable O consists of a Hermitian
operator acting on H that cannot be written as OA⊗OB, with OA and OB acting
42
1.3. Bell Inequalities and the No-Signalling Principle
Figure 1.4: In an instantaneous nonlocal quantum computation, a given nonlo-
cal unitary U is applied to a distributed quantum state |ψ〉 after a single round
of classical communication (CC) between the parties (Alice and Bob) sharing
the state. The protocol consists of local operations (LO), which include mea-
surements, on |ψ〉 and shared entanglement, communication of the measurement
results and further local operations.
on HA and HB, respectively. The measurement of the variable O gives as a result
the eigenvalue of O in the state |ψ〉, with a probability distribution given by the
Born rule if |ψ〉 is in a superposition of eigenstates of O. In general, the INLM
consists of local operations performed by Alice and Bob, and communication
between them or to a third party who learns the measurement outcome.
The no-signalling principle has implications for the nonlocal measurement
process. According to the no-signalling principle, the nonlocal measurement re-
quires at least one round of communication among the parties. As in an INLQC,
the word instantaneous in an INLM means that, in principle, the local opera-
tions applied on |ψ〉 can be performed in an arbitrarily short time. The nonlocal
measurement is conceptually easy to implement if two rounds of communication
are allowed: Bob sends his part of |ψ〉 to Alice through a quantum channel or
via quantum teleportation, then Alice measures the variable O in the state |ψ〉
localized at her site, and communicates the outcome to Bob. The no-signalling
principle implies too that in an INLM, if the quantum state |ψ〉 is in an eigen-
43
Chapter 1. Introduction
state of the measured variable O, the quantum state cannot in general remain
unchanged after the measurement is completed, that is, the measurement is of a
verification (also called demolition) type and not of a von-Neumann (also called
non-demolition) type [81–83].
An INLM can be implemented with an INLQC as follows. The unitary opera-
tion U mapping the eigenstates of the nonlocal variable to the computational basis
is instantaneously applied to |ψ〉 up to local uncontrollable errors. This refers to
the step in the INLQC corresponding to the local operations implemented by the
parties before they perform classical communication (see Figure 1.4). Then, each
party measures the transformed state in the computational basis and communi-
cate their outcomes to the other parties. The result of the nonlocal measurement,
which indicates the eigenstate of O in which |ψ〉 was previous to the nonlocal mea-
surement, is computed from the local measurement outcomes communicated by
the parties.
INLQC is possible if enough entanglement is previously distributed to the
participating parties. A recursive scheme based on standard teleportation im-
plements INLQC with an amount of entanglement growing double exponentially
with the number of qubits n of the input state |ψ〉 [82, 84]. However, another
scheme based on a different type of quantum teleportation protocol, denoted as
port-based teleportation, implements INLQC with an amount of entanglement
growing only exponentially with n [85].
INLQC has application to other distributed quantum information tasks. IN-
LQC allows an eavesdropper to break the security of position-based quantum
cryptography (PBQC) and some quantum tagging schemes [86–92]. Quantum
tagging [86–89,93] and PBQC [91,92] are cryptographic tasks that rely on quan-
tum information processing and relativistic constraints with the goals of verifying
the location of an object and providing secure communication with a party at a
given location, respectively.
In chapter 3, we derive another implication of the no-signalling principle for
an important quantum information task. The maximum success probability of
port-based teleportation is deduced from the no-signalling principle and a version
of the no-cloning theorem.
44
Chapter 2
Bloch Sphere Colourings and
Bell Inequalities
2.1 Introduction
As discussed in detail in section 1.3.1, the assumptions of local causality and the
criterion of physical reality made by Einstein, Podolsky and Rosen [1] in their
argument that quantum mechanics is an incomplete theory, led Bell [2] to in-
troduce a mathematical description for hypothetical physical theories based on
these assumptions. These local hidden variable theories (LHVT) make predic-
tions on experiments performed at space-like separations that are inconsistent
with the quantum predictions. Quantum mechanics predicts the violation of Bell
inequalities, which are satisfied by LHVT. One of the Bell inequalities that has
been investigated the most, both theoretically and experimentally, is the CHSH
inequality [42]. The CHSH inequality considers Bohm’s version [41] of the EPR
thought experiment.
In the EPR-Bohm experiment, a pair of qubits is created in the singlet state
|Ψ−〉 = 1√
2
(
|0〉|1〉 − |1〉|0〉
)
and then sent to distant laboratories. One of these
laboratories is controlled by Alice and the other one by Bob. Alice (Bob) ran-
domly chooses one of two possible measurements to perform on her (his) qubit,
which we denote by A ∈ {0, 1} (B ∈ {0, 1}). These are projective measurements
performed by Alice (Bob) on bases defined by the Bloch vectors ~aA
(~bB
)
. We
45
Chapter 2. Bloch Sphere Colourings and Bell Inequalities
denote Alice’s (Bob’s) outcome as a ∈ {−1, 1} (b ∈ {−1, 1}). An important con-
dition is that the events at which Alice (Bob) chooses her (his) measurement and
Bob (Alice) obtains his (her) outcome are space-like separated (see Figure 1.2 in
section 1.3.1).
According to deterministic LHVT, Alice’s (Bob’s) outcome is predetermined
by the value of a hidden variable λ and in general depends on her (his) measure-
ment choice, but does not depend on Bob’s (Alice’s) measurement choice. Thus,
according to deterministic LHVT, we have
a = a(A, λ), b = b(B, λ). (2.1)
Additionally, LHVT assume the existence of a probability distribution ρ(λ) that
does not have any dependence on the measurement choices and that satisfy
ρ(λ) ≥ 0,
∫
Λ
dλρ(λ) = 1, (2.2)
where Λ is the set of hidden variables. We restrict to consider deterministic
LHVT in this chapter, because probabilistic LHVT can be described by the same
equations if we extend the definitions of the hidden variables for probabilistic
measurement outcomes. The prediction of LHVT for the average product of
Alice’s and Bob’s outcomes, when their measurement choices are A and B, is
C(A,B) =
∫
Λ
dλρ(λ)a(A, λ)b(B, λ). (2.3)
The LHV correlations C(A,B) satisfy the CHSH inequality [42]:
|I2| ≡
∣
∣C(0, 0) + C(1, 1) + C(1, 0)− C(0, 1)
∣
∣ ≤ 2 . (2.4)
However, the correlations predicted by quantum mechanics for the singlet state
are
Q(θ) = − cos θ, (2.5)
where cos θ = ~aA ·~bB, which can violate the CHSH inequality for appropriate sets
of axes ~aA and ~bB, up to the value 2
√
2 given by Cirel’son’s bound [44].
46
2.2. Bloch Sphere Colourings and Correlation Functions
The quantum prediction that the CHSH inequality is violated has been ver-
ified in several experiments [47–55]. As discussed in section 1.3.1.5, there exist
loopholes in the Bell experiments performed so far, which do not allow us to
completely rule out descriptions of the experimental results in terms of LHVT.
An extension of the EPR-Bohm experiment in which Alice and Bob choose
their measurements from sets of N ≥ 2 elements leads to the Braunstein-Caves
inequality [45]:
|IN | ≡
∣
∣
∣
∣
∣
N−1∑
k=0
C(k, k) +
N−2∑
k=0
C(k + 1, k)− C(0, N − 1)
∣
∣
∣
∣
∣
≤ 2N − 2. (2.6)
The Braunstein-Caves inequality is satisfied by LHVT, but can be violated by
quantum mechanics, up to the value 2N cos
(
pi
2N
)
[46]. We see that the Braunstein-
Caves inequality generalizes the CHSH inequality for N ≥ 2.
In this chapter, we explore Bell inequalities that generalize the CHSH and
Braunstein-Caves inequalities, in the following sense. Alice’s and Bob’s choices
for their projective measurements are not restricted to be in a finite set, but
can take any values ~a and ~b on the Bloch sphere S2, with the condition that ~a
and ~b are separated by a fixed angle θ so that ~a · ~b = cos θ. Apart from this
condition, the measurement axes ~a and ~b are chosen randomly. The results of
this chapter correspond to work done in collaboration with Adrian Kent and has
been published in [94].
2.2 Bloch Sphere Colourings and Correlation Func-
tions
We explore LHVT in which Alice’s and Bob’s qubit projective measurement re-
sults are given by a(~a, λ) and b
(~b, λ
)
, respectively; where λ is a local hidden
variable common to both qubits. For fixed λ, we can describe the functions a
and b by two binary (black and white) colourings of spheres, associated to a and
b, respectively, where black (white) represents the outcome ‘1’ (‘-1’). Different
sphere colourings are associated with different values of λ. We investigate the
LHV predictions by analyzing the corresponding sphere colourings. Thus, we
47
Chapter 2. Bloch Sphere Colourings and Bell Inequalities
drop the λ-dependence and include a label x that indicates a particular pair of
colouring functions ax(~a) and bx(~b ).
A measurement along axis ~a with outcome 1 (−1) is equivalent to a measure-
ment along axis −~a with outcome −1 (1), and so a and b satisfy the antipodal
property:
ax(~a) = −ax(−~a), bx
(~b
)
= −bx
(
−~b
)
, (2.7)
for all ~a,~b ∈ S2. We define X as the set of all colourings x satisfying the antipodal
property.
The correlation for outcomes of measurements about randomly chosen axes
separated by θ for the pair of colouring functions labelled by x is
Cx(θ) =
1
8pi2
∫
S2
dAax(~a)
∫ 2pi
0
dωbx
(~b
)
, (2.8)
where dA is the area element of the sphere corresponding to Alice’s axis ~a and ω
is an angle in the range [0, 2pi] along the circle described by Bob’s axis ~b with an
angle θ respective to ~a (see Figure 2.1). A general correlation is of the form
C(θ) =
∫
X
dxµ(x)Cx(θ), (2.9)
where µ(x) is a probability distribution over X.
Let (, φ) be the spherical coordinates of ~a and (α, β) be those of ~b; where
, α ∈ [0, pi] are angles from the north pole and φ, β ∈ [0, 2pi] are azimuthal
angles. The spherical coordinates (α, β) for a point ~b with angular coordinate ω
on the circle around the axis ~a are:
α = arccos(cos θ cos − sin θ sin  cosω), (2.10)
β =
[
φ+ kω arccos
(cos  sin θ cosω + sin  cos θ
sinα
)]
mod 2pi, (2.11)
where kω = 1 if 0 ≤ ω ≤ pi and kω = −1 if pi < ω ≤ 2pi. Notice that β is
undefined for α ∈ {0, pi}.
Equations (2.10) and (2.11) can be obtained by applying three consecutive
rotations around the z, y and z axes, given by Rz(φ)Ry()Rz(ω), on the vectors
48
2.2. Bloch Sphere Colourings and Correlation Functions
~a′ = zˆ and ~b′ = sin θxˆ + cos θzˆ, which satisfy ~a′ · ~b′ = cos θ. After applying
the first two rotations, the rotated vector ~b′′ = Ry()Rz(ω)~b′ has the form ~b′′ =
sinα cos β′′xˆ + sinα sin β′′yˆ + cosαzˆ. Then, we see that α is obtained from the
z coordinate and β from the x coordinate by β = φ + β′′mod 2pi, which give
Equations (2.10) and (2.11).
Figure 2.1: Alice’s and Bob’s measurement axes ~a and ~b form an angle θ. The
spherical coordinates of ~a and~b are (, φ) and (α, β), respectively, related by (2.10)
and (2.11). Equation (2.8) computes the correlation Cx(θ) by (i) integrating the
colouring function bx
(~b
)
over the circle on the sphere generated by ~b (param-
eterized by the angle ω in (2.10) and (2.11)) and (ii) integrating the colouring
function ax(~a) over the sphere generated by ~a. Equation (2.9) computes a gen-
eral correlation C(θ) by integrating over the probability distribution µ(x) of the
colourings satisfying the antipodal property, Equation (2.7).
A simple colouring of the spheres satisfying the antipodal property is one in
which, for one sphere, one hemisphere is completely black and the other one is
completely white, and the colouring is reversed for the other sphere. We define
this to be colouring 1, with
a1 (, φ) = −b1 (, φ) =
{
1 if  ∈
[
0, pi2
]
,
−1 if  ∈
(
pi
2 , pi
]
,
(2.12)
for all φ ∈ [0, 2pi]; where we have used the spherical coordinates for vector ~a, as
introduced above.
If all colourings x ∈ X satisfy QρL(θ) < C
L(θ) ≤ Cx(θ) or Cx(θ) ≤ CU(θ) <
QρU(θ) for quantum correlations QρL(θ) and QρU(θ) obtained with particular two
qubit states ρU and ρL, and some identifiable lower and upper bounds, CL(θ)
49
Chapter 2. Bloch Sphere Colourings and Bell Inequalities
and CU(θ), respectively, then a general LHV correlation C(θ) must satisfy the
same inequalities. We aim here to explore this possibility via intuitive arguments
and numerical and analytic results. We focus on the case ρL = |Ψ−〉〈Ψ−|, for
which QρL(θ) ≡ Q(θ) = − cos θ, which is the maximum quantum anticorrelation
for a given angle θ (see section 2.5 for details). We begin with some general
observations.
First, we consider colouring functions x ∈ X for which the probability that Al-
ice and Bob obtain opposite outcomes when they choose the same measurement,
averaged uniformly over all measurement choices, is
P (ax = −bx|θ = 0) = 1− γ. (2.13)
In general, 0 ≤ γ ≤ 1. We first consider small values of γ and seek Bell inequalities
distinguishing the singlet state quantum correlations from LHV correlations for
which opposite outcomes are observed with probability 1 − γ when the same
measurement axis is chosen by Alice and Bob. Experimentally, we can verify the
violation of such Bell inequalities if the performed tests include some frequency of
tests for opposite outcomes for the same axis chosen randomly, and independently
for each test, by Alice and Bob. These tests of opposite outcomes allow statistical
bounds on γ, which imply statistical tests for the violation of the γ-dependent
Bell inequalities.
In the limiting case γ = 0, we have
ax (~a) = −bx (~a) , (2.14)
for all ~a ∈ S2. This case is very interesting theoretically because we expect to
prove stronger results for this case. We present some numerical results for this
case in section 2.4.
Second, for any pair of colourings x ∈ X and θ ∈ [0, pi], we have Cx(pi − θ) =
−Cx(θ). This can be seen as follows. For a fixed ~a, the circle with angle θ = θ′
around the axis ~a, defined by the angle ω in (2.8) contains a point ~b that is
antipodal to a point on the circle with angle θ = pi − θ′ around ~a. Since the
colouring is antipodal, we have that the value of the integral
∫ 2pi
0 dωbx
(~b
)
in (2.8)
50
2.2. Bloch Sphere Colourings and Correlation Functions
for θ = θ′ is the negative of the corresponding integral for θ = pi − θ′. It follows
that Cx(pi − θ′) = −Cx(θ′). Therefore, in the rest of this chapter, we restrict to
consider correlations for the range θ ∈
[
0, pi2
]
, unless otherwise stated. From the
previous argument, we have Cx
(
pi
2
)
= −Cx
(
pi
2
)
, which implies that Cx
(
pi
2
)
= 0.
We also have that Cx(0) = 1− 2P (ax = −bx|θ = 0), hence, from (2.13) we have
that the LHVT we consider give
Cx(0) = −1 + 2γ. (2.15)
The correlations for LHVT satisfying (2.8) and (2.13) in the case γ = 0 thus
coincide with the singlet state quantum correlations for θ = 0 and θ = pi2 , where
Q(0) = Cx(0) = −1 and Q
(
pi
2
)
= Cx
(
pi
2
)
= 0.
Third, consider colouring 1, defined by (2.12). We have
C1(θ) = −
(
1−
2θ
pi
)
, (2.16)
for θ ∈
[
0, pi2
]
. This is easily seen as follows. For any two different points on the
spheres defining colouring 1, ~a in one sphere and ~b in the oppositely coloured one,
an arc of angle pi of the great circle passing through ~a and ~b is completely black
and the other arc of angle pi is completely white. Thus, given that the pair of
vectors ~a and ~b are chosen randomly, subject to the constraint of angle separation
θ, the probability that both ~a and ~b are in oppositely coloured regions is P (a1 =
−b1|θ) = pi−θpi = 1−
θ
pi . Thus, the correlation C1(θ) = 1− 2P (a1 = −b1|θ) is given
by Equation (2.16). We see that C1(θ) linearly interpolates between the values
at C1(0) = −1, which is common to all colourings with γ = 0, and C1
(
pi
2
)
= 0,
which is common to all colourings x ∈ X, and we have 0 > C1(θ) > Q(θ) for
θ ∈
(
0, pi2
)
.
In the following sections, we motivate and explore the hypothesis that colour-
ing 1 gives the maximum LHV anticorrelation for a continuous range of θ > 0.
51
Chapter 2. Bloch Sphere Colourings and Bell Inequalities
2.3 The Hemispherical Colouring Maximality Hy-
pothesis
In this section, we motivate and state strong and weak forms of the hemispher-
ical colouring maximality hypothesis that, for a continuous range of θ > 0, the
maximum LHV anticorrelation is obtained by colouring 1, defined by (2.12).
We first consider the following lemmas.
Lemma 2.1. For any colouring x ∈ X satisfying (2.13) and any θ ∈
(
0, 2pi3
]
, we
have −1 + 23γ ≤ Cx(θ) ≤
1
3 +
2
3γ.
Proof. From the CHSH inequality,
∣
∣C(0, 0) + C(1, 1) + C(1, 0)− C(0, 1)
∣
∣ ≤ 2 ,
in the case in which the measurements A = 0, A = 1 and B = 0 correspond to
projections on states with Bloch vectors separated from each other by the same
angle θ ∈
(
0, 2pi3
]
, Bob’s measurement B = 1 is the same as Alice’s measurement
A = 0 and the outcomes are described by LHVT satisfying (2.8), we obtain after
averaging over random rotations of the Bloch sphere that
|3Cx(θ)− Cx(0)| ≤ 2.
Then, the result follows from (2.15):
Cx(0) = −1 + 2γ,
which holds for all x ∈ X satisfying (2.13).
Remark 2.1. Unsurprisingly, since small γ implies near-perfect anticorrelation at
θ = 0, we see that for θ ∈
(
0, 2pi3
]
and γ small there are no colourings with very
strong correlations. However, strong anticorrelations are possible for small θ. We
are interested in bounding these.
Lemma 2.2. For any colouring x ∈ X satisfying (2.13), any integer N > 2 and
any θ ∈
[
pi
N ,
pi
N−1
)
, we have Cx(θ) ≥ C1
(
pi
N
)
− 2γ.
52
2.3. The Hemispherical Colouring Maximality Hypothesis
Proof. From the Braunstein-Caves inequality (2.6), we have that
∣
∣
∣
∣
N−1∑
k=0
C(k, k) +
N−1∑
k=0
C(k + 1, k)
∣
∣
∣
∣ ≤ 2N − 2,
with the convention that measurement choice N is measurement choice 0 with
reversed outcomes. We consider the case in which Alice’s and Bob’s measurements
k are the same, for k = 0, 1, . . . , N − 1 and N > 2, and their outcomes are
described by LHVT satisfying (2.8) and (2.13), which then also satisfy (2.15). If
we take measurement k to be of the projection onto the state |ξk〉 so that the
states {|ξk〉}
N−1
k=0 are along a great circle on the Bloch sphere with a separation
angle θ = piN between |ξk〉 and |ξk+1〉 for k = 0, 1, . . . , N − 2, for example |ξk〉 =
cos
(
kpi
2N
)
|0〉+ sin
(
kpi
2N
)
|1〉, and average over random rotations of the Bloch sphere,
this gives
∣
∣NCx(0) +NCx(θ)
∣
∣ ≤ 2N − 2 .
Thus, from (2.15):
Cx(0) = −1 + 2γ,
it follows that
Cx(θ) ≥ −1 +
2
N
− 2γ.
Since C1
(
pi
N
)
= −1 + 2N , as follows from (2.16), we have
Cx(θ) ≥ C1
( pi
N
)
− 2γ.
Similarly, if we take the states {|ξk〉}
N−1
k=0 to be along a zigzag path crossing a
great circle on the Bloch sphere with a separation angle θ > piN between |ξk〉 and
|ξk+1〉 for k = 0, 1, . . . , N − 2, in such a way that the angle separation between
|ξN−1〉 and the state with Bloch vector antiparallel to that one of |ξ0〉 is also θ
(see Figure 2.2), we obtain after averaging over random rotations of the Bloch
sphere that Cx(θ) ≥ −1 + 2N − 2γ = C1
(
pi
N
)
− 2γ.
Remark 2.2. In other words, for small θ, C1(θ) is very close to the maximal
possible anticorrelation for LHVT when γ  θ.
53
Chapter 2. Bloch Sphere Colourings and Bell Inequalities
Figure 2.2: Diagram of the measurements performed by Alice and Bob that are
used in the proof of Lemma 2.2. Alice’s and Bob’s measurements k are the same,
for k = 0, 1, . . . , N − 1 and N > 2; these are projections onto the states |ξk〉
and correspond to points in the Bloch sphere with label k. These points form a
zigzag path crossing the dashed great circle. The state |ξN〉 is antipodal to |ξ0〉
and represents the measurement k = 0 with reversed outcomes. The solid lines
represent arcs of great circles with the same angle θ > piN that connect adjacent
points. If θ = piN , all these points are on the same great circle.
Geometric intuitions also suggest bounds on Cx(θ) that are maximised by
colouring 1 for small θ. Consider simple colourings, in which a set of (not nec-
essarily connected) piecewise differentiable curves of finite total length separate
black and white regions, with points lying on these curves having either colour.
Intuition suggests that, for small θ and simple colourings with γ = 0, the quan-
tity 1 + Cx(θ), which measures the deviation from pure anticorrelation, should
be bounded by a quantity roughly proportional to the length of the boundary
between the black and white areas of the sphere colouring x ∈ X. Since colour-
ing 1 has the smallest such boundary (the equator), this might suggest that
Cx(θ) ≥ C1(θ), for small θ and for all simple colourings x ∈ X with γ = 0.
Intuition also suggests that any non-simple colouring will produce less anticor-
relation than the optimal simple colouring, because regions in which black and
white colours alternate with arbitrarily small separation tend to wash out anti-
correlation. These intuitions are discussed further in section 2.6.
These various observations motivate us to explore what we call the Weak
54
2.3. The Hemispherical Colouring Maximality Hypothesis
Hemispherical Colouring Maximality Hypothesis (WHCMH ).
WHCMH. There exists an angle θwmax ∈
(
0, pi2
)
such that for every colouring
x ∈ X with γ = 0 and every angle θ ∈ [0, θwmax], Cx(θ) ≥ C1(θ).
The WHCMH considers models with perfect anticorrelation for θ = 0, because
we are interested in distinguishing LHV models from the quantum singlet state,
which produces perfect anticorrelations for θ = 0. Of course, there is a symmetry
in the space of LHV models given by exchanging the colours of one qubit’s sphere,
which maps γ → 1 − γ and Cx(θ) → −Cx(θ). The WHCMH thus also implies
that Cx(θ) ≤ −C1(θ) for all colourings x ∈ X with γ = 1 and θ ∈ [0, θwmax].
It is also interesting to investigate stronger versions of the WHCMH and re-
lated questions. For instance, is it the case that for every angle θ ∈
(
θwmax,
pi
2
)
there
exists a colouring x′ ∈ X with γ = 0 such that Cx′(θ) < C1(θ)? And does this
hypothesis still hold true (not necessarily for the same θwmax) if we consider general
local hidden variable models corresponding to independently chosen colourings
for the two qubits, not constrained by any choice of the correlation parameter γ?
The following theorem and lemmas give some relevant bounds.
Theorem 2.1. For any colouring x ∈ X, any integer N ≥ 2 and any θ ∈
[
pi
2N ,
pi
2(N−1)
)
, we have C1
(
pi
2N
)
≤ Cx(θ) ≤ −C1
(
pi
2N
)
.
Proof. Consider the Braunstein-Caves inequality (2.6):
∣
∣
∣
∣
N−1∑
k=0
C(k, k) +
N−1∑
k=0
C(k + 1, k)
∣
∣
∣
∣ ≤ 2N − 2,
with the convention that measurement choice N is measurement choice 0 with re-
versed outcomes, in the case in which Alice’s and Bob’s measurements’ outcomes
are described by LHVT satisfying (2.8). Let Alice’s and Bob’s measurements k
to correspond to the projections onto the states |ξk〉 and |χk〉, respectively, for
k = 0, 1, . . . , N − 1 and N ≥ 2. Let the angle along the great circle in the Bloch
sphere passing through the states |ξk〉 and |χk〉 be θ, for k = 0, 1, . . . , N − 1.
Similarly, let the angle along the great circle passing through |χk〉 and |ξk+1〉 be θ
for k = 0, 1, . . . , N − 1, with the convention that the state |ξN〉 has Bloch vector
antiparallel to that one of |ξ0〉. If θ = pi2N , all these states are on the same great
55
Chapter 2. Bloch Sphere Colourings and Bell Inequalities
circle beginning at |ξ0〉 and ending at |ξN〉. If θ > pi2N , the states can be accommo-
dated on a zigzag path crossing the great circle that goes from |ξ0〉 to |ξN〉 (see
Figure 2.3). Thus, from the Braunstein-Caves inequality, after averaging over
random rotations of the Bloch sphere, we have
−1 +
1
N
≤ Cx(θ) ≤ 1−
1
N
.
Since C1
(
pi
2N
)
= −1 + 1N , as follows from (2.16), we have
C1
( pi
2N
)
≤ Cx(θ) ≤ −C1
( pi
2N
)
,
for θ ≥ pi2N .
Figure 2.3: Diagram of the measurements performed by Alice and Bob that are
used in the proof of Theorem 2.1. Alice’s and Bob’s measurements A and B
are projections onto the states |ξA〉 and |χB〉 and correspond to points in the
Bloch sphere with labels A and B, respectively, for A,B ∈ {0, 1, . . . , N − 1} and
N ≥ 2. These points form a zigzag path crossing the dashed great circle. The
state |ξN〉 is antipodal to |ξ0〉 and represents Alice’s measurement A = 0 with
reversed outcomes. The solid lines represent arcs of great circles with the same
angle θ > pi2N that connect adjacent points. If θ =
pi
2N , all these points are on the
same great circle.
Remark 2.3. In particular, for small θ, −C1(θ) and C1(θ) are very close to the
maximal possible correlation and anticorrelation for any LHVT, respectively.
56
2.3. The Hemispherical Colouring Maximality Hypothesis
Lemma 2.3. If any colouring x ∈ X obeys Cx(θ) < C1(θ)
(
Cx(θ) > −C1(θ)
)
for
some θ ∈
(
pi
M+1 ,
pi
M
]
and an integer M ≥ 2 then there are angles θj ≡ piM+1−j − θ
with j = 1, 2, . . . ,M − 1, which satisfy 0 ≤ θj < θ if j < M2 + 1, and
pi
2 > θj > θ
if j ≥ M2 + 1, such that Cx(θj) > C1(θj)
(
Cx(θj) < −C1(θj)
)
.
Proof. Consider a colouring x ∈ X and an angle θ ∈
(
pi
M+1 ,
pi
M
]
for an integer
M ≥ 2 such that Cx(θ) < C1(θ) or Cx(θ) > −C1(θ). From Theorem 2.1 and the
fact that Cx
(
pi
2
)
= C1
(
pi
2
)
= 0, it must be that θ 6= piM if M is even. We define the
angles θj ≡ piM+1−j − θ with j = 1, 2, . . . ,M − 1. Considering the cases M even
and M odd, and using that θ 6= piM if M is even, it is straightforward to obtain
that 0 ≤ θj < θ if j < M2 + 1 and
pi
2 > θj > θ if j ≥
M
2 + 1. Now consider the
Braunstein-Caves inequality (2.6):
∣
∣
∣
∣
N−1∑
k=0
C(k, k) +
N−1∑
k=0
C(k + 1, k)
∣
∣
∣
∣ ≤ 2N − 2,
with the convention that measurement choice N is measurement choice 0 with re-
versed outcomes, in the case in which Alice’s and Bob’s measurements’ outcomes
are described by LHVT satisfying (2.8). Let Alice’s and Bob’s measurements k
to correspond to the projections onto the states |ξk〉 and |χk〉, respectively, for
k = 0, 1, . . . , N−1 and N ≡M+1−j. Since 1 ≤ j ≤M−1, we have 2 ≤ N ≤M .
Let all these states be on the great circle in the Bloch sphere that passes through
the states |ξ0〉 and |ξN〉, with the convention that the state |ξN〉 has Bloch vec-
tor antiparallel to that one of |ξ0〉. Let the angles between |ξk〉 and |χk〉, and
between |χk〉 and |ξk+1〉 along this great circle be θ and θj, respectively. For ex-
ample, |ξk〉 = cos
(
kpi
2N
)
|0〉+sin
(
kpi
2N
)
|1〉 and |χk〉 = cos
(
kpi
2N +
θ
2
)
|0〉+sin
(
kpi
2N +
θ
2
)
|1〉,
for k = 0, 1, . . . , N − 1. From the Braunstein-Caves inequality, after averaging
over random rotations of the Bloch sphere, we obtain
−1 +
1
N
≤
1
2
(
Cx(θ) + Cx(θj)
)
≤ 1−
1
N
.
Since the average angle θ¯j ≡ 12(θ+ θj) satisfies θ¯j =
pi
2(M+1−j) =
pi
2N and C1(
pi
2N ) =
57
Chapter 2. Bloch Sphere Colourings and Bell Inequalities
−1 + 1N , as follows from (2.16), we have
C1
(
θ¯j
)
≤
1
2
(
Cx(θ) + Cx(θj)
)
≤ −C1
(
θ¯j
)
.
Since C1(θ) is a linear function of θ, it follows that
Cx(θj) > C1(θj),
if Cx(θ) < C1(θ). Similarly,
Cx(θj) < −C1(θj),
if Cx(θ) > −C1(θ).
Remark 2.4. In this sense (at least), the anticorrelations defined by C1 and the
correlations defined by −C1 cannot be dominated by any other colourings.
Lemma 2.4. For any colouring x ∈ X and any θ ∈
(
0, pi3
)
, we have Q(θ) <
Cx(θ) < −Q(θ).
Proof. Let x ∈ X be any colouring and θ ∈
(
0, pi3
)
. We first consider the case
θ ∈
[
pi
4 ,
pi
3
)
. From Theorem 2.1, we have
C1
(pi
4
)
≤ Cx(θ) ≤ −C1
(pi
4
)
.
The quantum correlation for the singlet state is Q(θ) = − cos θ. Since Q(θ) is a
strictly increasing function of θ, we have Q(θ) < Q
(
pi
3
)
= −12 = C1
(
pi
4
)
for θ < pi3 ,
where the second equality follows from (2.16). Therefore,
Q(θ) < Cx(θ) < −Q(θ),
for θ ∈
[
pi
4 ,
pi
3
)
. Similarly, it is easy to see that Q(θ) < Cx(θ) < −Q(θ) for
θ ∈
[
pi
6 ,
pi
4
)
.
Now we consider the case θ ∈
(
0, pi6
)
. We define N = d pi2θe. It follows that
58
2.3. The Hemispherical Colouring Maximality Hypothesis
θ ∈
[
pi
2N ,
pi
2(N−1)
)
for an integer N ≥ 4. From Theorem 2.1 and (2.16), we have
−1 +
1
N
≤ Cx(θ) ≤ 1−
1
N
.
From the Taylor series Q(θ) = −1 + θ
2
2 −
θ4
4! +
θ6
6! − · · · , it is easy to see that
Q(θ) < −1 + θ
2
2 for 0 < θ <
√
30. Thus, we have
Q
(
pi
2(N − 1)
)
< −1 +
1
2
(
pi
2(N − 1)
)2
.
Since N2 >
(
pi2
8 + 2
)
N − 1, it follows that (N − 1)2 > pi
2
8 N , which implies that
−1 +
1
2
(
pi
2(N − 1)
)2
< −1 +
1
N
.
It follows that
Q
(
pi
2(N − 1)
)
< Cx(θ).
Since Q(θ) is a strictly increasing function of θ and θ < pi2(N−1) , we have
Q(θ) < Q
(
pi
2(N − 1)
)
.
Thus, we have
Q(θ) < Cx(θ).
Similarly, we have
Cx(θ) < −Q(θ).
Remark 2.5. This inequality separates all possible LHV correlations Cx(θ) from
the singlet state quantum correlations Q(θ) for all θ ∈
(
0, pi3
)
.
The previous observations motivate the Strong Hemispherical Colouring Max-
imality Hypothesis (SHCMH ).
SHCMH. There exists an angle θsmax ∈
(
0, pi2
)
such that for every colouring x ∈ X
and every angle θ ∈ [0, θsmax], C1(θ) ≤ Cx(θ) ≤ −C1(θ).
59
Chapter 2. Bloch Sphere Colourings and Bell Inequalities
Note that the SHCMH applies to all colourings, without any assumption of
perfect anticorrelation for θ = 0. If the SHCMH is true then so is the WHCMH.
In this case, we have that θsmax ≤ θ
w
max. Thus, an upper bound on θ
w
max implies an
upper bound on θsmax.
2.4 Numerical Results
We investigated the WHCMH numerically by computing the correlation Cx(θ) for
various colouring functions that satisfy the antipodal property (2.7), the condition
(2.14), and that have azimuthal symmetry. These colourings, which are illustrated
in Figure 2.4, are defined as
ax() ≡
{
1 if  ∈ Ex,
−1 if  ∈ [0, pi]/Ex,
(2.17)
where  ∈ [0, pi] is the polar angle in the sphere and
E1 ≡
[
0,
pi
2
]
,
E2 ≡
[
0,
pi
4
]
⋃
[
pi
2
,
3pi
4
]
,
E3 ≡
2⋃
k=0
[
k
pi
3
, (2k + 1)
pi
6
]
,
E4 ≡
3⋃
k=0
[
k
pi
4
, (2k + 1)
pi
8
]
,
E2∆ ≡
[
0,
pi
4
−∆
]
⋃
[
pi
2
,
3pi
4
+ ∆
]
,
E3δ ≡
[
0,
pi
6
+ δ
]
⋃
[
pi
3
,
pi
2
]
⋃
[
2pi
3
,
5pi
6
− δ
]
,
with 0 ≤ ∆ ≤ pi12 and −
pi
18 ≤ δ ≤
pi
24 . Notice that colourings 2∆ and 3δ reduce to
colourings 2 and 3 if ∆ = 0 and δ = 0, respectively.
Equation (2.10) was used to compute the double integral in (2.8). Equa-
tion (2.11) was not necessary because the colourings we considered have azimuthal
60
2.4. Numerical Results
Figure 2.4: Some antipodal colouring functions ax on the sphere defined by (2.17).
Their correlations Cx(θ), computed with Equation (2.8), subject to the constraint
(2.14), are plotted in Figure 2.5.
symmetry. The integral with respect to the angle ω was performed analytically.
Thus, the correlations Cx(θ) were reduced to a sum of terms that include single
integrals with respect to the polar angle ; the obtained expressions are given in
Appendix A. The single integrals with respect to  were computed numerically
with a computer program, whose code is given in Appendix B.
Our results are plotted in Figure 2.5; they are consistent with the WHCMH.
They also show that θwmax <
pi
2 , because they show that there exists a colouring x
with Cx(θ) < C1(θ) for some angles θ ∈
(
0, pi2
)
, namely colouring 3 for angles θ ∈
[
0.405pi, pi2
)
. They also show that there exist colourings x with Cx(θ) > −C1(θ)
for some angles θ ∈
(
0, pi2
)
, namely colouring 2 for angles θ ∈
[
0.375pi, pi2
)
and
colouring 4 for θ ∈
[
0.422pi, pi2
)
. Another interesting result is that there exist
colourings that produce correlations Cx(θ) < Q(θ) for θ close to pi2 : colouring
3 for angles θ ∈
[
0.467pi, pi2
)
. It is interesting to find other colourings whose
correlations satisfy Cx(θ) < C1(θ), Cx(θ) > −C1(θ) and Cx(θ) < Q(θ) for angles
θ closer to zero. For this purpose, we consider colourings 2∆ and 3δ, which are
defined in (2.17) and consist in small variations of colourings 2 and 3 in terms of
the parameters ∆ and δ, respectively. Colourings 2∆ and 3δ reduce to colourings
2 and 3 if ∆ = 0 and δ = 0, respectively. For values of δ in the range
[
− pi18 ,
pi
24
]
,
we obtained that the smallest angle θ for which C3δ(θ) < C1(θ) is achieved for
δ = −0.038pi, in which case we have that C3−0.038pi(θ) < C1(θ) for θ ∈
[
0.386pi, pi2
)
.
We also obtained that the smallest angle θ for which C3δ(θ) < Q(θ) is achieved for
δ = −0.046pi, in which case we have that C3−0.046pi(θ) < Q(θ) for θ ∈ [0.431pi,
pi
2
)
(see Figure 2.6). For values of ∆ in the range
[
0, pi12
]
, we obtained that the
smallest angle θ for which C2∆(θ) > −C1(θ) is achieved for ∆ = 0.035pi, in which
case we have that C20.035pi(θ) > −C1(θ) for θ ∈
[
0.345pi, pi2
)
(see Figure 2.7).
61
Chapter 2. Bloch Sphere Colourings and Bell Inequalities
Figure 2.5: Correlations computed with Equation (2.8), subject to the constraint
(2.14), for the colouring functions ax shown schematically in Figure 2.4 and de-
fined by (2.17). The correlations for colouring 2, 3 and 4 are blue dot-dashed, red
solid and green dashed curves, respectively. The black dot-dash-dotted curve rep-
resents the singlet state quantum correlation Q(θ). The dark red dotted and dark
green dash-dotted curves show respectively the colouring 1 correlation, C1(θ), and
anticorrelation, −C1(θ). The gray solid straight lines show the bounds given by
Theorem 2.1, for θ ≥ pi12 .
62
2.4. Numerical Results
Figure 2.6: Correlations obtained for colouring 3δ, defined in (2.17), for δ =
−0.038pi (a, green dashed curve) and δ = −0.046pi (b, blue dot-dashed curve);
for colourings 3, 1 and the singlet state quantum correlation Q(θ) (red solid, dark
red dotted and black dot-dash-dotted curves, respectively).



	 	
  
 





	




Figure 2.7: Correlations obtained for colouring 2∆, defined in (2.17), for ∆ =
0.035pi (a, black solid curve) and for colouring 2 (blue dot-dashed curve). The
colouring 1 anticorrelation is plotted too (-1, dark green dash-dotted curve).
63
Chapter 2. Bloch Sphere Colourings and Bell Inequalities
Our numerical results imply the bound θwmax ≤ 0.386pi. They also imply that
θsmax ≤ 0.345pi, because C20.035pi(θ) > −C1(θ) for θ ∈
(
0.345pi, pi2
)
, and C1(θ) ≤
Cx(θ) ≤ −C1(θ) for x = 2, 3, 4, 2∆, 3δ and θ ∈ [0, 0.345pi]. Notice that the weaker
upper bound θsmax ≤ 0.375pi is given in our publication [94] because colouring 2∆
is not considered there.
In order to confirm analytically the numerical observation that there exist
colouring functions x ∈ X such that Cx(θ) < Q(θ) for θ close to pi2 , we computed
C3
(
pi
2 − τ
)
for 0 ≤ τ  1 to order O(τ 2). The computation is presented in
Appendix A. We obtain
C3
(pi
2
− τ
)
= −1.5τ + O(τ 2). (2.18)
On the other hand, the singlet state quantum correlation gives Q
(
pi
2 − τ
)
=
− cos
(
pi
2 − τ
)
= −τ +O(τ 3). Thus, we see that for τ small enough, indeed C3
(
pi
2 −
τ
)
< Q
(
pi
2 − τ
)
.
2.5 Related Questions for Exploration
An interesting question is, for an arbitrary two qubit state ρ and qubit projec-
tive measurements performed by Alice and Bob corresponding to random Bloch
vectors separated by an angle θ, what are the maximum values of the quantum
correlations and anticorrelations Qρ(θ), and which states achieve them?
We show that the maximum quantum anticorrelations and correlations are
Qρ(θ) = − cos θ, achieved by the singlet state ρ = |Ψ−〉〈Ψ−|, and Qρ(θ) = 13 cos θ,
achieved by the other Bell states, ρ = |Φ±〉〈Φ±| and ρ = |Ψ+〉〈Ψ+|, respectively.1
1Notice that correlations equal to cos θ would be achieved with the singlet state, and cor-
relations equal to − 13 cos θ would be obtained with the other Bell states, if one of the parties
always flips their outcomes. This strategy corresponds only to relabeling the measurement
outcomes and is not included in our analysis.
At first sight, it could be surprising that the singlet state achieves the maximum quantum
anticorrelation, while the other Bell states do not. The fact that any Bell state can violate the
CHSH and Braunstein-Caves inequalities to the maximum value allowed by quantum mechanics,
for an appropriate set of measurement choices, would suggest that the maximum quantum
anticorrelations and correlations considered here could be achieved by the four Bell states too.
However, a fundamental difference between the singlet and the other Bell states is that the
singlet expressed in a given basis is the same in any other basis, while this property does not
64
2.5. Related Questions for Exploration
This result follows because, as we show below, we have
− cos θ ≤ Qρ(θ) ≤
1
3
cos θ. (2.19)
Another related question that we do not explore further here is, for a fixed
given angle θ separating Alice’s and Bob’s measurement axes, what are the max-
imum correlations and anticorrelations, if in addition to the two qubit state ρ,
Alice and Bob have other resources? For example, Alice and Bob could have an
arbitrary entangled state on which they perform arbitrary local quantum opera-
tions and measurements. In a different scenario, Alice and Bob could have some
amount of classical or quantum communication. Another possibility is for Alice
and Bob to share arbitrary non-signalling resources, not necessarily quantum,
with no communication allowed. It is interesting to note that in this case, the no-
signalling principle does not restrict the value of the correlations, because C = 1 is
achieved for all θ by a generalization of the PR-box [70], which is given by the fol-
lowing non-signalling outcome probabilities: P
(
1, 1|~a,~b
)
= P
(
−1,−1|~a,~b
)
= 12 ,
P
(
1,−1|~a,~b
)
= P
(
−1, 1|~a,~b
)
= 0 for all ~a,~b ∈ S2. Similarly, C = −1 is achieved
by the non-signalling outcome proabilities P
(
1,−1|~a,~b
)
= P
(
−1, 1|~a,~b
)
= 12 ,
P
(
1, 1|~a,~b
)
= P
(
−1,−1|~a,~b
)
= 0 for all ~a,~b ∈ S2. Different variations of the
task described above with continuous parameters can be investigated.
Some interesting related questions involving nonlocal games with continuous
inputs have been considered in [95]. In particular, in the third game considered
in [95], Alice and Bob are given uniformly distributed Bloch sphere vectors, ~rA and
~rB, and aim to maximise the probability of producing outputs that are opposite
if ~rA ·~rB ≥ 0 or equal if ~rA ·~rB < 0. It is suggested in [95] that the LHV strategy
defined by opposite hemispherical colourings is optimal, though no argument is
given. It is also suggested that the quantum strategy given by sharing a singlet
and carrying out measurements corresponding to the input vectors is optimal,
based on evidence from semi-definite programming. Equation (2.19) shows that
this is the case for all θ, and so in particular for the average advantage in the
hold for the other Bell states. Since the (anti) correlations we consider here are obtained after
averaging over all possible projective measurements on the Bloch sphere, given the constraint
of the angle separation θ between Alice’s and Bob’s measurement axes, the anticorrelation
achieved by the singlet is higher than the anticorrelation achieved by the other Bell states.
65
Chapter 2. Bloch Sphere Colourings and Bell Inequalities
game considered, if Alice and Bob are restricted to outputs defined by projective
measurements on a shared pair of qubits. Our results given by Lemma 2.4 also
prove that there is a quantum advantage for all θ in the range 0 < θ < pi3 ,
and hence for many versions of this game defined by a variety of probability
distributions for the inputs.
We complete this section by showing (2.19). First, we compute the average
outcome probabilities when Alice and Bob apply local projective measurements
on a two qubit state ρ, for measurement bases defined by Bloch vectors separated
by an angle θ. The average is taken over random rotations of these vectors in the
Bloch sphere, subject to the angle separation θ. Then, we compute the quantum
correlations.
Consider a fixed pair of pure qubit states |0〉 and |χ〉 = cos
(
θ
2
)
|0〉+ sin
(
θ
2
)
|1〉
for Alice’s and Bob’s measurements corresponding to outcomes ‘1’, respectively.
A general state for Bob’s measurement separated by an angle θ with respect to a
fixed state |0〉 for Alice’s measurement is obtained by applying the unitary Rz(ω)
that corresponds to a rotation of an angle ω ∈ [0, 2pi] around the z axis in the
Bloch sphere, which only adds a phase to the state |0〉. Then, after applying
Rz(ω), a general pure product state |ξ~a〉 ⊗ |χ~b〉 of two qubits with Bloch vectors
separated by an angle θ is obtained by applying the unitary Rz(φ)Ry() that
rotates the Bloch sphere around the y axis by an angle  ∈ [0, pi] and then around
the z axis by an angle φ ∈ [0, 2pi]. Thus, we have |ξ~a〉⊗|χ~b〉 = Uφ,,ω|0〉⊗Uφ,,ω|χ〉,
with Uφ,,ω = Rz(φ)Ry()Rz(ω). This is a general unitary acting on a qubit, up to
a global phase. Therefore, we can parameterize this unitary by the Haar measure
µ on SU(2), hence, we have |ξ~a〉⊗ |χ~b〉 = Uµ|0〉⊗Uµ|χ〉. After taking the average,
the probability that both Alice and Bob obtain the outcome ‘1’ is
P (1, 1|θ) =
∫
dµTr
(
ρ
(
|ξ~a〉〈ξ~a| ⊗ |χ~b〉〈χ~b|
))
=
∫
dµTr
(
ρ
(
Uµ ⊗ Uµ
)(
|0〉〈0| ⊗ |χ〉〈χ|
)(
U †µ ⊗ U
†
µ
))
= Tr
(∫
dµ
(
U †µ ⊗ U
†
µ
)
ρ
(
Uµ ⊗ Uµ
)(
|0〉〈0| ⊗ |χ〉〈χ|
)
)
= Tr
(
ρ˜
(
|0〉〈0| ⊗ |χ〉〈χ|
))
, (2.20)
66
2.6. Discussion
where in the third line we used the linearity and the cyclicity of the trace and in
the fourth line we used the definition ρ˜ ≡
∫
dµ
(
U †µ⊗U
†
µ
)
ρ
(
Uµ⊗Uµ
)
. The state ρ˜
is invariant under a unitary transformation U ⊗U , for any U ∈ SU(2). The only
states with this symmetry are the Werner states [96], which for the two qubit
case have the general form
ρ˜ = r|Ψ−〉〈Ψ−|+
1− r
3
(
|Ψ+〉〈Ψ+|+ |Φ+〉〈Φ+|+ |Φ−〉〈Φ−|
)
, (2.21)
with 0 ≤ r ≤ 1. Thus, from (2.20) and (2.21), we obtain
P (1, 1|θ) =
1− r
3
+
4r − 1
6
sin2
(θ
2
)
. (2.22)
Since the projectors corresponding to Alice and Bob obtaining outcomes ‘-1’
are obtained by a unitary transformation of the form U ⊗ U on |0〉 ⊗ |χ〉, with
U ∈ SU(2), then from (2.20) we see that after integrating over the Haar measure
on SU(2), we obtain P (−1,−1|θ) = P (1, 1|θ).
Thus, the average quantum correlation is Qρ(θ) = 4P (1, 1|θ) − 1, which
from (2.22) gives
Qρ(θ) = −
(4r − 1
3
)
cos θ. (2.23)
Then, Equation (2.19) follows because 0 ≤ r ≤ 1.
2.6 Discussion
In this chapter, we have investigated Bell inequalities for a pair of qubits in which
projective measurements are chosen randomly from the Bloch sphere, with the
constraint that the measurement axes are fixed by a given separation angle θ.
We have obtained Bell inequalities for θ ∈
[
0, pi2
]
, given by Theorem 2.1. These
inequalities allow us to distinguish any LHV correlations from the singlet state
quantum correlations for angles θ ∈
(
0, pi3
)
, as stated by Lemma 2.4. Nevertheless,
we have introduced a hypothesis, the SHCMH, which if were proven true would
imply that our Bell inequalities are not optimal.
The Strong Hemispherical Colouring Maximality Hypothesis (SHCMH) states
that colouring 1, in which for one sphere, one hemisphere is totally black and the
67
Chapter 2. Bloch Sphere Colourings and Bell Inequalities
other hemisphere is totally white, and the colours are opposite for the other
sphere, gives the maximum correlation and anticorrelation for θ ∈ [0, θsmax]. A
weak version of this hypothesis, the WHCMH, restricts to colourings in which the
spheres are coloured oppositely and states that, for these colourings, colouring 1
gives the maximum anticorrelation for θ ∈ [0, θwmax].
We have explored these hypotheses numerically for some simple colourings
(see Figure 2.4). Our numerical results are consistent with these hypotheses
for θwmax ≤ 0.386pi and θ
s
max ≤ 0.345pi. Notice that a smaller upper bound,
θsmax ≤ 0.375pi, is given in our publication [94], because colouring 2∆ is not
considered there.
It would be interesting to explore these hypotheses numerically for other
colourings. We have restricted to compute the correlations for a few simple
colourings. As discussed in detail in Appendix B, the main reason for this was
to obtain sufficiently high precision, of the order of 10−5, for the plotted values
so that the given precision in the bounds θwmax ≤ 0.386pi and θ
s
max ≤ 0.345pi is
guaranteed. We expect that more involved colourings can be investigated with
adequate numerical techniques. We describe some colourings that we consider
interesting to investigate. Before doing so, we discuss some geometric intuitions
that motivate our interest in such colourings.
As introduced in section 2.3, a geometric intuition supporting the WHCMH
is that, for colourings with oppositely coloured spheres and small values of θ, the
anticorrelation seems to decrease proportionally to the boundary between black
and white regions. Since colouring 1 has the shortest boundary, the equator,
according to the previous intuition, the colouring 1 anticorrelation should be
optimal for this class of colourings and for small values of θ. This intuition can be
challenged by colourings defined by some parameter ν such that Cν(θν) < C1(θν)
for some θν ∈
(
0, pi2
)
, while Cν(θ) > C1(θ) for 0 < θ << θν . If a set of colourings
satisfying these properties can be found for which the angle θν can be made
arbitrarily small then the above geometric intuition is clearly satisfied, but the
WHCMH is false, which implies that the SHCMH is false too.
Further analysis of the previous intuitions is made by distinguishing the one-
dimensional case of antipodal colourings for a pair of circles. In this case, the
analogue of colouring 1 is a colouring in which, for one circle, one half-circle is
68
2.6. Discussion
black and the other one is white, and the colours are opposite for the other circle.
The correlations for this colouring are given by the function C˜1(θ) = −1 + 2θpi ,
where the tilde is used to distinguish the one-dimensional case. We can find
antipodal colourings of the circle that violate the one-dimensional analogue of
the WHCMH, that is, for any arbitrarily small θ there exists a colouring A such
that C˜A(θ) < C˜1(θ). Consider the set of colourings Aν , for odd positive integers
ν, defined by the colouring functions aAν (φ) = −bAν (φ) = (−1)
⌊
νφ
pi
⌋
, where φ ∈
[0, 2pi] is the angular coordinate in the circle. The correlations for these colourings
satisfy C˜Aν (θ) = −1 +
2νθ
pi for θ ∈
[
0, piν
]
and C˜Aν (θ) = 3 −
2νθ
pi for θ ∈
[
pi
ν ,
2pi
ν
]
.
Thus, by defining θAν ≡
2pi
ν , we have that C˜Aν (θ) > C˜1(θ) for θ ∈
(
0, 2pi1+ν
)
, where
2pi
1+ν < θAν , and that C˜1(θAν ) > C˜Aν (θAν ) = −1. Therefore, the set of colourings
Aν are in agreement with the geometrical intuitions given above, in the sense
that C˜Aν (θ) > C˜1(θ) for 0 < θ << θAν . However, the analogue of colouring 1 in
one dimension does not give the maximum anticorrelation for any range of θ > 0,
because for any arbitrarily small ε > 0 we can find a positive angle θAν < ε for
which C˜Aν (θAν ) < C˜1(θAν ).
For the one-dimensional case, the high degree of symmetry for the colourings
Aν allows them to achieve perfect anticorrelation at the angles θAν . However, in
the two-dimensional case, which we have investigated in this chapter, colourings
that have similar symmetry properties along some directions are less symmetric
along other directions. For example, consider a natural extension of the one-
dimensional colourings Aν to the two-dimensional case, defined by the colouring
functions aAν (, φ) = −bAν (, φ) = (−1)
⌊
νφ
pi
⌋
for  ∈ [0, pi] and φ ∈ [0, 2pi], where
 and φ are the polar and the azimuthal angles in the sphere, respectively (see
Figure 2.8). These colourings are antipodal for odd positive integers ν. Notice
that colouring A1 is colouring 1, apart from a rotation of the spheres of an angle
pi
2 around an axis in the equatorial plane, which does not change the correlations.
As in the one-dimensional case, we define θAν ≡
2pi
ν . Perfect anticorrelations are
achieved at angles θ = θAν if we consider measurement axes constrained to be
along the equator, but this property is lost for measurement axes along other
great circles. Due to the periodic variations of the colourings Aν , we expect the
correlations CAν (θ) for these colourings to oscillate between local maximums and
minimums at intervals with values close to θAν2 .
69
Chapter 2. Bloch Sphere Colourings and Bell Inequalities
Figure 2.8: Colourings Aν . These colourings are defined by the colouring func-
tions aAν (, φ) = −bAν (, φ) = (−1)
⌊
νφ
pi
⌋
for  ∈ [0, pi] and φ ∈ [0, 2pi], where 
and φ are the polar and the azimuthal angles in the sphere, respectively. The
figure illustrates the range of angles φ ∈ [0, pi] for one of the pair of spheres.
These colourings are antipodal for ν odd. Colouring A1 is the same as colouring
1, apart from a rotation of the spheres, which does not change the correlations.
Another set of simple colourings is the set Bν , defined by the colouring
functions aBν (, φ) = −bBν (, φ) = (−1)
⌊
2ν
pi
⌋
for  ∈ [0, pi) and aBν (pi, φ) =
−bBν (pi, φ) = −1, for φ ∈ [0, 2pi] and integers ν ≥ 1. For ν = 1, 2, 3, 4, these
colourings correspond to the colourings illustrated in Figure 2.4, whose corre-
lations are plotted in Figure 2.5. Similar to the colourings Aν , we expect the
correlations for these colourings to oscillate between local maximums and mini-
mums at periodic intervals, close to the value pi2ν in this case. These intuitions are
confirmed to some extent, as observed in Figure 2.5, in particular for big ν and
small θ. For example, the first local maximum of C4(θ) is very close to θ = pi8 .
It would be interesting to investigate numerically colourings Aν , colourings
Bν that we have not explored here, and variations of these, for example, in the
lines of colourings 2∆ and 3δ defined by (2.17). However, we do not expect these
colourings to reduce drastically the upper bounds θwmax ≤ 0.386pi and θ
s
max ≤
0.345pi, for the reasons previously mentioned. The different class of colourings
Dν , illustrated in Figure 2.9, seems more promising in producing correlations
CDν (θ) < C1(θ) for smaller values of θ.
Our intuition about the colourings Dν is that for angles θDν , whose order
of magnitude is schematized in Figure 2.9, CDν (θDν ) is a local minimum. We
expect that CDν (θDν ) < C1(θDν ) for some values of ν. This intuition is based
on the observation from Figure 2.9 that for circles with angles close to θDν there
seems to be a considerably big area for which the centre of these circles, in
one sphere, and points on these circles, in the oppositely coloured sphere, are
70
2.6. Discussion
Figure 2.9: Colourings Dν . The colouring for one sphere is illustrated in the
figure. The other sphere is coloured oppositely. The equator is touched on a
single point by circular curves that define the boundary between black and white
regions. These curves correspond to identical half-circles that repeat periodically
at azimuthal angles with intervals of 2piν , for odd positive integers ν. That is,
an odd number ν of these half-circles go around the equator circumference (the
figure illustrates the case ν = 9). The colouring is antipodal. It is observed that
the white region of the southern hemisphere is obtained by reflecting the black
region of the northern hemisphere through the equatorial plane and then rotating
the obtained region by an azimuthal angle equal to piν . The small blue circle and
the red centre point represent Alice’s and Bob’s measurement axes separated by
an angle of the order of magnitude of θDν , for which we expect that CDν (θDν ) is
a local minimum.
71
Chapter 2. Bloch Sphere Colourings and Bell Inequalities
anticorrelated. Curves different to half-circles defining the boundary between
black and white regions could be considered in order to maximize the quantity
−CDν (θDν ) + C1(θDν ). We expect bigger anticorrelations to be achieved with
convex curves compared to concave curves, in the northern hemisphere. The
circular curves can be defined explicitly using (2.10) and (2.11). If the colourings
Dν were such that θDν is a strictly decreasing function of ν with limν→∞ θDν = 0,
as suggested by Figure 2.9, and CDν (θDν ) < C1(θDν ) for all ν big enough, the
WHCMH and the SHCMH would be false. It is perhaps more reasonable to
expect that CDν (θDν ) < C1(θDν ) for some values of ν, but not for ν arbitrarily
big. It would be interesting to investigate these colourings numerically and to
find formal analytic arguments that support or discard these geometric intuitions.
Further analytic investigation of the correlations produced by antipodal colour-
ings could give Bell inequalities stronger than those stated in Theorem 2.1. A
possible research direction is to consider sets of measurement choices that are
along curves on the sphere more complicated than those illustrated in Figure 2.3,
which were used to prove Theorem 2.1. Consider the set of measurement choices
corresponding to projections on states with Bloch vectors ~a(j)i , performed by Alice,
and ~b(l)k , performed by Bob, that are illustrated in Figure 2.10. The points ~a
(j)
i ,
~b(l)k on the unit spheres are defined so that any pair of adjacent points is separated
by the same angle θN,M,L ∈
(
0, pi2
)
and that two series of points ~a(j)i are antipodal.
Consider integers N ≥ 2, M ≥ 2 and L = 0, 1, . . . ,
⌈
M−3
2
⌉
. Let the spherical
coordinates for these points be ~a(j)i = (i, φj), ~b
(l)
k = (αk, βl), where i, αk ∈ [0, pi]
are polar angles and φj = 2jΓ mod 2pi, βl = (2l + 1)Γ mod 2pi are azimuthal
angles with Γ ≡ (2L+1)pi2M for i = 0, 1, . . . , N , j = 0, 1, . . . ,M , k = 0, 1, . . . , N − 1,
l = 0, 1, . . . ,M − 1, such that i+1 > i for i = 0, 1, . . . , N − 1, αk+1 > αk for
k = 0, 1, . . . , N − 2, 0 = 0 and N = pi; the four angles along the great circles
passing between ~b(l)k and the four points ~a
(j)
i with (i = k, j = l), (i = k + 1, j =
l), (i = k, j = l+1), (i = k+1, j = l+1) for k = 0, 1, . . . , N−1, l = 0, 1, . . . ,M−1
are equal and are defined as θN,M,L. Since 0 = 0, all points ~a
(j)
0 are the same and
correspond to the north pole; we define them as ~a0. Similarly, since N = pi, all
points ~a(j)N are the same and correspond to the south pole; we define them as ~aN .
It follows that
N−i = pi − i, αN−1−k = pi − αk, (2.24)
72
2.6. Discussion
for i = 0, 1, . . . , N and k = 0, 1, . . . , N − 1. Let the measurement outcomes be
described by an LHVT defined by an antipodal colouring x ∈ X. For simplicity,
we define a0 ≡ ax
(
~a0
)
, aN ≡ ax
(
~aN
)
, a(j)i ≡ ax
(
~a(j)i
)
for i = 1, 2, . . . , N − 1,
j = 0, 1, . . . ,M , and b(l)k ≡ bx
(~b(l)k
)
for k = 0, 1, . . . , N − 1, l = 0, 1, . . . ,M − 1.
We consider the products a(j)i b
(l)
k for all pairs of vectors ~a
(j)
i and ~b
(l)
k separated by
an angle θN,M,L. The correlation is
C =
1
(4N−2)M
M−1∑
l=0
[
N−2∑
k=1
b(l)k
(
a(l)k +a
(l+1)
k +a
(l)
k+1+a
(l+1)
k+1
)
+ b(l)0
(
a0+a
(l)
1 +a
(l+1)
1
)
+b(l)N−1
(
aN+a
(l)
N−1+a
(l+1)
N−1
)
]
. (2.25)
From (2.24) and the definition of ~a(j)i , we have that ~aN is antipodal to ~a0
and ~a(M)N−i is antipodal to ~a
(0)
i , for i = 1, 2, . . . , N − 1. Thus, since the colouring
x satisfies the antipodal property, we have aN = −a0 and a
(M)
N−i = −a
(0)
i , for
i = 1, 2, . . . , N − 1. Hence, after arranging terms, we have
C =
1
(4N−2)M
{
M−2∑
l=0
[
b(l)0
(
a0+a
(l)
1 +a
(l+1)
1
)
+
N−2∑
k=1
b(l)k
(
a(l)k +a
(l+1)
k +a
(l)
k+1+a
(l+1)
k+1
)
+b(l)N−1
(
−a0+a
(l)
N−1+a
(l+1)
N−1
)]
+ b(M−1)0
(
a0+a
(M−1)
1 −a
(0)
N−1
)
+
N−2∑
k=1
b(M−1)k
(
a(M−1)k −a
(0)
N−k+a
(M−1)
k+1 −a
(0)
N−k−1
)
+b(M−1)N−1
(
−a0+a
(M−1)
N−1 −a
(0)
1
)
}
. (2.26)
We expect that a bound C ≥ BN,M can be found from the previous expression.
If so, after averaging over random rotations of the Bloch sphere, this would imply
a bound Cx(θN,M,L) ≥ BN,M for all antipodal colourings x, which would also
imply the bound Cx(θN,M,L) ≤ −BN,M by reversing the colouring of one sphere.
We expect such bounds to be tighter than the bounds given by Theorem 2.1, for
appropriate values of N , M and L.
A different analytic approach to prove Bell inequalities stronger than those
73
Chapter 2. Bloch Sphere Colourings and Bell Inequalities
Figure 2.10: Diagram of points ~a(j)i ,~b
(l)
k in the unit sphere that define angles θN,M,L
for integers N ≥ 2, M ≥ 2 and L = 0, 1, . . . ,
⌈
M−3
2
⌉
. The figure represents the
case N even and L = 0. The blue and red curves are meridians with azimuthal
coordinates φj = 2jΓ mod 2pi and βl = (2l + 1)Γ mod 2pi, respectively, with
Γ ≡ (2L+1)pi2M for j = 0, 1, . . . ,M and l = 0, 1, . . . ,M − 1. The brown and green
curves are circles defined by constant polar angles i and αk, respectively, for
i = 1, 2, . . . , N − 1 and k = 0, 1, . . . , N − 1. The points ~a(j)i = (i, φj) are the
intersection of the corresponding blue and brown curves. Similarly, the points
~b(l)k = (αk, βl) are the intersection of the corresponding red and green curves.
The points ~a0 and ~aN correspond to the south and the north poles, respectively.
The black curves represent arcs of great circles with the same angles, defined as
θN,M,L.
74
2.6. Discussion
given by Theorem 2.1 and possibly to prove the WHCMH and the SHCMH
consists in using techniques of geometric combinatorics for sets in Rn [97, 98].
Finally, we would like to stress that the key idea considered in this chapter
is the investigation of Bell inequalities defined by continuous parameters. Some
interesting recent results have been obtained with this approach [72]. This work
can be extended to multipartite scenarios and to systems of higher dimension.
These ideas can also be explored in terms of nonlocal games with continuous
inputs [95].
75

Chapter 3
Bound on the Success Probability
of Port-Based Teleportation from
No-Cloning and No-Signalling
3.1 Introduction
It is interesting to investigate the limitations and possibilities on quantum infor-
mation processing tasks that can be derived directly from no-signalling and other
fundamental principles of quantum theory. There are important results that have
been obtained with this approach. The maximum fidelity achieved by quantum
cloning machines can be deduced from the no-signalling principle [37]. The secu-
rity of quantum key distribution can be guaranteed as long as the no-signalling
principle is satisfied [27]. The information causality principle [71] implies the
Cirel’son bound for the CHSH inequality. The maximum guessing probability in
quantum state discrimination can be derived from the no-signalling principle [73].
In this chapter, we present a proof, published by us in [99], of an upper
bound on the success probability of a class of teleportation protocols, denoted as
port-based teleportation. The proof is based on the no-cloning theorem and the
no-signalling principle.
77
Chapter 3. Bound on the Success Probability of Port-Based
Teleportation from No-Cloning and No-Signalling
3.1.1 Port-Based Teleportation
Quantum teleportation is a fundamental protocol of quantum information theory
in which an unknown quantum state |ψ〉 is destroyed at its original location
by Alice and reconstructed at another location by Bob. The original quantum
teleportation protocol [13] works as follows: Alice and Bob must initially share
a maximally entangled state, Alice applies a Bell measurement on her systems,
she communicates her measurement outcome to Bob, who then applies a unitary
correction operation according to Alice’s message (see section 1.2.5.2).
In this chapter, we consider a different type of teleportation protocol, de-
noted as port-based teleportation (PBT). PBT was devised by Ishizaka and Hi-
roshima [100, 101] with the purpose of implementing a universal programmable
quantum processor that succeeds with probability arbitrarily close to unity; this
task can be achieved using standard teleportation too, but with a very small
success probability for input states of big dimension [102].
We consider general PBT protocols, which allow Alice to teleport an unknown
quantum state |ψ〉 to one of N ports at Bob’s site. PBT requires that Alice and
Bob share quantum entanglement and consists in the following steps. Alice ap-
plies a measurement with outcome k ∈ {0, 1, . . . , N}; if k = 0, teleportation fails,
otherwise |ψ〉 is teleported to the kth port. Alice communicates k to Bob, who
then discards the states at ports with index distinct to k. No further correction
operations are required; this is an advantage over standard teleportation that
makes PBT useful in various quantum information tasks. In the probabilistic
version of PBT, |ψ〉 is teleported perfectly but with a success probability p < 1;
in the deterministic version, the outcome k = 0 never occurs but the fidelity of
the teleported state is smaller than unity [100,101].
Besides its use as a universal programmable quantum processor, PBT can
be used to implement instantaneous nonlocal quantum computation (INLQC),
reducing exponentially the amount of needed entanglement compared to schemes
based on standard teleportation [85]. INLQC is the application of a nonlocal
unitary operation U on a state |ψ〉 shared by two or more distant parties with a
single round of classical communication (CC); see Figure 1.4 and the discussion in
section 1.3.2.2. If two rounds of CC are allowed, U can be implemented trivially
78
3.2. The Bound
as follows: Alice teleports her part of |ψ〉 to Bob, who then applies U to |ψ〉, now
in his location, and then teleports Alice’s part of the state back to her. However,
it is not trivial to complete this task with only one round of CC. This task can
be implemented for a general unitary U and a state of arbitrary dimension using
a recursive scheme based on standard teleportation, which consumes an amount
of entanglement growing double exponentially with the number of qubits n of the
input state |ψ〉 [82,84]. However, a scheme based on PBT allows the implementa-
tion of INLQC with an amount of entanglement growing only exponentially with
n [85].
INLQC has application to other distributed quantum tasks: it allows the
implementation of instantaneous nonlocal measurements (INLM) and also breaks
the security of position-based quantum cryptography (PBQC) and some quantum
tagging schemes [86–92]. INLM is the measurement of a nonlocal observable in a
distributed quantum state with a single round of CC [77–84]. Quantum tagging
[86–89, 93] and PBQC [91, 92] are cryptographic tasks that rely on quantum
information processing and relativistic constraints with the goals of verifying the
location of an object and providing secure communication with a party at a given
location, respectively.
3.2 The Bound
In the rest of this chapter, we derive an upper bound on the success probability
p of probabilistic PBT of an n-qubits state as a function of n and the number of
ports N :
p ≤
N
4n +N − 1
. (3.1)
The proof is based on a version of the no-cloning theorem, which we state and
prove in section 3.4, and the no-signalling principle. Our bound agrees with the
maximum success probability obtained in [101] for the particular case n = 1:
pmax =
N
3 +N
. (3.2)
79
Chapter 3. Bound on the Success Probability of Port-Based
Teleportation from No-Cloning and No-Signalling
Thus, we confirm the hypothesis presented in [101] that (3.2) can be derived
from fundamental laws of physics. It is an interesting open problem to find a
probabilistic PBT protocol for the case n > 1 and to see whether our bound is
achievable.
Comparing (3.1) and (3.2), we see that (pmax)n can be bigger than the upper
bound on p, which means that applying PBT individually to each qubit of the
input state |ψ〉 can give a higher success probability than applying PBT glob-
ally to |ψ〉. However, we justify the restriction that |ψ〉 must be localized to a
single port by noting that the advantage of PBT as described here, at least for
implementing a universal programmable quantum processor and INLQC, is that,
before receiving Alice’s message, Bob can apply the desired quantum operation
on the state at every port, after which, |ψ〉 is transformed as desired. Clearly,
this advantage is lost if the qubits of |ψ〉 spread among different ports, as done,
for example, in the protocols presented in [103].
3.3 Summary of the Proof
Before summarizing our proof of the bound (3.1), it is useful to give a general
description of the PBT protocol (see Figure 3.1). For simplicity of the exposition
we consider a pure input state |ψ〉a ∈ Ha. Due to the linearity of quantum
theory, the protocol works for mixed states too. Alice and Bob share a fixed
entangled state |ξ〉AB ∈ HA ⊗ HB, which is independent of |ψ〉 because this is
arbitrary and unknown. Bob has N ports {Bj}Nj=1, hence HB =
N⊗
j=1
HBj , where
dimHBj = dimHa = 2
n ∀j ∈ {1, . . . , N}. The system A includes any ancilla
held by Alice and so has an arbitrarily big dimension. However, in [100, 101],
HA =
N⊗
j=1
HAj and dimHAj = dimHa ∀j ∈ {1, . . . , N}. We follow a notation
in which subindex a is written in |ψ〉a only when we wish to emphasize that the
system a is in the state |ψ〉, similarly for other states and systems. The initial
global state is
|G〉aAB = |ψ〉a|ξ〉AB. (3.3)
80
3.3. Summary of the Proof
Alice applies a generalized measurement, which in general can be decomposed
into a unitary operation U acting jointly on a and A, followed by a projective
measurement. Alice obtains the outcome k ∈ {1, . . . , N} with probability qk > 0
and k = 0 with probability 1 −
∑N
k=1 qk. Notice that since we consider that A
has arbitrary dimension, we can include any ancilla as part of A that purifies the
output states after any outcome k ∈ {0, 1, . . . , N}. If k 6= 0, the global state is
transformed into
|Gk〉aAB = |ψ〉Bk |Rk〉aAB˜k , (3.4)
where B˜k ≡ B1B2 · · ·Bk−1Bk+1Bk+2 · · ·BN , hence, the state |ψ〉 is teleported to
the port Bk. However, if k = 0, PBT fails; in this case we denote the final state
as
|G0〉aAB = |F
(ψ)〉aAB. (3.5)
The total success probability is
p ≡
N∑
j=1
qj. (3.6)
Now we are able to summarize the proof. First, in section 3.4, we present a
version of the no-cloning theorem that allows us to show that the probabilities qk
and the states |Rk〉 cannot depend on |ψ〉, while the state |F (ψ)〉 must do, as the
notation suggests. Second, in section 3.5, we use the no-signalling principle to
show that the state ηj of port Bj before implementing PBT must be of the form
ηj = qj|ψ〉〈ψ|+
N∑
i=1
i 6=j
qiγj,i + (1− p)ω
(ψ)
j , (3.7)
where γj,i and ω
(ψ)
j are the states to which Bj transforms into after the outcomes
k = i /∈ {0, j} and k = 0 are obtained, respectively. Since ηj, γj,i and ω
(ψ)
j are
reduced states of |ξ〉, |Ri〉 and |F (ψ)〉, respectively, ηj and γj,i do not depend on
|ψ〉, while ω(ψ)j does. Third, we use the independence of these states from |ψ〉
and Equation (3.7) to show that if there exists a protocol that achieves success
probability qj for some states ηj and γj,i then there exists a protocol that achieves
81
Chapter 3. Bound on the Success Probability of Port-Based
Teleportation from No-Cloning and No-Signalling
Figure 3.1: Probabilistic port-based teleportation. Alice teleports an unknown
n−qubits state |ψ〉 to one of N ports at Bob’s site. Alice applies a measurement
and sends Bob the outcome k. The outcome k = 0, indicating failure, is obtained
with a nonzero probability. If k 6= 0, Bob only needs to select the kth port in
order to obtain the state perfectly. Alice’s and Bob’s resource is an arbitrary
entangled state |ξ〉AB. Bob’s system B consists of N ports, which are n−qubits
systems in states ηj, for j = 0, 1, . . . , N − 1. The global system aAB transforms
into the state |F (ψ)〉aAB if k = 0, or into the states |ψ〉Bi |Ri〉aAB˜i if k = i 6= 0.
The states of the port Bj after an outcome k = 0 and k = i /∈ {0, j} are obtained
are denoted as ω(ψ)j and γj,i, respectively. The states |F
(ψ)〉 and ω(ψ)j depend on
|ψ〉, while the states |Ri〉 and γj,i do not.
82
3.3. Summary of the Proof
the same success probability and satisfies
ηj = γj,i =
I
2n
. (3.8)
Fourth, in section 3.6, we assume (3.8) and present a protocol in which Alice tries
to send Bob a random message of 2n bits. This protocol combines the superdense
coding protocol [12] and a modified PBT protocol in which Alice holds every port
except for Bj, which is held by Bob, but does not allow communication. We show
that this protocol succeeds with probability
p′j = qj +
1
4n
(p− qj) + (1− p)rj, (3.9)
for some probability rj. Since there is not communication in such a protocol,
the no-signalling principle implies that Bob cannot obtain any information about
Alice’s message. This means that Bob can only obtain the correct message with
the probability of making a random guess: p′j =
1
4n .
1 Thus, we have
qj +
1
4n
(p− qj) + (1− p)rj =
1
4n
. (3.10)
Summing over j ∈ {1, 2, . . . , N} and using (3.6), we obtain that
p = fn,N(R),
where R ≡
∑N
j=1 rj and
fn,N(R) ≡
(
1 +
4n − 1
N − 4nR
)−1
.
It is straightforward to obtain that the condition 0 ≤ p ≤ 1 is satisfied only if
R ≤ N4n . Since the function fn,N(R) decreases monotonically with R in the range[
0, N4n
]
, we have that fn,N(R) ≤ fn,N(0) = NN+4n−1 . Thus, we obtain the bound
(3.1):
p ≤
N
4n +N − 1
.
1This is shown in Equation (1.63) in section 1.3.2.
83
Chapter 3. Bound on the Success Probability of Port-Based
Teleportation from No-Cloning and No-Signalling
3.4 A More General No-Cloning Theorem
The following theorem is in the spirit of the no-cloning theorem [8,9], in a prob-
abilistic [33] and a stronger [10, 11] version, and tells us that it is impossible to
extract any information from a single copy of an unknown quantum state without
modifying it.
Theorem 3.1. Consider a physical operation O that consists in a unitary opera-
tion U acting on a single copy of an unknown pure quantum state |ψ〉a ∈ Ha and
a fixed initial state |ξ〉b ∈ Hb of an auxiliary system b of arbitrarily big dimension,
followed by a projective measurement on b (or a subsystem of b). Let O induce a
transformation Tk:
|ψ〉a|ξ〉b −→ |ψ〉a|R
(ψ)
k 〉b,
with probability q(ψ)k > 0, for k ∈ {1, 2, . . . , N}, and a transformation T0:
|ψ〉a|ξ〉b −→ |F
(ψ)〉ab,
with probability 1−
∑N
k=1 q
(ψ)
k , for all |ψ〉a ∈ Ha, in which the index j ∈ {0, 1, . . . , N}
of the induced transformation Tj is known after O is completed. The physical op-
eration O is possible only if
q(ψ)k = q
(φ)
k ≡ qk, |R
(ψ)
k 〉b = |R
(φ)
k 〉b ≡ |Rk〉b,
for all |ψ〉a, |φ〉a ∈ Ha and k ∈ {1, 2, . . . , N}. Additionally, if
∑N
k=1 qk < 1, the
operation O must satisfy
〈F (φ)|F (ψ)〉 = 〈φ|ψ〉,
for all |ψ〉a, |φ〉a ∈ Ha.
Proof. By definition, the physical operation O corresponds to a unitary operation
U acting on the input system ab, followed by a projective measurement on b (or a
subsystem of b). In principle, more than one measurement outcome could corre-
spond to a particular induced transformation. But, given that the output states
are pure, only one outcome must be associated to a particular transformation.
Consider for example that two or more outcomes induced T0. In that case, the
84
3.4. A More General No-Cloning Theorem
output system ab of the transformation T0 would be in a mixed state. In order to
exclude this possibility, the projective measurement must correspond exactly to
N + 1 outcomes. Let Ia and Ib be the identity acting on Ha and Hb, respectively,
and Πjb be the projector acting on Hb that induces the transformation Tj, for
j = 0, 1, . . . , N . We have that
∑N
j=0 Π
j
b = Ib, and that
(Ia ⊗ Π
0
b)U |ψ〉a|ξ〉b =
√
1− p(ψ)|F (ψ)〉ab, (3.11)
(Ia ⊗ Π
k
b )U |ψ〉a|ξ〉b =
√
q(ψ)k |ψ〉a|R
(ψ)
k 〉b, (3.12)
with q(ψ)k > 0 and p
(ψ) ≡
∑N
k=1 q
(ψ)
k , for all |ψ〉a ∈ Ha and k ∈ {1, 2, . . . , N}. The
unitary operation U is such that
U |ψ〉a|ξ〉b =
√
1− p(ψ)|F (ψ)〉ab +
N∑
k=1
√
q(ψ)k |ψ〉a|R
(ψ)
k 〉b, (3.13)
for all |ψ〉a ∈ Ha. Notice that Equations (3.11) – (3.13) apply to the general case
in which the projective measurement is implemented on any subsystem of b. The
only requirement for b is that its dimension is not smaller than N + 1. This can
always be satisfied because b has an arbitrarily big dimension, by definition.
Consider any pair of states |φ〉a, |ψ〉a ∈ Ha. There exists a state |τ〉a ∈ Ha
such that 〈ψ|τ〉 = 0, for which we have
|φ〉a = e
iω
(√
Q|ψ〉a +
√
1−Q|τ〉a
)
, (3.14)
for some ω ∈ R and 0 ≤ Q ≤ 1. It follows that
U |φ〉a|ξ〉b = U
[
eiω
(√
Q|ψ〉a +
√
1−Q|τ〉a
)]
|ξ〉b
= eiω
√
QU |ψ〉a|ξ〉b + e
iω
√
1−QU |τ〉a|ξ〉b,
= eiω
√
Q
(
√
1− p(ψ)|F (ψ)〉ab +
N∑
k=1
√
q(ψ)k |ψ〉a|R
(ψ)
k 〉b
)
+eiω
√
1−Q
(
√
1− p(τ)|F (τ)〉ab +
N∑
k=1
√
q(τ)k |τ〉a|R
(τ)
k 〉b
)
, (3.15)
85
Chapter 3. Bound on the Success Probability of Port-Based
Teleportation from No-Cloning and No-Signalling
where in the first line we used (3.14), in the second line we used the linearity of
unitary evolution, and in the third line we used (3.13). On the other hand, we
have
U |φ〉a|ξ〉b =
√
1− p(φ)|F (φ)〉ab +
N∑
k=1
√
q(φ)k |φ〉a|R
(φ)
k 〉b
=
√
1− p(φ)|F (φ)〉ab +
N∑
k=1
√
q(φ)k
[
eiω
(√
Q|ψ〉a +
√
1−Q|τ〉a
)]
|R(φ)k 〉b,
(3.16)
where in the first line we used (3.13) and in the second line we used (3.14).
Consider the following properties, which are shown at the end of this proof,
b〈R
(φ)
k |R
(ψ)
k′ 〉b = 0, for k, k
′ ∈ {1, 2, . . . , N} with k 6= k′, (3.17)
ab〈F
(φ)|
(
|ψ〉a|R
(ψ)
k 〉b
)
= 0, if p(φ) < 1, for k ∈ {1, 2, . . . , N}, (3.18)
for all |ψ〉a, |φ〉a ∈ Ha. Taking the inner product a〈ψ| b〈R
(ψ)
k |U |φ〉a|ξ〉b in (3.15)
and (3.16), and using (3.17) and (3.18), together with the condition 〈ψ|τ〉 = 0,
we have that satisfaction of both expressions (3.15) and (3.16) requires
√
Qq(ψ)k =
√
Qq(φ)k 〈R
(ψ)
k |R
(φ)
k 〉. (3.19)
Thus, if 〈φ|ψ〉 6= 0, which from (3.14) means that Q > 0, it follows that
q(ψ)k = q
(φ)
k
∣
∣〈R(ψ)k |R
(φ)
k 〉
∣
∣
2
. (3.20)
Since this is valid for any pair of states |ψ〉a, |φ〉a ∈ Ha such that 〈φ|ψ〉 6= 0, a
similar expression is obtained by interchanging the roles of |ψ〉 and |φ〉 in (3.20),
which is
q(φ)k = q
(ψ)
k
∣
∣〈R(ψ)k |R
(φ)
k 〉
∣
∣
2
. (3.21)
From (3.19) – (3.21), it follows that
q(ψ)k = q
(φ)
k , |R
(ψ)
k 〉b = |R
(φ)
k 〉b, (3.22)
86
3.4. A More General No-Cloning Theorem
for all |ψ〉a, |φ〉a ∈ Ha satisfying 〈φ|ψ〉 6= 0 and all k ∈ {1, 2, . . . , N}. Since the
input state |ψ〉a is arbitrary and unknown, this property also applies for |ψ〉a, |φ〉a
with 〈φ|ψ〉 = 0. To see this consider arbitrary states |ψ〉a, |φ〉a, |φ
′〉a ∈ Ha such
that 〈φ′|ψ〉 = 0, 〈φ|ψ〉 6= 0 and 〈φ|φ′〉 6= 0. Applying (3.22) to the three different
pairs of states from the set {|ψ〉a, |φ〉a, |φ
′〉a} we obtain that (3.22) holds for all
|ψ〉a, |φ〉a ∈ Ha and all k ∈ {1, 2, . . . , N}, as claimed.
Consider the case p(ψ) < 1, which from (3.22) implies that p(φ) = p(ψ) < 1.
Taking the inner product of U |ψ〉a|ξ〉b and U |φ〉a|ξ〉b, applying (3.13), and using
(3.17), (3.18) and (3.22), we obtain
〈F (φ)|F (ψ)〉 = 〈φ|ψ〉, (3.23)
for all |ψ〉a, |φ〉a ∈ Ha, as claimed.
Now we show (3.17). Let |ψ〉a, |φ〉a ∈ Ha and k, k′ ∈ {1, 2, . . . , N}, with
k 6= k′. Let Va be the unitary operation acting on Ha that satisfies |φ〉a = Va|ψ〉a.
From (3.12), we have
√
q(φ)k q
(ψ)
k′ 〈R
(φ)
k |R
(ψ)
k′ 〉 =
√
q(φ)k q
(ψ)
k′ a〈φ| b〈R
(φ)
k |
(
|φ〉a|R
(ψ)
k′ 〉b
)
=
√
q(φ)k q
(ψ)
k′ a〈φ| b〈R
(φ)
k |(Va ⊗ Ib)|ψ〉a|R
(ψ)
k′ 〉b
= a〈φ| b〈ξ|U
†(Ia ⊗ Π
k
b )(Va ⊗ Ib)(Ia ⊗ Π
k′
b )U |ψ〉a|ξ〉b
= a〈φ| b〈ξ|U
†(Va ⊗ Π
k
bΠ
k′
b )U |ψ〉a|ξ〉b
= 0, (3.24)
where in the last line we used that ΠkbΠ
k′
b = 0 because {Π
j
b}
N
j=0 are projectors and
k 6= k′. Equation (3.17) follows because q(φ)k > 0 and q
(ψ)
k′ > 0.
Finally, we show (3.18). Let |ψ〉a, |φ〉a ∈ Ha and k ∈ {1, 2, . . . , N}, with
p(φ) < 1. From (3.11) and (3.12), we have
√
(
1− p(φ)
)
q(ψ)k ab〈F
(φ)|
(
|ψ〉a|R
(ψ)
k 〉b
)
= a〈φ| b〈ξ|U
†(Ia ⊗ Π
0
b)(Ia ⊗ Π
k
b )U |ψ〉a|ξ〉b
= a〈φ| b〈ξ|U
†(Ia ⊗ Π
0
bΠ
k
b )U |ψ〉a|ξ〉b
= 0, (3.25)
87
Chapter 3. Bound on the Success Probability of Port-Based
Teleportation from No-Cloning and No-Signalling
where we used that Π0bΠ
k
b = 0 because {Π
j
b}
N
j=0 are projectors and k > 0. Equation
(3.18) follows because p(φ) < 1 and q(ψ)k > 0, which completes the proof.
Notice that a general quantum operation acting on a single copy of an un-
known pure quantum state |ψ〉a ∈ Ha and an ancilla b in some fixed state, in
which the output state includes |ψ〉a with some nonzero probability, can be ob-
tained from the class of operations O considered in Theorem 3.1. This is because
a general quantum operation can be implemented by including an ancilla c of
sufficiently big dimension, applying a unitary operation followed by a projective
measurement, and then discarding the ancilla. Since b can have an arbitrarily big
dimension, the ancilla c can be included in b. Thus, after discarding c, by tracing
out over its Hilbert space, a general quantum operation satisfying the properties
mentioned above can be obtained from O. Also notice that the fact that the
system b is not transformed into general mixed states, but into pure states, does
not restrict the type of physical operations O. This follows because, given that
the system b has arbitrary dimension, any system that purifies the output states
can be included as part of b.
The no-cloning theorem [8,9] is easily obtained from Theorem 3.1. For conve-
nience, we can set N = 1. We see that the state |R1〉 cannot contain any copies
of the input state |ψ〉 because |R1〉 cannot have any relation to |ψ〉 at all. Thus,
the probabilistic version of the no-cloning theorem is obtained. The deterministic
version follows if we set q1 = 1.
Theorem 3.1 implies the following lemma.
Lemma 3.1. In a PBT protocol, as described by (3.3) – (3.5), for every input
state |ψ〉a the following is true. The probability qk of successful teleportation to
port Bk and the residual state |Rk〉aAB˜k when |ψ〉 is teleported to port Bk do not
depend on |ψ〉. However, the global state |F (ψ)〉aAB obtained after a failed PBT
protocol depends on |ψ〉.
Proof. We can identify the physical operation O in Theorem 3.1 with the PBT
protocol described by (3.3) – (3.5) followed by a swap operation of systems a and
Bk when outcome k 6= 0 is obtained. Therefore, according to Theorem 3.1, the
probability qk and the state |Rk〉 in (3.4) do not depend on |ψ〉, while the state
|F (ψ)〉 in (3.5) does. This is done precisely in the following way.
88
3.4. A More General No-Cloning Theorem
Let UaA be the unitary operation and {P
j
aA}
N
j=0 be the projectors acting on
the joint system aA corresponding to the PBT protocol given by (3.3) – (3.5).
Let IB be the identity acting on HB. We have that
(P 0aA ⊗ IB)UaA|ψ〉a|ξ〉AB =
√
1− p(ψ)|F (ψ)〉aAB, (3.26)
(P kaA ⊗ IB)UaA|ψ〉a|ξ〉AB =
√
q(ψ)k |ψ〉Bk |R
(ψ)
k 〉aAB˜k , (3.27)
for k ∈ {1, 2, . . . , N}, where p(ψ) ≡
∑N
k=1 q
(ψ)
k . We introduce a system a
′ with
dimHa′ = dimHa in a fixed pure state |χ〉a′ . We identify the physical operation
O in Theorem 3.1 with the PBT protocol given by (3.3) – (3.5) as follows. We
define b ≡ a′AB, |ξ′〉b ≡ |χ〉a′|ξ〉AB and
U ′ ≡
(
P 0a′A ⊗ IaB +
N∑
k=1
P ka′A ⊗ Sa,Bk
)
Ua′ASa,a′ , (3.28)
where Sc,d is a unitary operation that swaps the states of systems c and d. Notice
that we have primed the states and the unitary corresponding to the operation O
in order to avoid confusion with the states and unitary of the PBT protocol, which
remain unprimed. We also define the projectors acting on Hb corresponding to O
by Πjb ≡ P
j
a′A⊗IB, for j = 0, 1, . . . , N . Using the definitions for |ξ
′〉b and {Π
j
b}
N
j=0,
and using the expressions (3.26) – (3.28), it is straightforward to obtain that
(Ia ⊗ Π
0
b)U
′|ψ〉a|ξ
′〉b =
√
1− p(ψ)|χ〉a|F
(ψ)〉b, (3.29)
(Ia ⊗ Π
k
b )U
′|ψ〉a|ξ
′〉b =
√
q(ψ)k |ψ〉a|χ〉Bk |R
(ψ)
k 〉a′AB˜k , (3.30)
for k = 1, 2, . . . , N . By identifying the output states of the physical operation
O with |F ′(ψ)〉ab ≡ |χ〉a|F (ψ)〉b and |R
′(ψ)
k 〉b ≡ |χ〉Bk |R
(ψ)
k 〉a′AB˜k , it follows from
Theorem 3.1 that the success probabilities q(ψ)k and the residual states |R
(ψ)
k 〉 of
the PBT protocol cannot depend on the input state |ψ〉. Additionally, it follows
that the failure states in the PBT protocol satisfy 〈F (φ)|F (ψ)〉 = 〈φ|ψ〉 for all
|φ〉, |ψ〉 ∈ Ha, and thus depend on the input state.
The fact that the states |Rk〉aAB˜k do not depend on |ψ〉, together with the fact
that Alice knows the resource state |ξ〉AB, her unitary operation U , her projective
89
Chapter 3. Bound on the Success Probability of Port-Based
Teleportation from No-Cloning and No-Signalling
measurement and its result k, implies that Alice knows the states |Rk〉aAB˜k . This
is useful in section 3.6.
3.5 Conditions on the Port States
In this section, we deduce the form of the states in the port-based teleportation
protocol. In particular, we prove (3.7) and (3.8).
We consider the state ηj of the port Bj held by Bob before the PBT protocol
begins. From (3.3), we have that
ηj ≡ TraAB˜j (|G〉〈G|)aAB
= TrAB˜j (|ξ〉〈ξ|)AB . (3.31)
The no-signalling principle implies that from Bob’s point of view, his state does
not change with Alice’s local operations if she does not send him any information.
However, from Alice’s point of view, after she applies her operations, Bob’s state
changes according to her measurement result k.
1) With probability qj, k = j and ηj changes to |ψ〉〈ψ|, as can be seen from
(3.4):
TraAB˜j (|Gj〉〈Gj|)aAB = (|ψ〉〈ψ|)Bj .
2) With probability qi, k = i /∈ {0, j} and ηj changes to some state that we
denote as γj,i. From (3.4), we have that
γj,i ≡ TraAB˜j (|Gi〉〈Gi|)aAB
= TraAB˜j,i (|Ri〉〈Ri|)aAB˜i , (3.32)
where B˜j,i ≡ B1 · · ·Bj−1Bj+1 · · ·Bi−1Bi+1 · · ·BN . In this case, |ψ〉 is successfully
teleported to port Bi 6= Bj.
3) With probability 1 − p, k = 0 and ηj changes to some state that we call
ω(ψ)j . From (3.5), we have that
ω(ψ)j ≡ TraAB˜j (|G0〉〈G0|)aAB
= TraAB˜j
(
|F (ψ)〉〈F (ψ)|
)
aAB
. (3.33)
90
3.5. Conditions on the Port States
This is the failure result, hence ω(ψ)j 6= |ψ〉〈ψ| in general.
Since the resource state |ξ〉 is fixed for any input state |ψ〉, we see from (3.31)
that ηj does not depend on |ψ〉. Equations (3.32), (3.33) and Lemma 3.1 imply
that qj and γj,i do not depend on |ψ〉, while ω
(ψ)
j does.
Due to the no-signalling principle, Bob cannot learn Alice’s outcome before
he receives any information from her. Therefore, from Bob’s point of view, before
receiving any information from Alice, his state is
ηj = qj|ψ〉〈ψ|+
N∑
i=1
i 6=j
qiγj,i + (1− p)ω
(ψ)
j ,
which is Equation (3.7).
Now we show that if there exists a protocol that achieves success probability qj
for some states ηj and γj,i then there exists a protocol with corresponding states
η′j, γ
′
j,i and ω
′(ψ)
j that achieves the same success probability and satisfies η
′
j =
γ′j,i =
I
2n , that is, Equation (3.8). The claimed protocol, which for convenience
we call primed, is the following.
We define the set of unitary operations {Vl}4
n
l=1 ≡ {σ0, σ1, σ2, σ3}
⊗n, where σ0
is the identity acting on C2 and {σi}3i=1 are the Pauli matrices. Below, we use
the identity
I
2n
≡
1
4n
4n∑
l=1
VlρV
†
l , (3.34)
which is satisfied for any quantum state ρ of dimension 2n, and which follows from
the identity (1.12) for a qubit state. Consider an ancilla a′ with Hilbert space Ha′
of dimension 4n at Alice’s site, which in general can be included as part of the
system A, but which for clarity of the presentation is distinguished as a different
system. Let {|µl〉}4
n
l=1 be an orthonormal basis of Ha′ . The ancilla a
′ is prepared
in the state |φ〉 ≡ 12n
∑4n
l=1|µl〉. Conditioned on a
′ being in the state |µl〉, the
following operations are performed. Before implementing PBT, Bob’s system Bj
is prepared in the state VlηjV
†
l . Then, if Alice applies the PBT protocol described
by (3.3) – (3.5), which satisfies (3.7), on her system aA then with probability qj
the state of the system Bj transforms into Vl|ψ〉 and with probability qi transforms
into Vlγj,iV
†
l , where i /∈ {j, 0}; this is clear from (3.7) and follows from the fact
91
Chapter 3. Bound on the Success Probability of Port-Based
Teleportation from No-Cloning and No-Signalling
that the operations on Bj commute with those on aA (no-signalling) and from
the linearity of quantum theory (see details in Appendix C). Thus, consider that
before doing this, Alice applies V †l on her input state |ψ〉a. In this case, the
state |ψ〉 is teleported without error. Since the states of the system Bj before
implementing PBT and after an outcome k = i /∈ {j, 0} is obtained do not depend
on the teleported state, these states remain the same. Hence, in the primed PBT
protocol, we obtain that, after discarding the ancilla a′, by taking the partial
trace over Ha′ , the initial state of the system Bj is η′j =
1
4n
∑4n
l=1 VlηjV
†
l and its
final state after an outcome k = i /∈ {j, 0} is obtained is γ′j,i =
1
4n
∑4n
l=1 Vlγj,iV
†
l ,
both of which equal I2n , as follows from the identity (3.34). Therefore, we see
from (3.7) that this protocol satisfies
I
2n
= qj|ψ〉〈ψ|+
N∑
i=1
i 6=j
qi
I
2n
+ (1− p)ω′(ψ)j , (3.35)
where
ω′(ψ)j ≡
1
4n
4n∑
l=1
Vlω
(ψl)
j V
†
l , (3.36)
and ψl refers to dependence on the state V
†
l |ψ〉.
We have shown that the previous PBT protocol succeeds with probability qj
and satisfies (3.8). Thus, we assume (3.8) in the following section. Without loss
of generality, we consider that the system a′ is included in A, hence, we do not
need to mention it again.
3.6 Implications from Superdense Coding
Now we present a protocol in which Alice tries to send Bob a random message of
2n bits that succeeds with the probability p′j given by Equation (3.9). Note that,
so far, we have considered the input system a to be in a pure state. However,
the previous arguments work if a is in a mixed state too. Thus, consider that
a is in a bipartite maximally entangled state |φ〉ab with system b, held by Bob,
and that Alice and Bob perform superdense coding [12] using this state. Alice
wants to communicate Bob a 2n-bits random message x ∈ {1, 2, . . . , 4n}. Alice
92
3.6. Implications from Superdense Coding
applies a local unitary operation Ux on a, after which, the system ab transforms
into the state Ux ⊗ I|φ〉ab. For clarity of the presentation, let us consider that
|ψ〉ab = Ux⊗I|φ〉ab. The set of unitary operations {Uy}4
n
y=1 is such that it generates
an orthonormal basis, B = {Uy ⊗ I|φ〉}4
n
y=1. Instead of sending the system a
directly to Bob, Alice teleports its state to the system Bj, at Bob’s location,
using the modified PBT protocol described below in which no communication
is allowed (see Figure 3.2). Bob completes the superdense coding protocol by
measuring the system Bjb in the basis B. Let y be Bob’s measurement outcome.
Bob obtains Alice’s message correctly, that is y = x, if his outcome corresponds
to the state |ψ〉.
Figure 3.2: A superdense coding protocol without communication. Alice is given
a random message x of 2n bits that she encodes in the n−qubits system a by
performing the unitary operation of the superdense coding protocol. Alice’s sys-
tem a is in a bipartite maximally entangled state (MES) |φ〉ab with Bob’s system
b. Then, Alice teleports the state of system a to Bob’s port Bj, using a modified
port-based teleportation protocol in which there is no communication. In this
modified PBT protocol, Bob has the system Bjb and Alice has the system aAB˜j,
where B˜j = B1B2 · · ·Bj−1Bj+1Bj+2 · · ·BN . Similar to the original PBT protocol,
the resource state is |ξ〉AB. The state of port Bj is completely mixed, which
means that Bj is in a bipartite MES with Alice’s system. Finally, Bob applies
the projective measurement of the superdense coding protocol on his system bBj.
Bob’s 2n−bits measurement outcome y is his guess for Alice’s input message x.
93
Chapter 3. Bound on the Success Probability of Port-Based
Teleportation from No-Cloning and No-Signalling
Bob has the system Bjb and Alice has the system aAB˜j. Similar to the
original PBT protocol, the initial global state is given by |ψ〉ab|ξ〉AB, and Alice
applies the same local operations on the systems a and A, only. Therefore,
if Alice’s measurement result is k 6= 0, the final state is |ψ〉Bkb|Rk〉aAB˜k and
the residual state |Rk〉aAB˜k is known by her. However, Alice is not allowed to
communicate with Bob. From (3.8), we assume that Bob’s system Bj is initially
in the completely mixed state, meaning that it is maximally entangled with its
purifying system, at Alice’s site. Consider the possible situations according to
Alice’s outcome k.
1) Alice obtains k = j, so the system Bjb is transformed into the state |ψ〉Bjb;
this occurs with probability qj. Bob measures the system Bjb in the basis B and
obtains Alice’s message correctly.
2) Alice obtains k = i /∈ {0, j}. Thus, the state of the system a is teleported
to the system Bi, at Alice’s site. This occurs with probability qi. We denote the
composite system aAB˜j,i as Aj,i. Alice has the systems Aj,i and Bi, while Bob has
the system Bjb. The global system is in the state |ψ〉Bib|Ri〉Aj,iBj . Equation (3.8)
tells us that the system Bj is completely mixed, meaning that it is maximally
entangled with its purifying system Aj,i. It follows that
|Ri〉Aj,iBj =
1
√
2n
2n∑
l=1
|li〉Aj,i |li〉Bj ,
where {|li〉}2
n
l=1 is the Schmidt basis of the state |Ri〉Aj,iBj . As mentioned in section
3.4, Alice knows the obtained state |Ri〉Aj,iBj , which means that she knows its
Schmidt basis. Therefore, Alice can apply the local operations of the standard
teleportation protocol [13] on the systems Bi and Aj,i in order to teleport the state
of the system Bi to the system Bj. Then, Bob completes the superdense coding
protocol by measuring his system Bjb in the basis B. Since communication is
not allowed, the no-signalling principle implies that Bob obtains Alice’s message
correctly with probability 14n . Thus, if k /∈ {0, j}, the success probability is
1
4n
N∑
i=1
i 6=j
qi =
1
4n
(p− qj).
94
3.7. Discussion
3) Alice obtains k = 0, hence the protocol fails; this occurs with probability 1−
p. In this case, we should allow for the possibility that the final state of the system
Bjb has nonzero overlap with the input state |ψ〉. Hence, after measuring in the
basis B, the system Bjb transforms into the state |ψ〉Bjb with some probability
rj. Thus, if k = 0, Bob obtains Alice’s message with probability rj.
It follows that the total success probability p′j of the previous protocol is given
by (3.9):
p′j = qj +
1
4n
(p− qj) + (1− p)rj.
Thus, Equation (3.10) and our main result Equation (3.1), follow.
3.7 Discussion
Port-based teleportation (PBT) was introduced by Ishizaka and Hiroshima in
[100, 101]. The case n = 1 in which a single qubit state is input to port-based
teleportation was discussed in [101]. By direct optimization, the maximum suc-
cess probability pmax in the probabilistic version and the maximum average fidelity
fmax in the deterministic version were obtained: pmax = NN+3 (Equation (3.2)) and
fmax = 23 +
1
3 cos
2pi
N+2 . Given the simplicity of these expressions, it was hypothe-
sized that they can be derived from fundamental laws of physics. In this chapter,
we have confirmed such a hypothesis for the probabilistic version of PBT.
We have shown an upper bound on the success probability p of probabilistic
PBT of an unknown n-qubits state as a function of n and the number of ports
N , Equation (3.1). This bound implies a lower bound on the number of ports
that are needed to achieve a given success probability p:
N ≥
4n − 1
p−1 − 1
. (3.37)
Our proof of (3.1) is based on the no-signalling principle and a version of
the no-cloning theorem, Theorem 3.1, which we have presented and proven in
section 3.4. Our bound on p agrees with the maximum success probability for
the case n = 1, Equation (3.2). A probabilistic PBT protocol for the case n > 1
has not been developed explicitly; it would be interesting to know whether our
95
Chapter 3. Bound on the Success Probability of Port-Based
Teleportation from No-Cloning and No-Signalling
bound can be achieved in this case too. It would also be interesting to investigate
whether our techniques are useful to show an upper bound on the maximum
average fidelity achieved in the deterministic version of PBT.
An interesting question is, how much entanglement do Alice and Bob need
to share to perform PBT with a given success probability p in the probabilistic
version, or a given fidelity f in the deterministic version? An explicit deterministic
PBT protocol for a general n−qubits state was given by Ishizaka and Hiroshima
[100]. The resource state |ξ〉AB of this protocol was not optimized to achieve the
maximum average fidelity. It consists of Nn singlets, each port consisting of n
qubits in singlet states. The optimal PBT protocol was given only for the case
n = 1 [101], where it was shown that the resource state |ξ〉AB optimizing this
protocol is not a maximally entangled state.
As mentioned in section 3.1, an important application of PBT is that it allows
the implementation of instantaneous nonlocal quantum computation (INLQC)
with an amount of entanglement that is only exponential in the number of qubits
n of the input state [85], compared to the double exponential amount of schemes
based on standard teleportation [82, 84]. An INLQC consists in the application
of a nonlocal unitary operation on a state |ψ〉 distributed among distant parties,
with a single round of communication by the parties. INLQC has application
to position-based quantum cryptography (PBQC) [91, 92] and some quantum
tagging schemes [86–90] because it allows an eavesdropper to break their security,
as described below.
In PBQC [91], a set of M distant parties called verifiers initially share a quan-
tum state |ψ〉v1v2···vM . The verifiers collaborate with the goal of authenticating
that communication is performed with another party, the prover, who is at a
particular position P in space. The verifiers send their respective systems vj to
the prover. The prover computes a nonlocal unitary U on the received systems
and then sends the transformed systems back to the verifiers. The communica-
tion between the verifiers and the prover is performed at the speed of light. The
verifiers should share the state U |ψ〉 at the end of this protocol. Each verifier
requires that he receives his system back from the prover in the time that light
takes to travel from his location to P and back to his location. The verifiers
collaborate to verify that they share the state U |ψ〉 and that the communication
96
3.7. Discussion
times are as expected.
A set of M collaborating eavesdroppers can break the security of PBQC in
the following way [91]. Each eavesdropper is located between P and a verifier.
The eavesdroppers intersect the systems vj and then collaborate to implement
an INLQC, after which they share the state U |ψ〉. Then the eavesdroppers send
the corresponding systems back to the verifiers. If necessary, they wait some
time before sending the systems so that the verifiers receive them at the correct
times. Since the INLQC takes only a single round of communication among the
eavesdroppers, the verifiers receive the systems at the right times and they share
the correct state U |ψ〉. However, the prover does not communicate with the
verifiers at all. Thus, this scheme breaks the security of PBQC.
We consider it useful to describe explicitly how an INLQC is implemented
using port-based teleportation [85]. Alice and Bob share a quantum state |ψ〉a1b
on which they want to implement an arbitrary nonlocal unitary operation U .
Alice has the system a1 of n1 qubits and Bob has the system b of n2 qubits,
with n = n1 + n2. We notice that in general, a bipartite INLQC on a state
|ψ〉 ∈ H1 ⊗ H2 requires that the dimension of H1 ⊗ H2 is d ≥ 4, that is, the
Hilbert spaces H1 and H2 correspond to at least one qubit each. Alice and Bob
also share singlet states and the PBT resource state |ξ〉AB, with B = B1B2 · · ·BN
denoting the N ports of n qubits each at Bob’s site.
First, Bob applies Bell measurements on his system b and his part of n2
singlets shared with Alice and obtains the outcome ~x = (x1, x2, . . . , xn2), with
xj ∈ {0, 1}2. He does not communicate his outcome to Alice. Thus, his state
is teleported to a system a2 at Alice’s location, up to some Pauli errors σ~x =
σx1 ⊗ σx2 ⊗ · · · ⊗ σxn2 .
Second, Alice applies the measurement of PBT on his sytem aA and obtains
the outcome k, where a denotes the n−qubits joint system a1a2. If k = i 6= 0, the
port Bi is transformed into the state σ~x|ψ〉. Notice that, before receiving Alice’s
message, Bob does not know whether PBT has been successful and if so to which
port the state has been teleported.
Third, Bob applies the string of Pauli operations σ~x followed by the n−qubits
unitary operation U on each of his N ports. If Alice’s measurement outcome is
k = i 6= 0, the final state of port Bi is U |ψ〉.
97
Chapter 3. Bound on the Success Probability of Port-Based
Teleportation from No-Cloning and No-Signalling
Finally, Bob applies the standard teleportation protocol [13] to the corre-
sponding n1−qubits system for each of his N ports, which includes communica-
tion of his outcomes to Alice. Alice communicates her outcome k to Bob. Thus,
if k = i 6= 0, Bob selects the port Bi and the state U |ψ〉 is shared by Alice and
Bob, as desired. Notice that the communication from Alice to Bob and from Bob
to Alice is made at the same time and thus this protocol requires a single round
of communication, which, by definition, is a requirement for INLQC.
We hope the reader finds the previous discussion useful in clarifying what we
see as the key benefit of PBT, that the only operation that Bob needs to apply in
order to obtain the teleported state is to select a single of a set of N ports, whose
identity is indicated by Alice in a single message. This important advantage of
PBT requires a considerably big number of ports, as given by (3.37), in order
to teleport an n−qubits state with a success probability p, in the probabilistic
version, as follows from the bound (3.1) proven in this chapter. This follows from
the no-signalling principle and the no-cloning theorem, as we have shown.
98
Chapter 4
Quantum Information Causality
4.1 Introduction
Quantum physics satisfies the no-signalling principle. The no-signalling principle
states that a party, Alice, cannot communicate any information to a distant party,
Bob, if she does not send him any physical systems, independently of any physical
resources that they share [69], as we discussed in section 1.3.2. It implies that Bob
must necessarily receive a physical system from Alice, or possibly from another
party sharing correlations with her, in order to obtain information about data at
Alice’s location. Thus, an interesting question to ask is, how much information
can a transmitted physical system fundamentally communicate?
Different scenarios for answering the previous question can be considered ac-
cording to the properties of the transmitted system, the type of communicated
information, the way in which information is quantified, the number of involved
parties, the resources allowed among the parties, etc. The Holevo theorem states
an upper bound on the amount of classical information that a transmitted quan-
tum system can communicate, if shared quantum entanglement is not used by the
communicating parties [7]. The principle of information causality can be stated
in the context of hypothetical probabilistic theories more general than quantum
mechanics. It states an upper bound on the amount of information that a trans-
mitted classical system can communicate as a function of its dimension, indepen-
dently of any non-signalling physical resources allowed by the considered theory
99
Chapter 4. Quantum Information Causality
that the communicating parties previously shared [71]. In the case of quantum
theory, to which we restrict in this thesis, unless otherwise stated, information
causality applies to the scenario in which the communicating parties share arbi-
trary quantum resources. In this chapter, the principle of quantum information
causality, published by us in [104], is presented. Quantum information causality
is the quantum version of information causality. It states an upper bound on the
amount of quantum information that a transmitted quantum system can com-
municate as a function of its dimension, independently of any quantum physical
resources previously shared by the communicating parties.
4.1.1 The Holevo Bound
The Holevo theorem considers the following scenario involving two parties, Alice
and Bob. Alice has a classical random variable X that takes the value x ∈
{0, 1, . . . , d − 1} with probability Px. Alice encodes X in a quantum system
T as follows: if X = x, Alice prepares T in the quantum state ρx. Then, Alice
sends Bob the system T . Bob applies a positive operator valued measure on T and
obtains the outcome y, whose probability distribution defines the random variable
Y . Bob tries to guess Alice’s value x from his outcome y. A good measure of how
much information Bob has obtained about X from his measurement is given by
the classical mutual information between X and Y , H(X : Y ) ≡ H(X)+H(Y )−
H(XY ), where H(X) ≡ −
∑d−1
x=0 Px log2 Px is the classical entropy of X, etc. The
Holevo bound is the following:
H(X : Y ) ≤ S(ρ)−
d−1∑
x=0
PxS(ρx), (4.1)
where ρ =
∑ d−1
x=0Pxρx. In this scenario, Alice and Bob do not use shared quantum
entanglement. In particular, the transmitted system T is not entangled with
Bob’s system [5,7].
The Holevo bound implies that Bob cannot obtain more than m bits of in-
formation about Alice’s data, if Alice sends Bob a quantum system of m qubits.
100
4.1. Introduction
More precisely, if the transmitted system T is an m-qubits system then
H(X : Y ) ≤ m. (4.2)
The proof is as follows. From (4.1), we have that, in general, H(X : Y ) ≤ S(ρ).
Since the dimension of T is 2m, the quantum entropy of any state of T cannot be
bigger than m. Thus, S(ρ) ≤ m and Equation (4.2) follows. It is required that T
is not entangled with Bob’s system for this result to hold. If T were allowed to be
entangled with Bob’s system, by performing the superdense coding protocol, Bob
could learn 2m of Alice’s bits perfectly, in which case a value of H(X : Y ) = 2m
could be achieved.
4.1.2 Information Causality
Information causality was introduced in the context of the following information
task [71], which for convenience is denoted here as the information causality (IC )
game (see Figure 4.1).
The IC game. Consider two parties at different locations, Alice and Bob. Alice
is given a string of n random bits ~x ≡ (x0, x1, . . . , xn−1) in a physical system A.
Bob is given a random number k ∈ {0, 1, . . . , n−1}. After Alice and Bob play the
game, Bob outputs a bit yk. If yk = xk, Alice and Bob win the game. Alice and
Bob may play any strategy allowed by the theory, as long as their communication
is limited to a single message ~τ , encoded in a system T of m bits, that Alice sends
Bob, with m < n. In particular, Alice and Bob may use arbitrary non-signalling
physical resources allowed by the theory, which they share in the bipartite system
A′B, where A′ is held by Alice and B is held by Bob. The probability to win the
IC game is:
PIC ≡
1
n
n−1∑
k=0
P (yk = xk). (4.3)
We can consider the more general situation in which Alice’s inputs are dits,
which are variables of d > 2 possible values, instead of bits. In this case Alice’s
message is an m-dits number, with m < n, and Bob’s output is a dit. For
convenience, we only discuss the bit case, unless otherwise stated.
101
Chapter 4. Quantum Information Causality
Figure 4.1: The information causality game. Alice and Bob share an arbitrary
non-signalling resource in the physical systems A′ and B. Alice is given a system
A containing a string of n random bits, x0, x1, . . . , xn−1. Bob is given a random
number k ∈ {0, 1, . . . , n− 1}. After Alice and Bob play the game, Bob outputs a
bit yk. Alice and Bob win the game if yk = xk. They may play any strategy al-
lowed by the theory, as long as their communication is limited to a single message
from Alice to Bob only encoded in a system T of m bits, with m < n.
The IC game is an extension of quantum random access coding. The idea of
quantum random access coding was first considered by Wiesner with the name
of conjugate coding [6]. Formally introduced in [105], an (n,m, p) quantum ran-
dom access code (QRAC) is a map, applied by Alice, that encodes n bits into
a quantum state of m qubits that Alice sends Bob, with m < n, and a set of n
possible measurements that Bob applies on the received state. If Bob applies the
kth measurement, he decodes Alice’s kth bit with a probability not smaller than
p. It is a requirement that p > 12 , because p =
1
2 can be achieved by a random
guess. In a (classical) random access code (RAC), Alice sends Bob m bits instead
of m qubits, Bob applies an operation that depends on the message received
from Alice and on the bit that he wants to learn from Alice’s inputs. In general,
QRACs achieve higher success probabilities than RACs. RACs and QRACs do
not use entanglement shared by Alice and Bob; the use of shared randomness is
discussed in [106]. RACs in which Alice and Bob use shared entanglement are de-
noted as entanglement-assisted random access codes (EARACs) [107]. Generally,
EARACs achieve higher success probabilities than QRACs. A general strategy
102
4.1. Introduction
to play the IC game is an EARAC.
Some interesting questions that can be asked regarding the information causal-
ity game are, how much information can Bob obtain about ~x from the message ~τ
and his local resources in the system B? Is there any strategy that allows Alice
and Bob to achieve PIC = 1, given that m < n? What is the maximum value of
PIC that can be achieved?
The no-signalling principle provides simple answers to the previous questions
in the case m = 0, in which Alice does not send Bob any bits. It says that, in
this case, Bob cannot learn anything about Alice’s bit-string ~x. Thus, for any
strategy with m = 0, Bob outputs the correct number with the probability of
making a random guess: PIC = 12 .
1 Therefore, PIC = 12 , if m = 0.
The principle of information causality states an upper bound on the amount
of information that a message of m bits can communicate:
I(A : TB) ≤ m, (4.4)
where I(A : TB) is the mutual information between the system A containing
Alice’s data ~x and the joint system TB at Bob’s location, which includes the
received system T encoding the m-bits message ~τ and Bob’s local resources B [71].
Information causality holds in more general probabilistic theories in which the
definition of mutual information I(X : Y ) between two systems X and Y satisfies
a few properties. In the paper that introduced information causality [71], it was
shown that together with symmetry and non-negativity, sufficient conditions on
the mutual information for satisfaction of information causality are:
Consistency If both systems X and Y are classical, I(X : Y ) reduces to the
classical mutual information.
Data-processing inequality Acting locally on one of the systems cannot in-
crease the mutual information (see Equation (1.30) for the quantum case).
That is, if a physical transformation allowed by the theory acts on the
system Y only, transforming it to Y ′, then I(X : Y ′) ≤ I(X : Y ).
1This is shown in Equation (1.63) in section 1.3.2.
103
Chapter 4. Quantum Information Causality
Chain rule There exists a conditional mutual information I(X : Y |Z) that sat-
isfies the following: I(X : Y Z) ≡ I(X : Z) + I(X : Y |Z). This condition,
together with the symmetry property of the mutual information, implies
the identity I(X : Y Z)− I(X : Z) = I(X : Y |Z) = I(Y : XZ)− I(Y : Z).
It can also be shown that information causality follows from a few physical
conditions on the measure of entropy H [108]:
Consistency If X is a classical system then H(X) is the classical entropy.
Evolution with an ancilla Consider a system composed of two subsystems X
and Y . If a transformation is performed only on a single subsystem, Y →
Y ′, then the increase of entropy of the composite system, ∆H(XY ) ≡
H(XY ′) − H(XY ), cannot be smaller than the increase of entropy of the
transformed subsystem, ∆H(Y ) ≡ H(Y ′) − H(Y ). That is, ∆H(XY ) ≥
∆H(Y ).
The previous sets of conditions are satisfied by quantum theory if the mu-
tual information and the entropy are defined as the quantum mutual information
and the quantum entropy, respectively. Therefore, information causality is satis-
fied by quantum theory. Similarly, information causality is satisfied by classical
probabilistic theory. This follows straightforwardly from the fact that the quan-
tum entropy of a classical system reduces to the classical entropy, which is the
consistency condition.
Although originally defined in [71] as satisfaction of (4.4), information causal-
ity is usually stated as satisfaction of the following bound:
n−1∑
k=0
H(xk : yk) ≤ m, (4.5)
where H(xk : yk) is the classical mutual information between Bob’s output yk and
Alice’s bit xk, when Bob receives the number k [109]. An advantage of (4.5) over
(4.4) is that the quantity
∑n−1
k=0 H(xk : yk) is completely defined in classical terms,
while the quantity I(A : TB) requires a definition for the mutual information in
the particular theory that is being considered. On the other hand, if the theory is
104
4.1. Introduction
restricted to be quantum, we consider the bound (4.4) to be more advantageous
than the bound (4.5), because the former applies to a more general scenario than
the information causality game. Moreover, (4.4) implies (4.5), because it was
shown in [71] that
n−1∑
k=0
H(xk : yk) ≤ I(A : TB), (4.6)
while the inverse does not necessarily hold. We consider more convenient to define
information causality as satisfaction of the bound (4.4), hence, we adopt such a
convention here. In fact, we present a more general bound in section 4.2.2, from
which (4.4) is obtained.
The bound (4.5) is saturated by the following strategy in the IC game, which
here is denoted as naive. As previously agreed by Alice and Bob, Alice sends
Bob the message ~τ = (x0, x1, . . . , xm−1). Bob receives the number k. If k < m,
Bob outputs the correct bit yk = xk, otherwise he outputs a random bit. It is
straightforward to obtain that this strategy succeeds with a probability PIC =
1
2(1 +
m
n ).
The naive strategy does not achieve the maximum success probability, because
a different strategy achieves a higher success probability PIC = 12(1 +
1√
n) in the
case m = 1 [108]. Therefore, the quantity
∑n−1
k=0 H(xk : yk) does not necessarily
quantify how well Alice and Bob have played the IC game. But, it has the
advantage of having a simple and achievable upper bound that applies in the
general case m < n. On the other hand, the maximum value of PIC is only known
if m = 1 [107,108]. However, an upper bound on PIC that applies in the general
case m < n can be derived from information causality.
It is shown in [71] that satisfaction of (4.5) implies
n−1∑
k=0
h(Pk) ≥ n−m, (4.7)
where Pk ≡ P (yk = xk) and h(x) ≡ −x log2 x− (1− x) log2(1− x) is the binary
entropy. The concavity property of the classical entropy implies that h(PIC) ≥
105
Chapter 4. Quantum Information Causality
1
n
∑n−1
k=0 h(Pk), which from (4.7) implies the bound:
PIC ≤ P
′
IC, (4.8)
where P ′IC is defined as the maximum solution of the equation h(P
′
IC) = 1 −
m
n .
The lower bound 1−P ′IC ≤ PIC follows from the bound (4.8): if a strategy achieved
a success probability PIC = p such that p < 1−P ′IC then a strategy that flips the
outcomes would achieve a success probability 1 − p that violates the inequality
(4.8).
Information causality has important implications for the set of quantum cor-
relations [71,110–113]. It implies [71] the Cirel’son bound, while the no-signalling
principle does not [70]. The Cirel’son bound [44] states the maximum violation
of the CHSH inequality [42] that can be achieved by quantum theory. It can
be equivalently stated as the maximum success probability achieved by quantum
theory in the CHSH game. The CHSH game involves two parties at different
locations, Alice and Bob. Alice has a physical system A on which she applies one
of two measurements, chosen randomly and labelled by a ∈ {0, 1}. Similarly, Bob
has a physical system B on which he applies a random measurement b ∈ {0, 1}.
Alice and Bob obtain the outcomes r and s, respectively, where r, s ∈ {0, 1}.
The game’s goal is that Alice and Bob output numbers r and s that satisfy
r⊕ s = ab, where ⊕ denotes sum modulo 2. The success probability is defined as
PCHSH ≡ P (r ⊕ s = ab). The Cirel’son bound is:
PCHSH ≤
1
2
(
1 +
1
√
2
)
. (4.9)
The no-signalling principle does not imply satisfaction of Cirel’son’s bound, be-
cause there exist theoretical correlation boxes, the PR boxes, for which PCHSH = 1,
that still satisfy the no-signalling principle [70] (see section 1.3.2.1 for details).
However, information causality does imply Cirel’son’s bound. The proof con-
siders a particular protocol to play the IC game in the case m = 1 that uses
hypothetical non-signalling correlation boxes, not restricted to be described by
quantum theory. If boxes violating (4.9) were available in the protocol then the
bound (4.5) would be violated for a sufficiently big value of n, which would imply
106
4.2. Quantum Information Causality
violation of information causality, Equation (4.4) [71].
It was hypothesized that the complete set of correlations allowed by quan-
tum theory could be derived from information causality [71]. However, it was
shown that information causality, being a bipartite information principle, could
not imply the complete set of quantum correlations for an arbitrary number of
parties [112,113].
4.2 Quantum Information Causality
The principle of quantum information causality states that the maximum amount
of quantum information that a quantum system can communicate is limited by its
dimension, independently of any quantum physical resources previously shared
by the communicating parties. It considers the following scenario, which is illus-
trated in Figure 4.2. Alice, Bob and Charlie have quantum systems A, B and C,
respectively, of any dimension. The total system ABC is in an arbitrary quantum
state. Alice applies local operations on her system in order to obtain a quantum
system T of m qubits that she sends Bob. Apart from its dimension, there are
no other constraints on the transmitted system. In particular, the transmitted
system can be entangled in any way with Alice’s, Bob’s and Charlie’s systems.
Bob receives T and applies local operations on the system BT , which is denoted
as B′ after Bob’s operations.
The principle of quantum information causality states an upper bound on the
amount of quantum information that m qubits can communicate:
∆I(C : B) ≤ 2m, (4.10)
where ∆I(C : B) ≡ I(C : B′) − I(C : B) is Bob’s gain of quantum information
about C, I(C : B) ≡ S(C) + S(B) − S(CB) is the quantum mutual informa-
tion between C and B, S(C) is the quantum entropy [4, 5] of C, etc. Since
the quantum mutual information quantifies the total correlations between two
quantum systems [30–32], we consider ∆I(C : B) to be a good measure for the
communicated quantum information.1
1Note that there are measures [30–32] for the purely classical and purely quantum parts
107
Chapter 4. Quantum Information Causality
Figure 4.2: Setting for quantum information causality. Alice, Bob and Charlie
have respective quantum systems A, B and C in an arbitrary quantum state,
which can be entangled in any possible way. After applying local operations on
her system, Alice obtains a quantum system T of m qubits that she sends Bob.
Bob receives T and applies local operations on the system BT , which is denoted
as B′ after Bob’s operations.
The proof of quantum information causality is very simple. It follows from
three properties of the quantum entropy: subadditivity [28], the triangle inequal-
ity (also called the Araki-Lieb inequality) [29] and the data-processing inequal-
ity [5]. By definition of the quantum mutual information,
I(C : BT ) = S(C) + S(BT )− S(CBT ). (4.11)
From the subadditivity property, we have that
S(BT ) ≤ S(B) + S(T ). (4.12)
The triangle inequality, |S(CB)− S(T )| ≤ S(CBT ), implies that
− S(CBT ) ≤ S(T )− S(CB). (4.13)
of the correlations between two quantum systems, whose sum is equal to the quantum mutual
information (see [114] for a review). We do not consider such a classification in our discussion.
108
4.2. Quantum Information Causality
Equations (4.11) – (4.13) imply
I(C : BT ) ≤ 2S(T ) + I(C : B). (4.14)
The data-processing inequality states that local operations cannot increase the
quantum mutual information [5]. Thus, I(C : B′) ≤ I(C : BT ), which from (4.14)
implies that I(C : B′) ≤ 2S(T ) + I(C : B). It follows that
∆I(C : B) ≤ 2S(T ). (4.15)
Finally, since S(T ) ≤ log2 (dim T ), the quantum information that T can com-
municate is limited by its dimension. Therefore, if T is a system of m qubits,
Equation (4.10) follows from Equation (4.15), because S(T ) ≤ m in this case.
The previous proof does not require to mention Alice’s system. This means
that Equation (4.10) is valid independently of how much quantum entanglement
Alice and Bob share. This also means that Equation (4.10) is valid too if we
consider that Alice and Charlie are actually the same party. Thus, quantum
information causality shows: the maximum possible increase of the quantum mu-
tual information between Charlie’s and Bob’s systems is only a function of the
dimension of the system T received by Bob, independently of whether it is Alice
or Charlie who sends Bob the system T and of how much quantum entanglement
Bob shares with them.
4.2.1 Achievability of the Bound
Achievability of equality in (4.10) requires that the transmitted system T is max-
imally entangled with Charlie’s system C, as shown below.
Following the proof of the bound (4.10), we note that equality requires the
following conditions to be satisfied. First, the transmitted system T has to be in
a product state with Bob’s system B in order to satisfy S(BT ) = S(B) + S(T ).
Second, the system T can only be entangled with the joint system CB, so that we
have −S(CBT ) = S(T )−S(CB), as shown below. Third, the state of the system
T has to be completely mixed so that its entropy is maximum: S(T ) = m; this
means that T has to be maximally entangled with its purifying system. Together,
109
Chapter 4. Quantum Information Causality
the previous conditions imply that T has to be maximally entangled with C. It
is also required that the quantum mutual information between BT and C does
not decrease by Bob’s operations, that is, I(C : B′) = I(C : BT ).
Now we show the second condition: satisfaction of−S(CBT ) = S(T )−S(CB)
is achieved if and only if T is entangled only with the joint system CB [5]. Recall
that the original system at Alice’s site is denoted as A, from which Alice obtains
the transmitted system T after performing local operations. It is convenient to
consider that Alice initially has both systems A and T and then applies some local
operation on the joint system AT . The systems A, B and C are arbitrarily big.
Thus, without loss of generality, we can consider that the global system ACBT
is in a pure state. Alice’s quantum operation on the system AT can in general be
represented by a unitary operation followed by a projective measurement. Thus,
after Alice’s operation, the global system ACBT remains in a pure state. Due to
the Schmidt decomposition of a bipartite pure state, it follows that
S(CB) = S(AT ),
S(A) = S(CBT ). (4.16)
From subadditivity, it is obtained that
S(AT ) ≤ S(A) + S(T ), (4.17)
which from (4.16) implies that
S(CB) ≤ S(CBT ) + S(T ). (4.18)
Equality in (4.18) is achieved if and only if equality in (4.17) is satisfied, which
occurs if and only if T is in a product state with A. Therefore, the relation
−S(CBT ) = S(T ) − S(CB) is satisfied if and only if T is entangled only with
the system CB, as claimed.
110
4.3. The Quantum Information Causality Game
4.2.2 The Case of Information Causality
If the transmitted system T is classical, equality in (4.10) cannot be achieved. In
this case, the following bound is satisfied:
∆I(C : B) ≤ m, (4.19)
where C and B are quantum systems and I(C : B) denotes their quantum mu-
tual information. From now on, this bound is denoted as information causality.
It considers a situation more general than the information causality principle
originally introduced in [71], Equation (4.4). It reduces to Equation (4.4) if the
following conditions are satisfied. First, Charlie and Bob do not share correla-
tions initially, hence, I(C : B) = 0. Second, Charlie and Alice are actually the
same party, who are identified as Alice. Third, Alice’s systems are re-labelled as
follows: A → A′ and C → A. Fourth, the (re-labelled) system A is restricted to
be classical and correspond to a string ~x of n bits. Finally, the system B′ denotes
the system BT before Bob applies any operations.
The proof of the information causality bound, Equation (4.19), is similar to
the proof of the quantum information causality bound, Equation (4.10). The only
difference is that if T is classical then the bound −S(CBT ) ≤ S(T ) − S(CB)
cannot be achieved. In fact, in this case the smaller upper bound −S(CBT ) ≤
−S(CB) is satisfied. A way to see this is that, if T is a classical variable, the
state of the joint system CBT is a distribution over all possible values ~τ of T and
states of CB for each ~τ . Therefore, there exists a transformation ~τ → (CB)~τ .
Thus, the data-processing inequality implies that I(CB : T ) ≤ I(T : T ). Hence,
since I(CB : T ) = S(CB) + S(T ) − S(CBT ) and I(T : T ) = S(T ), it follows
that −S(CBT ) ≤ −S(CB) [71, 108].
4.3 The Quantum Information Causality Game
Quantum information causality implies an upper bound on the success probability
of a new quantum information task, the quantum information causality (QIC )
game. We present two version of this game.
111
Chapter 4. Quantum Information Causality
The QIC game (version I). This task is illustrated in Figure 4.3. Initially,
Alice and Bob may share an arbitrary entangled state. However, they do not
share any correlations with Charlie. Let A′ and B denote the quantum systems
at Alice’s and Bob’s locations, respectively. Charlie prepares the qubits Aj and
Cj in the singlet state |Ψ−〉, for j = 0, 1, . . . , n − 1. Charlie keeps the system
C ≡ C0C1 · · ·Cn−1 and sends Alice the system A ≡ A0A1 · · ·An−1. Charlie
generates a random integer k ∈ {0, 1, . . . , n − 1} and gives it to Bob. Bob gives
Charlie a qubit Bk, whose joint state with the qubit Ck, denoted as ωk, must
be as close as possible to the singlet. Alice and Bob may play any strategy
allowed by quantum physics as long as the following constraint is satisfied: their
communication is limited to a single message from Alice to Bob only, encoded in
a quantum system T of m qubits, with m < n. Extra classical communication is
not allowed. Let B′ denote the joint system BT after Bob’s quantum operations.
In general, the qubit Bk is obtained by Bob from B′. Charlie applies a Bell
measurement on the joint system CkBk. Alice and Bob win the game if Charlie
obtains the outcome corresponding to the singlet. The success probability is
P ≡
1
n
n−1∑
k=0
〈Ψ−|ωk|Ψ
−〉. (4.20)
The QIC game (version II). This version is similar to version I, with the
following differences. Charlie does not prepare singlet states. Instead, Charlie
prepares n qubits in the pure states {|ψj〉}
n−1
j=0 , completely randomly. Charlie
sends Alice the qubit Aj in the quantum state |ψj〉, for j = 0, 1, . . . , n − 1, and
keeps a classical record of the states. We denote the global system that Alice
receives from Charlie as A ≡ A0A1 · · ·An−1. Bob gives Charlie a qubit Bk in
the state ρk, which must be as close as possible to |ψk〉. Charlie measures the
received state ρk in the orthonormal basis {|ψk〉, |ψ⊥k 〉}, where |ψ
⊥
k 〉 is the qubit
state with Bloch vector antiparallel to that one of |ψk〉. Alice and Bob win
the game if Charlie’s measurement outcome corresponds to the state |ψk〉. The
112
4.3. The Quantum Information Causality Game
Figure 4.3: The QIC game (version I). Alice and Bob share an arbitrary entangled
state in the quantum systems A′ and B. Alice is given n qubits, A0, A1, . . . , An−1,
which are in singlet states with Charlie’s respective qubits, C0, C1, . . . , Cn−1. Bob
is given a random integer k ∈ {0, 1, . . . , n − 1} by Charlie. Bob gives Charlie
a qubit Bk, whose joint state with the qubit Ck must be as close as possible
to the singlet. Alice and Bob may play any quantum strategy as long as their
communication is limited to a single message from Alice to Bob only, encoded in a
quantum system T of m qubits, with m < n. Charlie applies a Bell measurement
(BM) on the joint system CkBk. Alice and Bob win the game if Charlie obtains
the outcome corresponding to the singlet.
113
Chapter 4. Quantum Information Causality
success probability is
p ≡
∫
dµ0
∫
dµ1 · · ·
∫
dµn−1
(
1
n
n−1∑
k=0
〈ψk|ρk|ψk〉
)
, (4.21)
where
∫
dµj is the normalized integral over the Bloch sphere corresponding to
the state |ψj〉.
We show in section 4.4.1 that the two version of the QIC game are equivalent
and that their success probabilities satisfy the relation
p =
1
3
(1 + 2P ). (4.22)
For convenience, we only refer to version I from now on, unless otherwise stated.
Consider the following naive strategy to play the QIC game. Alice simply
sends Bob m of the n qubits received from Charlie without applying any opera-
tions on these. Alice and Bob previously agree on which qubits Alice would send
Bob, for example, those with index 0 ≤ j < m. If Bob receives from Charlie
a number k < m, he outputs the correct state; in this case, 〈Ψ−|ωk|Ψ−〉 = 1.
However, if k ≥ m, Bob does not have the correct state, hence, he can only give
Charlie a fixed state, say |0〉; in this case, 〈Ψ−|ωk|Ψ−〉 = 14 . Thus, this strategy
succeeds with probability
PN =
1
4
(
1 + 3
m
n
)
, (4.23)
where the label N stands for naive. This strategy saturates the quantum infor-
mation causality bound, as can be easily seen to achieve ∆I(C : B) = 2m. But,
it does not achieve the maximum success probability, because there are other
strategies that achieve success probabilities higher than PN (see section 4.5).
There is not an obvious relation between the quantum information causality
bound and the maximum success probability in the QIC game. However, quantum
information causality implies an upper bound on P . In particular, P < 1, if
m < n.
114
4.4. Upper Bound on the Success Probability in the QIC Game
4.4 Upper Bound on the Success Probability in
the QIC Game
Quantum information causality implies an upper bound on the success probability
in the QIC game:
P ≤ P ′, (4.24)
where P ′ is defined as the maximum solution of the equation
h(P ′) + (1− P ′) log2 3 = 2
(
1−
m
n
)
, (4.25)
and h(x) ≡ −x log2 x− (1− x) log2(1− x) denotes the binary entropy. The value
of P ′ is a strictly increasing function of the ratio mn . It achieves P
′ = 14 , if m = 0
and P ′ = 1, if m = n. It follows that P < 1, if m < n. Some values of P ′ are
plotted in Figure 4.6. The proof of the bound (4.24) requires several steps.
First, we show in section 4.4.1 that versions I and II of the QIC game are
equivalent and that their success probabilities satisfy the relation (4.22).
Second, using version II of the QIC game, we show in section 4.4.2 that for
any strategy that Alice and Bob may play that achieves success probability p,
there exists a covariant strategy achieving the same value of p that Alice and
Bob can perform. By covariance, we mean the following: if, when Alice’s input
qubit Ak is in the state |ψk〉, Bob’s output qubit state is ρk, then, when Ak is
in the state U |ψk〉, Bob’s output state is UρkU †, for any qubit state |ψk〉 ∈ C2
and unitary operation U ∈ SU(2). Recall that k is the number that Charlie gives
Bob. Therefore, without loss of generality, we consider that a covariant strategy
is implemented. This means that the Bloch sphere of the qubit Ak is contracted
uniformly and output in the qubit Bk. This implies that, in version II of the QIC
game, the joint system CkBk is transformed into the state
ωk = λkΨ
− +
1− λk
3
(
Ψ+ + Φ+ + Φ−
)
, (4.26)
where 14 ≤ λk ≤ 1 and Ψ
− denotes |Ψ−〉〈Ψ−|, etc. That is, the depolarizing
map [5] is applied to the qubit Ak and output by Bob in the qubit Bk.
Third, the data-processing inequality and the fact that the qubits Cj and Cj′
115
Chapter 4. Quantum Information Causality
are in a product state for every j 6= j′ are used in section 4.4.3 to show that
n−1∑
k=0
I(Ck : Bk) ≤ I(C : B
′). (4.27)
In the QIC game, Charlie’s and Bob’s systems are initially uncorrelated. Thus,
quantum information causality, Equation (4.10), reduces to I(C : B′) ≤ 2m in
this case. From this bound and (4.27), we obtain that
n−1∑
k=0
I(Ck : Bk) ≤ 2m. (4.28)
Finally, below we use (4.26) and (4.28) to obtain an upper bound on 1n
∑n−1
k=0 λk,
which equals P , as easily seen from (4.20) and (4.26).
We obtain from (4.26) that I(Ck : Bk) = 2 − S(ωk). Thus, from (4.28), we
have that
1
n
n−1∑
k=0
S(ωk) ≥ 2
(
1−
m
n
)
. (4.29)
Consider the state ω ≡ 1n
∑n−1
k=0 ωk. From the concavity property of the quantum
entropy [5], we obtain that S(ω) ≥ 1n
∑n−1
k=0 S(ωk), which together with Equa-
tion (4.29) implies
S(ω) ≥ 2
(
1−
m
n
)
. (4.30)
From the definition of P , Equation (4.20), and the form of the states ωk, Equa-
tion (4.26), we obtain that
P =
1
n
n−1∑
k=0
λk. (4.31)
From the definition of the state ω and Equations (4.26) and (4.31), we have that
ω = PΨ− +
1− P
3
(
Ψ+ + Φ+ + Φ−
)
. (4.32)
The quantum entropy of this state is S(ω) = h(P ) + (1−P ) log2 3, where h(x) ≡
116
4.4. Upper Bound on the Success Probability in the QIC Game
−x log2 x− (1−x) log2(1−x) is the binary entropy. Thus, from (4.30), we obtain
h(P ) + (1− P ) log2 3 ≥ 2
(
1−
m
n
)
. (4.33)
Satisfaction of (4.33) implies satisfaction of (4.24). This can be seen as follows.
The function h(P )+(1−P ) log2 3 corresponds to the classical entropy of a random
variable taking four values, one with probability P and the others with probability
1
3(1 − P ) [5]. It is a strictly increasing function of P in the range
[
0, 14
]
and a
strictly decreasing function in the range
[
1
4 , 1
]
. It takes the values log2 3 at P = 0
and P = 0.609, 2 at P = 14 , and 0 at P = 1. If 2
(
1− mn
)
≥ log2 3, Equation (4.25)
has two solutions, one in the range
[
0, 14
]
and the other one in the range
[
1
4 , 0.609
]
.
Otherwise, Equation (4.25) has a single solution in the range (0.609, 1]. Therefore,
the maximum solution of Equation (4.25) is in the range
[
1
4 , 1
]
. Since in this range
the function h(P ) + (1− P ) log2 3 is strictly decreasing, Equation (4.33) implies
Equation (4.24). In particular, it is easy to see from (4.30) that if m < n then
S(ω) > 0. Thus, in this case, ω cannot be a perfect singlet, which from (4.32)
implies that P < 1.
4.4.1 Equivalence of the Two Versions of the Game
Versions I and II of the QIC game are equivalent. This means that, if Alice and
Bob play a strategy in version I of the QIC game that achieves a success proba-
bility P , the same strategy applied to version II achieves a success probability p
that satisfies
p =
1
3
(1 + 2P ), (4.34)
for any strategy that they may play, and vice versa. We present the proof below.
For convenience, we use the notation |↑~rk〉 ≡ |ψk〉, |↓~rk〉 ≡ |ψ
⊥
k 〉, in order to
make clear that |↑~rk〉 and |↓~rk〉 correspond to pure qubit states with Bloch vectors
~rk and −~rk, respectively.
Version II of the QIC game is equivalent to the following. For every j ∈
{0, 1, . . . , n − 1}, Charlie prepares the pair of qubits Aj and Cj in the singlet
state |Ψ−〉, he chooses a vector ~rj completely randomly from the Bloch sphere,
and he measures the qubit Cj in the orthonormal basis {|↑~rj〉, |↓~rj〉}. Due to the
117
Chapter 4. Quantum Information Causality
properties of the singlet, if Cj projects into the state with Bloch vector ±~rj then
Aj projects into the state with Bloch vector ∓~rj. Charlie sends Alice the system
A ≡ A0A1 · · ·An−1 and keeps the system C ≡ C0C1 · · ·Cn−1. Charlie generates a
random integer k ∈ {0, 1, . . . , n − 1} that he gives Bob. Alice and Bob play the
QIC game. Bob outputs a qubit Bk that he gives Charlie. Charlie measures Bk in
the basis {|↑~rk〉, |↓~rk〉}. Alice and Bob win the game if Bk projects into the state
with Bloch vector antiparallel to that one of the state of Ck. Since Charlie’s initial
measurements on his system C commute with Alice’s and Bob’s operations, it is
equivalent to consider that Charlie does not apply any measurements on C before
sending A to Alice, that he waits until receiving the qubit Bk to measure the
joint system CkBk in the basis B~rk ≡ {|↑~rk〉|↑~rk〉, |↓~rk〉|↓~rk〉, |↑~rk〉|↓~rk〉, |↓~rk〉|↑~rk〉}.
Opposite outcomes correspond to success. Therefore, the success probability p
that Alice and Bob achieve in version II of the QIC game, defined by (4.21),
equals the following:
p =
∫
dµ0
∫
dµ1 · · ·
∫
dµn−1
[
1
n
n−1∑
k=0
(
〈↑~rk |〈↓~rk |ωk|↑~rk〉|↓~rk〉+ 〈↓~rk |〈↑~rk |ωk|↓~rk〉|↑~rk〉
)
]
,
(4.35)
where
∫
dµj is the normalized integral over the Bloch sphere corresponding to the
Bloch vector ~rj and we denote ωk to the state of the joint system CkBk.
The Bell states defined in the basis B~rk are
|Φ±~rk〉 ≡
1
√
2
(
|↑~rk〉|↑~rk〉 ± |↓~rk〉|↓~rk〉
)
,
|Ψ±~rk〉 ≡
1
√
2
(
|↑~rk〉|↓~rk〉 ± |↓~rk〉|↑~rk〉
)
. (4.36)
Consider that instead of measuring the state ωk in the basis B~rk , Charlie mea-
sures it in this Bell basis. Since the singlet state is the same in any basis, this
corresponds to version I of the QIC game. Therefore, versions I and II of the QIC
game are equivalent. We show below that their success probabilities satisfy the
claimed relation.
118
4.4. Upper Bound on the Success Probability in the QIC Game
Using the previous Bell basis, we obtain from (4.35) that
p =
∫
dµ0
∫
dµ1 · · ·
∫
dµn−1
[
1
n
n−1∑
k=0
(
〈Ψ−~rk |ωk|Ψ
−
~rk
〉+ 〈Ψ+~rk |ωk|Ψ
+
~rk
〉
)
]
. (4.37)
Since the singlet state |Ψ−~rk〉 is the same in any basis, from the definition of P ,
Equation (4.20), we have that
∫
dµ0
∫
dµ1 · · ·
∫
dµn−1
1
n
n−1∑
k=0
〈Ψ−~rk |ωk|Ψ
−
~rk
〉 = P. (4.38)
On the other hand, we obtain that
∫
dµ0
∫
dµ1 · · ·
∫
dµn−1〈Ψ
+
~rk
|ωk|Ψ
+
~rk
〉 =
∫
dµ0
∫
dµ1 · · ·
∫
dµn−1Tr
(
ωk|Ψ
+
~rk
〉〈Ψ+~rk |
)
= Tr
(∫
dµ0
∫
dµ1 · · ·
∫
dµn−1ωk|Ψ
+
~rk
〉〈Ψ+~rk |
)
= Tr
(
ωk
∫
dµ0
∫
dµ1 · · ·
∫
dµn−1|Ψ
+
~rk
〉〈Ψ+~rk |
)
= Tr
(
ωk
∫
dµk|Ψ
+
~rk
〉〈Ψ+~rk |
)
, (4.39)
where in the second line we used the linearity of the trace; in the third line we
used the fact that ωk does not depend on the Bloch vector ~rk because Charlie
chooses it completely randomly to define the measurement basis B~rk , and can do
so after Bob gives him the qubit Bk, and naturally does not depend on the Bloch
vectors ~rj with j 6= k for the same reason; and in the last line we used that the
state |Ψ+~rk〉 is defined in terms of the Bloch vector ~rk, which is parameterized by
µk, and so is independent of the parameters µj with j 6= k.
Using the expressions
|↑~r〉 = cos
(θ
2
)
|0〉+ eiφ sin
(θ
2
)
|1〉,
|↓~r〉 = sin
(θ
2
)
|0〉 − eiφ cos
(θ
2
)
|1〉, (4.40)
for ~r ≡ (sin θ cosφ, sin θ sinφ, cos θ), and computing the integral over the Bloch
119
Chapter 4. Quantum Information Causality
sphere,
∫
dµ = 14pi
∫ pi
0dθ sin θ
∫ 2pi
0 dφ, it is straightforward to obtain that
∫
dµk|Ψ
+
~rk
〉〈Ψ+~rk | =
1
3
(
I − |Ψ−〉〈Ψ−|
)
, (4.41)
where |Ψ−〉 ≡ 1√
2
(
|0〉|1〉 − |1〉|0〉
)
is the singlet state in the computational basis
and I is the identity operator acting on C4. From (4.39), (4.41) and the definition
of P , given by (4.20), we have that
1
n
n−1∑
k=0
∫
dµ0
∫
dµ1 · · ·
∫
dµn−1〈Ψ
+
~rk
|ωk|Ψ
+
~rk
〉 =
1
3
−
1
3
P. (4.42)
Finally, we substitute (4.38) and (4.42) into (4.37) to obtain the claimed
relation, Equation (4.34).
4.4.2 Reduction to a Covariant Strategy
We show that for any strategy to play the QIC game there exists a covariant
strategy that achieves the same success probability. For convenience, we consider
version II of the QIC game.
Recall version II of the QIC game. Charlie gives Alice n qubits in the state
~ψ ≡ ⊗n−1j=0
(
|ψj〉〈ψj|
)
Aj
∈ D
((
C2
)⊗n
)
, where we define D(H) to be the set of
density operators acting on the Hilbert space H. Charlie gives Bob the number
k. Let Γk : D
((
C2
)⊗n
)
→ D
(
C2
)
be the map that Alice and Bob apply to the
state ~ψ. Bob outputs the state ρk ≡ Γk
(~ψ
)
and gives it to Charlie. Taking the
average over all possible input pure product states of qubits with index j 6= k,
Bob’s output only depends on the state ψk ≡ |ψk〉〈ψk|, as follows:
Γ¯k(ψk) ≡
∫
dµ0
∫
dµ1· · ·
∫
dµk−1
∫
dµk+1
∫
dµk+2· · ·
∫
dµn−1Γk
(~ψ
)
, (4.43)
where
∫
dµj is the normalized integral over the Bloch sphere corresponding to
the state |ψj〉. From the definitions (4.21) and (4.43), we see that the success
probability is
p =
1
n
n−1∑
k=0
∫
dµk〈ψk|Γ¯k(ψk)|ψk〉. (4.44)
120
4.4. Upper Bound on the Success Probability in the QIC Game
We notice that the success probability only depends on the averaged output
Γ¯k(ψk). Thus, the same success probability is achieved if we consider that Bob’s
output state is Γ¯k(ψk). We show below that for any map Γk
(~ψ
)
that achieves
success probability p, there exits a covariant averaged map Γ¯covk (ψk) that Alice
and Bob can implement achieving the same success probability p.
Consider the following map:
Γ¯covk (φ) ≡
∫
dνU †ν Γ¯k
(
UνφU
†
ν
)
Uν , (4.45)
where φ ∈ D
(
C2
)
, Uν ∈ SU(2) and dν is the Haar measure on SU(2). It is
easy to see that this map is covariant, that is, Γ¯covk
(
UφU †
)
= U Γ¯covk (φ)U
†, for all
φ ∈ D
(
C2
)
and U ∈ SU(2). In principle, for any map Γk that Alice and Bob
perform, they can implement the covariant map Γ¯covk as follows. Alice and Bob
initially share randomness. With uniform probability, they obtain the random
number ν in the range dν that corresponds to an, ideally, infinitesimal region of
the Haar measure on SU(2). This can be done, for example, if Alice and Bob
share a maximally entangled state of arbitrarily big dimension and they both
apply a local projective measurement in the Schmidt basis on their part of the
state, with their measurement outcome indicating the number ν. Alice applies
the unitary operation Uν parameterized by the obtained number ν on each of
her input qubit states |ψj〉. Then, Alice and Bob apply the map Γk to the input
state ⊗n−1j=0
(
Uν |ψj〉〈ψj|U †ν
)
Aj
. Finally, Bob applies the unitary U †ν to his output
qubit. From the definitions (4.43) and (4.45), we see that, taking the average
over all shared random numbers ν and over all possible input pure qubit states
with index distinct to k, Bob’s output is Γ¯covk (ψk).
It is straightforward to see that the map Γ¯covk satisfies
∫
dµk〈ψk|Γ¯
cov
k (ψk)|ψk〉 =
∫
dµk〈ψk|Γ¯k(ψk)|ψk〉. (4.46)
From (4.44) and (4.46), we see that the map Γ¯covk (ψk) achieves the same success
probability as Γ¯k(ψk). Thus, for convenience, we consider that Alice and Bob
121
Chapter 4. Quantum Information Causality
implement the covariant map Γ¯covk (ψk). In general, this is the depolarizing map [5]:
Γ¯covk (φ) =
3∑
i=0
EiφE
†
i ,
where φ ∈ D
(
C2
)
, E0 =
√
λkI, Ei =
√
1
3(1− λk)σi,
1
4 ≤ λk ≤ 1 and σi are the
Pauli matrices, for i = 1, 2, 3. Application of the depolarizing map to a qubit
that is in the singlet state with another qubit, as in version I of the QIC game,
gives the output state
ωk = λkΨ
− +
1− λk
3
(
Ψ+ + Φ+ + Φ−
)
, (4.47)
where 14 ≤ λk ≤ 1 and Ψ
− denotes |Ψ−〉〈Ψ−|, etc., as in (4.26).
4.4.3 A Useful Bound
We show the following bound:
n−1∑
k=0
I (Ck : Bk) ≤ I (C : B
′) . (4.48)
The proof is equivalent to the one for classical bits [71].
From the definition of the quantum mutual information, we obtain
I (C : B′) ≡ I (C0C1 · · ·Cn−1 : B
′)
= I (C0 : B
′) + I (C1C2 · · ·Cn−1 : B
′C0)− I (C1C2 · · ·Cn−1 : C0) .
(4.49)
Since Charlie’s qubits are in a product state with each other, we have that
I (C1C2 · · ·Cn−1 : C0) = 0. (4.50)
The data-processing inequality implies that discarding a quantum system cannot
122
4.5. Strategies in the QIC Game
increase the quantum mutual information. Thus, it follows that
I (C1C2 · · ·Cn−1 : B
′C0) ≥ I (C1C2 · · ·Cn−1 : B
′) . (4.51)
From (4.49) – (4.51), we obtain that
I (C0C1 · · ·Cn−1 : B
′) ≥ I (C0 : B
′) + I (C1C2 · · ·Cn−1 : B
′) .
After iterating these steps n− 1 times, we have
I (C : B′) ≥
n−1∑
k=0
I (Ck : B
′) . (4.52)
Since the system Bk is output by Bob after local operations on his system B′,
applying the data-processing inequality, we obtain I(Ck : B′) ≥ I(Ck : Bk), which
from (4.52) implies the bound (4.48).
4.5 Strategies in the QIC Game
As mentioned in section 4.3, a simple strategy in the QIC game is the naive
strategy. For completeness of this section, we present it again.
The naive strategy in the QIC game. Alice sends Bob m of the n qubits
received from Charlie without applying any operations on these. Alice and Bob
previously agree on which qubits Alice would send Bob, for example, those with
index 0 ≤ j < m. If Bob receives from Charlie a number k < m, he outputs the
correct state; in this case, 〈Ψ−|ωk|Ψ−〉 = 1. However, if k ≥ m, Bob does not
have the correct state, hence, he can only give Charlie a fixed state, say |0〉; in
this case, 〈Ψ−|ωk|Ψ−〉 = 14 . Thus, this strategy succeeds with probability
PN =
1
4
(
1 + 3
m
n
)
, (4.53)
where the label N stands for naive.
The naive strategy saturates the quantum information causality bound, Equa-
tion (4.10), because it achieves I(C : B) = 2m. Bob obtains complete quantum
123
Chapter 4. Quantum Information Causality
information about m of the n qubit states prepared by Charlie, but he does not
know anything about the other ones. This strategy is extremely simple. It does
not require that Alice and Bob share any entanglement. There exists a more
powerful class of strategies, which requires entanglement and uses the protocol of
quantum teleportation.
4.5.1 Teleportation Strategies
The teleportation strategies aim to teleport the n input qubit states from Alice’s
to Bob’s location. This cannot be done perfectly due to the restricted amount
of communication. The quantum states of the n input qubits are immediately
accessible to Bob after Alice has applied the corresponding Bell measurements,
but with some teleportation errors. Alice and Bob perform a protocol, which
involves the transmission of m qubits from Alice to Bob, so that, with high
probability, Bob can correct the teleportation error of the qubit state asked by
Charlie. The correction stage involves a task of classical inputs and outputs, the
IC-2 game.
The IC-2 game is similar to the IC game, described in section 4.1.2, with the
difference that the inputs and outputs are two bit numbers (see Figure 4.4).
The IC-2 game. Alice is given a random string of n two bit numbers, ~x ≡
(x0, x1, . . . , xn−1), where xj ≡ (x0j , x
1
j) and x
0
j , x
1
j ∈ {0, 1}, for j = 0, 1, . . . , n− 1.
Bob is given a random value of k = 0, 1, . . . , n− 1. The game’s goal is that Bob
outputs xk. Alice and Bob can perform any strategy allowed by quantum physics
with the only condition that communication is limited to a single message of 2m
bits from Alice to Bob, with m < n. In particular, Alice and Bob may use an
arbitrary entangled state. Let yk ≡ (y0k, y
1
k) be Bob’s output, where y
0
k, y
1
k ∈ {0, 1}.
The success probability in the IC-2 game is defined as
Q ≡
1
n
n−1∑
k=0
P (yk = xk) . (4.54)
The teleportation strategies combine the protocols of quantum teleportation
[13], superdense coding [12] and the IC-2 game (see Figure 4.5).
124
4.5. Strategies in the QIC Game
Figure 4.4: The IC-2 game. Alice is given a random string of n two bit numbers,
x0, x1, . . . , xn−1. Bob is given a random number k = 0, 1, . . . , n− 1. Bob outputs
a two bit number yk. Alice and Bob win the game if yk = xk. Alice and Bob
can play any strategy allowed by quantum physics with the only condition that
communication is limited to a single message of 2m bits from Alice to Bob, with
m < n. In particular, Alice and Bob may use an arbitrary entangled state.
Teleportation strategies in the QIC game. Alice and Bob share a singlet
state in the qubits A′j, at Alice’s site, and Bj, at Bob’s site, for j = 0, 1, . . . , n−1.
Alice applies a Bell measurement on her qubits AjA′j and obtains the two bit
outcome xj ≡ (x0j , x
1
j). Thus, the state of the qubit Aj is teleported to Bob’s
qubit Bj, up to the Pauli error σxj . This means that the joint state of the
system CjBj transforms into one of the four Bell states, according to the value
of xj. Alice and Bob play the IC-2 game, with Alice’s and Bob’s inputs being
~x = (x0, x1, . . . , xn−1) and k, respectively. However, instead of sending Bob the
2m−bits message directly, Alice encodes it in m qubits via superdense coding.
Bob receives the m qubits and decodes the correct 2m-bits message, which he
inputs to his part of the IC-2 game. Bob outputs the two bit number yk ≡ (y0k, y
1
k)
and applies the Pauli correction operation σyk on the qubit Bk, which then he
outputs and gives to Charlie. If yk = xk, the output state ωk of the system CkBk
is the singlet; otherwise, we have that 〈Ψ−|ωk|Ψ−〉 = 0. Thus, from the definition
of P , Equation (4.20), we see that P = Q, where Q is defined by (4.54).
The best strategy that we have found to play the QIC game in the case
m = 1 is a teleportation strategy in which the IC-2 game is played with two
125
Chapter 4. Quantum Information Causality
Figure 4.5: Teleportation strategies in the QIC game. Alice and Bob share sin-
glets in their respective qubits A′0, A
′
1, . . . , A
′
n−1 and B0, B1, . . . , Bn−1. Alice ap-
plies a Bell measurement on her qubits AjA′j and obtains the two bit outcome xj,
for j = 0, 1, . . . , n− 1. Alice and Bob play the IC-2 game, with Alice’s and Bob’s
inputs being x0, x1, . . . , xn−1 and k, respectively. Alice encodes her 2m−bits mes-
sage in m qubits via superdense coding (SDC). Bob outputs the two bit number
yk and applies the encoded Pauli correction operation σyk on the qubit Bk, which
then he outputs and gives to Charlie. If yk = xk, the state of the qubit Ak is
teleported to the qubit Bk without error, which means that the system CkBk is
transformed into the singlet, and hence that Alice and Bob win the QIC game.
126
4.5. Strategies in the QIC Game
equivalent and independent protocols in the IC-1 game.1 In both protocols Bob
inputs the number k, while Alice inputs the bits {x0j}
n−1
j=0 in the first protocol
and the bits {x1j}
n−1
j=0 in the second one. If Bob outputs the correct value of
x0k with probability q in the first protocol, and similarly, he outputs the correct
value of x1k with probability q in the second protocol, for any k, then the success
probability in the IC-2 game is Q = q2. The maximum value of q that has been
shown [107, 108] is q = 12
(
1 + 1√n
)
. With this value of Q, Alice and Bob achieve
a success probability in the QIC game of
PT =
1
4
(
1 +
1
√
n
)2
, (4.55)
where the label T stands for teleportation. Some values of PT, PN and P ′, in the
case m = 1, are plotted in Figure 4.6.
Figure 4.6: Success probability (P ) in the QIC game for m = 1 achieved with
the naive strategy, PN (circles), and with the best teleportation strategy that
we have found, PT (triangles). The upper bound on P obtained from quantum
information causality, P ′ (squares), is plotted too.
1We denote the IC game described in section 4.1.2 as the IC-1 game in order to make clear
that the inputs and outputs are one bit numbers.
127
Chapter 4. Quantum Information Causality
An explicit strategy to achieve the success probability q = 12
(
1 + 1√n
)
in
the IC-1 game for m = 1 is given by an EARAC in the case n = 2h3l with
h, l nonnegative integers [107]. This protocol consists in a concatenation of two
primitive EARACs,
(
2, 1, 12
(
1 + 1√
2
))
and
(
3, 1, 12
(
1 + 1√
3
))
. Recall from section
4.1.2 that in an (n,m, p) EARAC, Alice has n input bits, she sends Bob m bits,
Bob outputs any, but only one, of Alice’s bits with probability at least p and
shared entanglement is used.
We review the
(
2, 1, 12
(
1 + 1√
2
))
EARAC presented in [107]. Alice and Bob
share a singlet state |Ψ−〉 = 1√
2
(
|0〉|1〉 − |1〉|0〉
)
. Let Alice’s input bits be x0 and
x1. Alice and Bob measure their qubits in the orthonormal bases {|exr 〉}
1
r=0 and
{|fks 〉}
1
s=0, respectively. Alice measures in the basis with label x = x0⊕x1 ∈ {0, 1},
where ⊕ denotes sum modulo 2. Bob measures in the basis with label k ∈ {0, 1}
in order to learn the bit xk. Alice and Bob obtain the bit outcomes r and s,
respectively. Alice sends Bob the one bit message τ = x0 ⊕ r. Bob outputs the
bit yk = τ ⊕ s. It is easy to see that yk = xk with probability P (r ⊕ s = xk).
Recall from section 4.1.2 that P (r⊕ s = xk) is the probability to win the CHSH
game, which satisfies Cirel’son’s bound, Equation (4.9). Thus, its maximum value
is P (r ⊕ s = xk) = 12
(
1 + 1√
2
)
. It is achieved with the following basis states:
|exr 〉 =
1
√
2
(
|0〉+ eiγ
x
r pi|1〉
)
,
|fks 〉 =
1
√
2
(
|0〉+ eiδ
k
spi|1〉
)
,
where γ00 =
1
4 , γ
0
1 = −
3
4 , γ
1
0 = −
1
4 , γ
1
1 =
3
4 , δ
0
0 = 1, δ
0
1 = 0, δ
1
0 = −
1
2 , δ
1
1 =
1
2 .
4.5.2 An Optimal Strategy
An optimal strategy in the QIC game is a teleportation strategy in which the
IC-2 game is played optimally. For a fixed value of m and n such that m < n,
we define Qmax and Pmax to be the maximum values of Q and P over all possible
strategies to play the IC-2 game and the QIC game, respectively. We show that
Pmax = Qmax. (4.56)
128
4.5. Strategies in the QIC Game
It was shown above that a teleportation strategy achieves a success probability
P = Q, where Q is the success probability in the IC-2 game played within the
teleportation strategy. Thus, a teleportation strategy in which the IC-2 game
is played optimally achieves a success probability P = Qmax. Therefore, Equa-
tion (4.56) follows if
P ≤ Qmax, (4.57)
for a general strategy in the QIC game. To show (4.57), we consider the following
class of strategies to play the IC-2 game (see Figure 4.7).
Superdense coding strategies in the IC-2 game. Alice and Bob initially
share a singlet state in the qubits Aj and Cj, for j = 0, 1, . . . , n − 1. Alice has
the system A ≡ A0A1 · · ·An−1. Bob has the system C ≡ C0C1 · · ·Cn−1. Alice
and Bob share an arbitrary entangled state in the system A′, at Alice’s location,
and B at Bob’s location. Alice applies the unitary operation σxj on the qubit Aj,
for j = 0, 1, . . . , n − 1, where σ0,0 ≡ I is the identity operator acting on C2 and
σ0,1 ≡ σ1, σ1,0 ≡ σ2, σ1,1 ≡ σ3 are the Pauli matrices. Then, Alice and Bob play
the QIC game. Alice applies operations on the input system A and her ancilla
A′, and obtains a system T of m qubits. But, instead of sending T directly to
Bob, Alice teleports [13] its state, using more entanglement shared with Bob.
Thus, communication consists of 2m bits only, as required by the IC-2 game.
Bob applies operations on the teleported state and his system B to obtain the
output qubit Bk. At this stage, for consistency with the QIC game, Bob does not
apply any operations on the system C. As previously indicated, we can consider
that in a general strategy in the QIC game, the depolarizing map is applied on
the qubit Ak and output by Bob in the qubit Bk. Since this map commutes
with the operation σxk applied by Alice on Ak, Bob outputs Bk in the following
joint state with the qubit Ck: Ωk = (σxk ⊗ I)ωk(σxk ⊗ I). From the form of ωk,
Equation (4.47), we have that
Ωk = σxk ⊗ I
[
λkΨ
− +
1− λk
3
(
Ψ+ + Φ+ + Φ−
)]
σxk ⊗ I. (4.58)
Bob applies a Bell measurement on the state Ωk. The outcome yk ∈ {0, 1}2
indicates that Ωk projects into the Bell state σyk ⊗ I|Ψ
−〉BkCk . From (4.58), the
129
Chapter 4. Quantum Information Causality
probability that yk equals the value xk encoded by Alice is P (yk = xk) = λk.
Thus, from (4.54), we have that Q = 1n
∑n−1
k=0 λk. It follows from (4.31) that
Q = P . That is, the success probability Q achieved by this class of strategies in
the IC-2 game equals the success probability P in the QIC game played within.
Since the QIC game is played arbitrarily and Q ≤ Qmax, we obtain (4.57).
Figure 4.7: Superdense coding strategies in the IC-2 game. Alice and Bob share
singlets in the qubits A0, A1, . . . , An−1 and C0, C1, . . . , Cn−1. Alice encodes her
two bit inputs xk by applying the Pauli operations σxk on her qubits Ak. Then,
Alice and Bob play the QIC game using their ancillary system A′B. Alice’s
m−qubits message is sent to Bob via quantum teleportation. Bob outputs a
qubit Bk in the QIC game. Then, Bob applies a Bell measurement (BM) on
CkBk, and obtains the two bit outcome yk. The probability to win the IC-2 game
equals the probability to win the QIC game played within.
The value ofQmax and a strategy that achieves it remain as open problems. We
have obtained an upper bound on Q for a particular class of strategies, nonlocal
strategies, in the case m = 1.
130
4.5. Strategies in the QIC Game
4.5.3 Nonlocal Strategies
We consider teleportation strategies in the case m = 1 for which the IC-2 game
is played as follows.
Nonlocal strategies in the IC-2 game. Alice and Bob share an entangled
state |ψ〉 ∈ HA ⊗ HB. Alice has the system A and Bob has the system B.
Alice and Bob measure their systems in the orthonormal bases {|ν~xr,s〉}
1
r,s=0 and
{|wkt,u〉}
1
t,u=0, respectively. They choose their measurements according to the value
of ~x = (x0, x1, . . . , xn−1) and k. Recall that xj ≡ (x0j , x
1
j) is a two bit number,
for j = 0, 1, . . . , n − 1. Alice’s and Bob’s measurement outcomes are the two
bit numbers (a0k, a
1
k) and (b
0
k, b
1
k), respectively. That is, after the measurement
is completed, |ψ〉 projects into the state |ν~xa0k,a1k
〉|wkb0k,b1k
〉. Alice sends Bob her
outcome. Bob outputs the two bit value yk ≡ (y0k, y
1
k), where y
j
k = a
j
k ⊕ b
j
k, for
j = 0, 1, and ⊕ denotes sum modulo 2. The success probability is
Q =
1
n
n−1∑
k=0
P
(
y0k = x
0
k, y
1
k = x
1
k
)
. (4.59)
There is an upper bound on Q for this class of strategies:
Q ≤ Q′, (4.60)
where Q′ ≡ 14
(
1 + 3√n
)
. This bound does not imply that Qmax ≤ Q′, because
there can be strategies more general than the nonlocal strategies, for example,
those in which Bob uses Alice’s message in order to choose his measurement.
Moreover, quantum information causality implies that the previous bound cannot
be achieved for n ≥ 50. It can easily be computed that if m = 1 and n ≥ 50,
P ′ < Q′, where P ′ is defined by (4.25). Therefore, if the bound (4.60) were
saturated, a teleportation strategy achieving P = Q′ would satisfy P ′ < P ,
violating the inequality (4.24), and hence quantum information causality.
The proof of (4.60) is presented in Appendix D. It is an extension of the one
given in [108] for the IC-1 game.
131
Chapter 4. Quantum Information Causality
4.6 Discussion
In this chapter, we have introduced the principle of quantum information causal-
ity. Quantum information causality, Equation (4.10), states the maximum amount
of quantum information that a transmitted quantum system can communicate as
a function of its dimension, independently of any quantum physical resources
previously shared by the communicating parties. Its proof follows from three
properties of the quantum entropy: subadditivity, the data-processing inequal-
ity and the triangle inequality. The triangle inequality provides the main dif-
ference compared to the proof of information causality. Information causality,
Equation (4.19), considers the particular case in which the transmitted system
is classical. If the transmitted system is classical, the triangle inequality, Equa-
tion (4.13), in the proof of quantum information causality cannot be saturated.
The concept of entropy in mathematical frameworks for general probabilistic
theories [33, 115, 116] and its implication for information causality have been re-
cently investigated [108,117–119]. Particularly, it has been shown that a physical
condition on the measure of entropy implies subadditivity and the data-processing
inequality, and hence that information causality follows from this condition [108].
It would be interesting to investigate whether physically-sensible definitions of
entropy for more general probabilistic theories satisfy subadditivity, the data-
processing and the triangle inequalities, and hence a generalized version of quan-
tum information causality. A different version of information causality in more
general probabilistic theories has been considered in [120].
We have presented a new quantum information task, the QIC game. We have
shown an upper bound, Equation (4.24), on the success probability P in the QIC
game from quantum information causality. The bound implies, in particular,
that P < 1, if m < n. This means that Bob is unable to perfectly reproduce
the kth state from a set of n unknown qubit states at Alice’s location, if Alice
does not know the number k and Bob only receives a message of less than n
qubits from Alice, independently of any quantum physical resources previously
shared by them. We have presented two versions of the QIC game, which we have
shown are equivalent and whose success probabilities satisfy an equality relation,
Equation (4.34). In version I, Charlie prepares singlets; in version II, he prepares
132
4.6. Discussion
pure qubit states, totally randomly. It would be interesting to investigate more
general versions of the game, for example, versions in which the input states
have higher dimension or in which they are not generated completely randomly.
It would also be interesting to investigate possible extensions to multipartite
scenarios.
We have presented a class of strategies in the QIC game, teleportation strate-
gies, that combine the protocols of quantum teleportation, superdense coding
and a task with classical inputs and outputs, the IC-2 game. The IC-2 game is
intimately related to the QIC game, as suggested by (4.56). Moreover, as easily
seen from Figures 4.5 and 4.7, the teleportation strategies in the QIC game and
the superdense coding strategies in the IC-2 game are related in a way that re-
sembles the relation between the quantum teleportation and superdense coding
protocols. We have shown that an optimal strategy in the QIC game is a telepor-
tation strategy in which the IC-2 game is played optimally. An optimal strategy
in the IC-2 game remains as an interesting open problem.
133

Chapter 5
Conclusions
Quantum theory predicts the existence of correlations that violate the Bell in-
equalities, and hence that cannot be described by local hidden variable theories
(LHVT) [2]. However, these nonlocal quantum correlations cannot be used for
instantaneous communication because of the no-signalling principle [69]. The
no-signalling principle states that two distant parties cannot communicate any
information without the transmission of any physical systems, despite any quan-
tum physical resources shared by them. This is a fundamental physical principle
that allows quantum theory to be consistent with relativistic causality. The no-
signalling principle imposes important constraints on quantum information tasks.
Our PhD research focused on the investigation of Bell inequalities from a new
perspective [94], the implications of the no-signalling principle for quantum in-
formation tasks [99], and extensions of the no-signalling principle [104].
The problems that we have discussed along this thesis can be presented in a
unifying scenario. Consider a general information task performed by two distant
parties, Alice and Bob. This task can be extended to include multiple parties,
but we restrict here to consider the bipartite case. The information properties
of different physical theories, quantum or non-quantum, can be investigated in
this scenario. The goal of the considered task is that Bob outputs some spe-
cific information originally at Alice’s location. Alice and Bob can perform any
strategy allowed by the theory that is being analyzed, for which they can use
physical resources of certain type, and are required to satisfy specific constraints.
The information task is defined by the type of information originally at Alice’s
135
Chapter 5. Conclusions
location, the type of information that Bob has to output, the physical resources
that Alice and Bob have access to, the constraints that they need to satisfy, and
the physical theory ruling the task.
Physical theories satisfying the principle of local causality are investigated by
tasks in which Alice and Bob cannot transmit any physical systems to each other,
the resources they have access to are described by LHVT, and the task goal is
that, after they input respective random numbers A ∈ A and B ∈ B to their
resources, Alice and Bob output numbers a and b that are correlated in a specific
way defined by the task. Bounds on the probability to succeed in these tasks
define Bell inequalities, which characterize the information limitations of LHVT.
Bell inequalities in which A and B are finite sets have been investigated in
great detail. Important Bell inequalities are the CHSH [42] and the Braunstein-
Caves [45] inequalities, which consider the cases |A| = |B| = 2 and |A| = |B| = N ,
respectively, for which a and b take one of two possible values. Our contribution
in chapter 2 has been to introduce a setting in which A and B are chosen from
a continuous set, thus generalizing the settings considered by the CHSH and the
Braunstein-Caves inequalities. We have considered the particular case in which
this set is a pair of unit spheres, which follows from the constraint studied by us in
which a and b can have only two possible values. In the case we have considered,
A and B are chosen randomly, but are fixed to satisfy a given separation angle
θ in the spheres. We obtained Bell inequalities for all values of θ ∈ [0, pi2 ], given
by Theorem 2.1. These inequalities distinguish all LHV correlations from the
singlet state quantum correlations for θ ∈
(
0, pi3
)
, as stated by Lemma 2.4. We
have motivated and explored, numerically and analytically, hypotheses implying
that our obtained Bell inequalities are not optimal for all range of θ. The strong
hemispherical colouring maximality hypothesis (SHCMH) states that an LHVT in
which the possible outcomes a correspond to a sphere with opposite hemispheres
coloured oppositely and the possible outcomes b correspond to a sphere with
the reverse colouring, in which different colours determine different measurement
outcomes, maximizes the LHV anticorrelations for a continuous range of θ > 0.
The weak hemispherical colouring maximality hypothesis (WHCMH) states that
such a colouring maximizes the LHV anticorrelations for a continuous range of
θ > 0 among restricted colourings in which the pair of spheres are coloured
136
oppositely. The investigation of different geometries for continuous sets A and
B, corresponding to bigger numbers of possible values that a and b can take, will
lead to Bell inequalities generalizing the ones we have investigated.
Different quantum information tasks can be investigated in the general setting
introduced above. An important class of quantum tasks corresponds to quantum
teleportation protocols [13]. In quantum teleportation, Bob needs to output an
unknown quantum state originally at Alice’s location, Alice and Bob have access
to an arbitrary quantum state that they share, and they can perform arbitrary
communication, as long as this is classical.
Port-based teleportation (PBT) protocols [100, 101] were considered in chap-
ter 3. These protocols have the particular characteristic that, in order to obtain
the teleported state, the only operation that Bob needs to perform after receiv-
ing Alice’s message consists in selecting the particular port informed by Alice.
We analyzed PBT protocols in a general setting in which the teleported state is
an unknown n−qubits state, the resources used by Alice and Bob consist of an
arbitrary quantum state |ξ〉AB, and Alice’s operations correspond to a general
quantum measurement. In the PBT protocols we considered, the state is tele-
ported perfectly, which requires the protocol to fail with some finite probability.
Our contribution is a proof of the upper bound (3.1) on the success probability
of PBT from the no-signalling principle and a version of the no-cloning theo-
rem, Theorem 3.1, introduced by us. It is an interesting open problem to find
whether the obtained bound is achievable. It would also be interesting to inves-
tigate whether the no-signalling principle implies an upper bound on the average
fidelity of deterministic PBT protocols, in which the protocol always succeeds
but the state is teleported with a fidelity smaller than unity.
Physical theories restricted by the no-signalling principle are investigated by
tasks in which Alice’s and Bob’s resources are constrained by this principle, Alice
and Bob cannot transmit any physical systems to each other, and the task goal
is that Bob outputs some specific information previously unknown to him that is
at Alice’s location. Theories satisfying particular extensions of the no-signalling
principle can be investigated if Alice and Bob can transmit a finite amount of
physical systems restricted to be of a given dimension. The information causality
principle [71] considers the scenario in which the information originally at Alice’s
137
Chapter 5. Conclusions
location that Bob needs to output and the transmitted physical system are classi-
cal. The quantum information causality principle, introduced by us in chapter 4,
considers the scenario in which the information at Alice’s location that Bob needs
to output and the transmitted system are quantum.
Quantum information causality states the maximum amount of quantum in-
formation that a transmitted quantum system can communicate as a function of
its dimension, independently of any quantum physical resources previously shared
by the communicating parties, Equation (4.10). We have found that an important
application of quantum information causality is that it imposes an upper bound,
Equation (4.24), on the success probability in the QIC game, a new quantum in-
formation task introduced in section 4.3. This bound implies, in particular, that
Bob is unable to perfectly reproduce the kth state from a set of n unknown qubit
states at Alice’s location, if Alice does not know the number k and Bob only re-
ceives a message of less than n qubits from Alice, independently of any quantum
physical resources previously shared by them. It was shown in section 4.5.2 that
the maximum success probability in the QIC game is achieved by a strategy that
combines quantum teleportation, superdense coding and the IC-2 game, a task
with classical inputs, in which the IC-2 game is played optimally. An optimal
strategy in the IC-2 game remains as an interesting open problem.
The principles of information causality and quantum information causality
can be generalized even further by considering that the inputs by Alice, the
outputs by Bob, and the transmitted systems are arbitrary information systems,
described by more general probabilistic theories, that generalize the concept of a
quantum state. An example of such a more general principle has been considered
in [120].
138
Appendix A
Details of Numerical Work
In this appendix, we present some details of the numerical work corresponding to
section 2.4. We present the expressions for the correlations Cx(θ) corresponding
to the colourings x = 2, 3, 4, 2∆, 3δ shown in Figure 2.4 and defined in (2.17).
We computed these correlations numerically using a computer program coded in
Mathematica. This code is given and described in Appendix B. The results are
plotted in Figures 2.5, 2.6 and 2.7. We also show Equation (2.18).
We use the azimuthal symmetry of the colourings x = 2, 3, 4, 2∆, 3δ defined
in (2.17), the antipodal property (2.7) and the constraint (2.14) to reduce the
correlation given by (2.8) to:
Cx(θ) = −
1
pi
∫ pi
2
0
d sin ax()
∫ pi
0
dωax[α(θ, , ω)], (A.1)
where α(θ, , ω) is given by (2.10). The integrals with respect to ω are computed
analytically in the previous expression. We define the function
χ(θ, a, b, α) ≡
2
pi
∫ b
a
d sin  arccos
(cos θ cos − cosα
sin θ sin 
)
, (A.2)
where a, b, α ∈ [0, pi] and θ ∈
[
0, pi2
]
. The obtained expressions for the correlations
Cx(θ) include terms of the form (A.2). These expressions are:
139
Appendix A. Details of Numerical Work
C2(θ) =
{
h12(θ) if θ ∈ [0, pi/4],
h22(θ) if θ ∈ (pi/4, pi/2],
C3(θ) =



h13(θ) if θ ∈ [0, pi/6],
h23(θ) if θ ∈ (pi/6, pi/4],
h33(θ) if θ ∈ (pi/4, pi/3],
h43(θ) if θ ∈ (pi/3, pi/2],
C4(θ) =



h14(θ) if θ ∈ [0, pi/8],
h24(θ) if θ ∈ (pi/8, pi/4],
h34(θ) if θ ∈ (pi/4, 3pi/8],
h44(θ) if θ ∈ (3pi/8, pi/2],
C2∆(θ) =
{
R1∆(θ) if θ ∈
[
pi
4 + ∆,
pi
2 − 2∆
]
,
R2∆(θ) if θ ∈
[
pi
2 − 2∆,
pi
2
]
,
C3δ(θ) =



r1δ(θ) if δ ∈
[
− pi18 , 0
]
and θ ∈
[
pi
3 ,
pi
3 − δ
]
,
r2δ(θ) if δ ∈
[
− pi18 , 0
]
and θ ∈
(
pi
3 − δ,
pi
2 + δ
]
,
r3δ(θ) if δ ∈
[
− pi18 , 0
]
and θ ∈
(
pi
2 + δ,
pi
2
]
,
r4δ(θ) if δ ∈
(
0, pi24
]
and θ ∈
[
pi
3 ,
pi
3 + 2δ
]
,
r2δ(θ) if δ ∈
(
0, pi24
]
and θ ∈
(
pi
3 + 2δ,
pi
2 − δ
]
,
r5δ(θ) if δ ∈
(
0, pi24
]
and θ ∈
(
pi
2 − δ,
pi
2
]
,
(A.3)
for ∆ ∈
[
0, pi12
]
, and where
h12(θ)≡ −1 + 2
[
cos
(pi
4
)
−cos
(pi
4
+θ
)]
+χ
(
θ,
pi
4
−θ,
pi
4
,
pi
4
)
−χ
(
θ,
pi
4
,
pi
4
+θ,
pi
4
)
+χ
(
θ,
pi
2
−θ,
pi
2
,
pi
2
)
,
h22(θ)≡ 1 + 2
[
cos
(pi
4
)
−cos
(
θ−
pi
4
)]
+χ
(
θ,θ−
pi
4
,
pi
4
,
pi
4
)
−χ
(
θ,
pi
2
−θ,
pi
4
,
pi
2
)
+χ
(
θ,
pi
4
,
pi
2
,
pi
2
)
−χ
(
θ,
pi
4
,
pi
2
,
pi
4
)
−χ
(
θ,
3pi
4
−θ,
pi
2
,
3pi
4
)
;
140
h13(θ)≡ −1 + 2
[
cos
(pi
6
)
−cos
(pi
6
+θ
)
+cos
(pi
3
)
−cos
(pi
3
+θ
)]
+χ
(
θ,
pi
6
−θ,
pi
6
,
pi
6
)
−χ
(
θ,
pi
6
,
pi
6
+θ,
pi
6
)
+χ
(
θ,
pi
3
−θ,
pi
3
,
pi
3
)
−χ
(
θ,
pi
3
,
pi
3
+θ,
pi
3
)
+χ
(
θ,
pi
2
−θ,
pi
2
,
pi
2
)
,
h23(θ)≡ 1 + 2
[
cos
(pi
6
)
−cos
(
θ−
pi
6
)
+cos
(pi
6
+θ
)
−cos
(pi
3
)]
+χ
(
θ,θ−
pi
6
,
pi
6
,
pi
6
)
−χ
(
θ,
pi
3
−θ,
pi
6
,
pi
3
)
+χ
(
θ,
pi
6
,
pi
2
−θ,
pi
3
)
−χ
(
θ,
pi
6
,
pi
3
,
pi
6
)
+χ
(
θ,
pi
2
−θ,
pi
3
,
pi
3
)
−χ
(
θ,
pi
2
−θ,
pi
3
,
pi
2
)
+ χ
(
θ,
pi
3
,
pi
6
+θ,
pi
6
)
+χ
(
θ,
pi
3
,
pi
2
,
pi
2
)
−χ
(
θ,
pi
3
,
pi
2
,
pi
3
)
−χ
(
θ,
2pi
3
−θ,
pi
2
,
2pi
3
)
,
h33(θ)≡ 1 + 2
[
cos
(pi
6
)
−cos
(
θ−
pi
6
)
+cos
(pi
6
+θ
)
−cos
(pi
3
)]
−χ
(
θ,
pi
3
−θ,
pi
6
,
pi
3
)
+χ
(
θ,θ−
pi
6
,
pi
6
,
pi
6
)
+χ
(
θ,
pi
6
,
pi
3
,
pi
3
)
−χ
(
θ,
pi
6
,
pi
3
,
pi
6
)
−χ
(
θ,
pi
2
−θ,
pi
3
,
pi
2
)
+χ
(
θ,
pi
3
,
pi
2
,
pi
2
)
−χ
(
θ,
pi
3
,
pi
2
,
pi
3
)
+χ
(
θ,
pi
3
,
pi
6
+θ,
pi
6
)
−χ
(
θ,
2pi
3
−θ,
pi
2
,
2pi
3
)
,
h43(θ)≡ −1 + 2
[
cos
(
θ−
pi
3
)
−cos
(pi
6
)
+cos
(
θ−
pi
6
)
−cos
(pi
3
)]
−χ
(
θ,θ−
pi
3
,
pi
6
,
pi
3
)
+χ
(
θ,
pi
2
−θ,
pi
6
,
pi
2
)
+χ
(
θ,
pi
6
,
pi
3
,
pi
3
)
−χ
(
θ,
pi
6
,
pi
3
,
pi
2
)
−χ
(
θ,θ−
pi
6
,
pi
3
,
pi
6
)
+χ
(
θ,
2pi
3
−θ,
pi
3
,
2pi
3
)
−χ
(
θ,
pi
3
,
pi
2
,
2pi
3
)
+χ
(
θ,
pi
3
,
pi
2
,
pi
2
)
−χ
(
θ,
pi
3
,
pi
2
,
pi
3
)
+χ
(
θ,
pi
3
,
pi
2
,
pi
6
)
+χ
(
θ,
5pi
6
−θ,
pi
2
,
5pi
6
)
;
141
Appendix A. Details of Numerical Work
h14(θ)≡−1+2
[
cos
(pi
8
)
−cos
(pi
8
+θ
)
+cos
(pi
4
)
−cos
(pi
4
+θ
)
+cos
(3pi
8
)
−cos
(3pi
8
+θ
)]
+χ
(
θ,
pi
8
−θ,
pi
8
,
pi
8
)
−χ
(
θ,
pi
8
,
pi
8
+θ,
pi
8
)
+χ
(
θ,
pi
4
−θ,
pi
4
,
pi
4
)
−χ
(
θ,
pi
4
,
pi
4
+θ,
pi
4
)
+χ
(
θ,
3pi
8
−θ,
3pi
8
,
3pi
8
)
−χ
(
θ,
3pi
8
,
3pi
8
+θ,
3pi
8
)
+χ
(
θ,
pi
2
−θ,
pi
2
,
pi
2
)
,
h24(θ)≡ 1+2
[
cos
(pi
8
)
−cos
(
θ−
pi
8
)
+cos
(
θ+
pi
8
)
−cos
(pi
4
)
+cos
(
θ+
pi
4
)
−cos
(3pi
8
)]
+χ
(
θ,θ−
pi
8
,
pi
8
,
pi
8
)
−χ
(
θ,
pi
4
−θ,
pi
8
,
pi
4
)
+χ
(
θ,
pi
8
,
pi
4
,
pi
4
)
−χ
(
θ,
pi
8
,
pi
4
,
pi
8
)
−χ
(
θ,
3pi
8
−θ,
pi
4
,
3pi
8
)
+χ
(
θ,
pi
4
,
pi
8
+θ,
pi
8
)
+χ
(
θ,
pi
4
,
3pi
8
,
3pi
8
)
−χ
(
θ,
pi
4
,
3pi
8
,
pi
4
)
−χ
(
θ,
pi
2
−θ,
3pi
8
,
pi
2
)
+χ
(
θ,
3pi
8
,
pi
4
+θ,
pi
4
)
+χ
(
θ,
3pi
8
,
pi
2
,
pi
2
)
−χ
(
θ,
3pi
8
,
pi
2
,
3pi
8
)
−χ
(
θ,
5pi
8
−θ,
pi
2
,
5pi
8
)
,
h34(θ)≡−1+2
[
cos
(
θ−
pi
4
)
−cos
(pi
8
)
+cos
(
θ−
pi
8
)
−cos
(pi
4
)
+cos
(3pi
8
)
−cos
(
θ+
pi
8
)]
−χ
(
θ,θ−
pi
4
,
pi
8
,
pi
4
)
+χ
(
θ,
3pi
8
−θ,
pi
8
,
3pi
8
)
−χ
(
θ,
pi
8
,
pi
4
,
3pi
8
)
+χ
(
θ,
pi
8
,
pi
4
,
pi
4
)
−χ
(
θ,θ−
pi
8
,
pi
4
,
pi
8
)
+χ
(
θ,
pi
2
−θ,
pi
4
,
pi
2
)
−χ
(
θ,
pi
4
,
3pi
8
,
pi
2
)
+χ
(
θ,
pi
4
,
3pi
8
,
3pi
8
)
−χ
(
θ,
pi
4
,
3pi
8
,
pi
4
)
+χ
(
θ,
pi
4
,
3pi
8
,
pi
8
)
+χ
(
θ,
5pi
8
−θ,
3pi
8
,
5pi
8
)
−χ
(
θ,
3pi
8
,
pi
2
,
5pi
8
)
+χ
(
θ,
3pi
8
,
pi
2
,
pi
2
)
−χ
(
θ,
3pi
8
,
pi
2
,
3pi
8
)
+χ
(
θ,
3pi
8
,
pi
2
,
pi
4
)
−χ
(
θ,
3pi
8
,
pi
8
+θ,
pi
8
)
+χ
(
θ,
3pi
4
−θ,
pi
2
,
3pi
4
)
,
h44(θ)≡ 1+2
[
cos
(pi
8
)
−cos
(
θ−
3pi
8
)
+cos
(pi
4
)
−cos
(
θ−
pi
4
)
+cos
(3pi
8
)
−cos
(
θ−
pi
8
)]
+χ
(
θ,θ−
3pi
8
,
pi
8
,
3pi
8
)
−χ
(
θ,
pi
2
−θ,
pi
8
,
pi
2
)
+χ
(
θ,
pi
8
,
pi
4
,
pi
2
)
−χ
(
θ,
pi
8
,
pi
4
,
3pi
8
)
+χ
(
θ,θ−
pi
4
,
pi
4
,
pi
4
)
−χ
(
θ,
5pi
8
−θ,
pi
4
,
5pi
8
)
+χ
(
θ,
pi
4
,
3pi
8
,
5pi
8
)
−χ
(
θ,
pi
4
,
3pi
8
,
pi
2
)
+χ
(
θ,
pi
4
,
3pi
8
,
3pi
8
)
−χ
(
θ,
pi
4
,
3pi
8
,
pi
4
)
+χ
(
θ,θ−
pi
8
,
3pi
8
,
pi
8
)
−χ
(
θ,
3pi
4
−θ,
3pi
8
,
3pi
4
)
+χ
(
θ,
3pi
8
,
pi
2
,
3pi
4
)
−χ
(
θ,
3pi
8
,
pi
2
,
5pi
8
)
+χ
(
θ,
3pi
8
,
pi
2
,
pi
2
)
−χ
(
θ,
3pi
8
,
pi
2
,
3pi
8
)
+χ
(
θ,
3pi
8
,
pi
2
,
pi
4
)
−χ
(
θ,
3pi
8
,
pi
2
,
pi
8
)
−χ
(
θ,
7pi
8
−θ,
pi
2
,
7pi
8
)
;
142
R1∆(θ)≡ 1 + 2
[
− cos
(
−
pi
4
+ ∆ + θ
)
+cos
(pi
4
−∆
)]
+χ
(
θ,−
pi
4
+∆+θ,
pi
4
−∆,
pi
4
−∆
)
−χ
(
θ,
pi
2
−θ,
pi
4
−∆,
pi
2
)
+χ
(
θ,
pi
4
−∆,
pi
2
,
pi
2
)
−χ
(
θ,
pi
4
−∆,
pi
2
,
pi
4
−∆
)
−χ
(
θ,
3pi
4
+∆−θ,
pi
2
,
3pi
4
+∆
)
,
R2∆(θ)≡ 1 + 2
[
−cos
(pi
4
−∆
)
+ cos
(
θ −
pi
4
+ ∆
)]
−χ
(
θ,
pi
2
−θ,
pi
4
−∆,
pi
2
)
+χ
(
θ,
pi
4
−∆,
pi
2
,
pi
2
)
−χ
(
θ, θ−
pi
4
+∆,
pi
2
,
pi
4
−∆
)
−χ
(
θ,
3pi
4
+∆−θ,
pi
2
,
3pi
4
+∆
)
;
r1δ(θ)≡ −1 + 2
[
cos
(
θ−
pi
3
)
−cos
(pi
6
+δ
)
+cos
(
θ−
pi
6
−δ
)
−cos
(pi
3
)
+cos
(
θ+
pi
6
+δ
)]
−χ
(
θ,θ−
pi
3
,
pi
6
+δ,
pi
3
)
+χ
(
θ,
pi
6
+δ,
pi
3
,
pi
3
)
−χ
(
θ,
pi
2
−θ,
pi
3
,
pi
2
)
−χ
(
θ,θ−
pi
6
−δ,
pi
3
,
pi
6
+δ
)
+χ
(
θ,
2pi
3
−θ,
pi
3
,
2pi
3
)
−χ
(
θ,
pi
3
,
pi
2
,
2pi
3
)
+χ
(
θ,
pi
3
,
pi
2
,
pi
2
)
−χ
(
θ,
pi
3
,
pi
2
,
pi
3
)
+χ
(
θ,
pi
3
,θ+
pi
6
+δ,
pi
6
+δ
)
,
r2δ(θ)≡ −1 + 2
[
cos
(
θ−
pi
3
)
−cos
(pi
6
+δ
)
+cos
(
θ−
pi
6
−δ
)
−cos
(pi
3
)]
−χ
(
θ,θ−
pi
3
,
pi
6
+δ,
pi
3
)
+χ
(
θ,
pi
2
−θ,
pi
6
+δ,
pi
2
)
−χ
(
θ,
pi
6
+δ,
pi
3
,
pi
2
)
+χ
(
θ,
pi
6
+δ,
pi
3
,
pi
3
)
−χ
(
θ,θ−
pi
6
−δ,
pi
3
,
pi
6
+δ
)
+χ
(
θ,
2pi
3
−θ,
pi
3
,
2pi
3
)
−χ
(
θ,
pi
3
,
pi
2
,
2pi
3
)
+χ
(
θ,
pi
3
,
pi
2
,
pi
2
)
−χ
(
θ,
pi
3
,
pi
2
,
pi
3
)
+χ
(
θ,
pi
3
,
pi
2
,
pi
6
+δ
)
+χ
(
θ,
5pi
6
−δ−θ,
pi
2
,
5pi
6
−δ
)
,
r3δ(θ)≡ −1 + 2
[
cos
(pi
6
+δ
)
−cos
(
θ−
pi
3
)
+cos
(pi
3
)
−cos
(
θ−
pi
6
−δ
)]
+χ
(
θ,
pi
2
−θ,
pi
6
+δ,
pi
2
)
−χ
(
θ,
pi
6
+δ,
pi
3
,
pi
2
)
+χ
(
θ,θ−
pi
3
,
pi
3
,
pi
3
)
+χ
(
θ,
2pi
3
−θ,
pi
3
,
2pi
3
)
−χ
(
θ,
pi
3
,
pi
2
,
2pi
3
)
+χ
(
θ,
pi
3
,
pi
2
,
pi
2
)
−χ
(
θ,
pi
3
,
pi
2
,
pi
3
)
+χ
(
θ,θ−
pi
6
−δ,
pi
2
,
pi
6
+δ
)
+χ
(
θ,
5pi
6
−δ−θ,
pi
2
,
5pi
6
−δ
)
,
143
Appendix A. Details of Numerical Work
r4δ(θ)≡ −1 + 2
[
cos
(
θ−
pi
3
)
−cos
(
θ−
pi
6
−δ
)
+cos
(pi
6
+δ
)
−cos
(pi
3
)]
−χ
(
θ,θ−
pi
3
,
pi
6
+δ,
pi
3
)
+χ
(
θ,θ−
pi
6
−δ,
pi
6
+δ,
pi
6
+δ
)
+χ
(
θ,
pi
2
−θ,
pi
6
+δ,
pi
2
)
−χ
(
θ,
pi
6
+δ,
pi
3
,
pi
2
)
+χ
(
θ,
pi
6
+δ,
pi
3
,
pi
3
)
−χ
(
θ,
pi
6
+δ,
pi
3
,
pi
6
+δ
)
+χ
(
θ,
2pi
3
−θ,
pi
3
,
2pi
3
)
−χ
(
θ,
pi
3
,
pi
2
,
2pi
3
)
+χ
(
θ,
pi
3
,
pi
2
,
pi
2
)
−χ
(
θ,
pi
3
,
pi
2
,
pi
3
)
+χ
(
θ,
pi
3
,
pi
2
,
pi
6
+δ
)
+χ
(
θ,
5pi
6
−δ−θ,
pi
2
,
5pi
6
−δ
)
,
r5δ(θ)≡ −1 + 2
[
cos
(
θ−
pi
3
)
−cos
(pi
6
+δ
)
+cos
(
θ−
pi
6
−δ
)
−cos
(pi
3
)]
+χ
(
θ,
pi
2
−θ,
pi
6
+δ,
pi
2
)
−χ
(
θ,θ−
pi
3
,
pi
6
+δ,
pi
3
)
−χ
(
θ,
2pi
3
−θ,
pi
6
+δ,
2pi
3
)
+χ
(
θ,
pi
6
+δ,
pi
3
,
2pi
3
)
−χ
(
θ,
pi
6
+δ,
pi
3
,
pi
2
)
+χ
(
θ,
pi
6
+δ,
pi
3
,
pi
3
)
−χ
(
θ,θ−
pi
6
−δ,
pi
3
,
pi
6
+δ
)
−χ
(
θ,
5pi
6
−δ−θ,
pi
3
,
5pi
6
−δ
)
+χ
(
θ,
pi
3
,
pi
2
,
5pi
6
−δ
)
−χ
(
θ,
pi
3
,
pi
2
,
2pi
3
)
+χ
(
θ,
pi
3
,
pi
2
,
pi
2
)
−χ
(
θ,
pi
3
,
pi
2
,
pi
3
)
+χ
(
θ,
pi
3
,
pi
2
,
pi
6
+δ
)
.
Now we show (2.18):
C3
(pi
2
− τ
)
= −1.5τ + O(τ 2). (A.4)
Let 0 ≤ τ  1. We expand C3
(
pi
2 − τ
)
in its Taylor series to obtain
C3
(pi
2
− τ
)
= C3
(pi
2
)
+ τ
[ d
dτ
C3
(pi
2
− τ
)]
τ=0
+ O(τ 2). (A.5)
As shown in section 2.2, the correlation satisfies Cx(pi2 ) = 0 for every colouring
x ∈ X. Thus, we have that C3
(
pi
2
)
= 0. From (A.3), we have that C3
(
pi
2 − τ
)
=
h43
(
pi
2 − τ
)
for 0 ≤ τ  1. Thus, we only need to show that
[ d
dθ
h43(θ)
]
θ=pi/2
= 1.5. (A.6)
144
The function h43(θ) has terms of the form
χ(θ, a, b, α) ≡
∫ b
a
dξ(θ, , α), (A.7)
where
ξ(θ, , α) ≡
2
pi
sin  arccos
(cos θ cos − cosα
sin θ sin 
)
, (A.8)
as defined by (A.2). Differentiating the function χ, we obtain
d
dθ
χ(θ, a, b, α) = ξ(θ, b, α)
db
dθ
− ξ(θ, a, α)
da
dθ
+
∫ b
a
d
∂
∂θ
ξ(θ, , α). (A.9)
We have that [
∂
∂θ
ξ(θ, , α)
]
θ=pi/2
=
2 cos 
pi
√
1−
(
cosα
sin 
)2
. (A.10)
We obtain that
2
pi
b∫
a
d cos 
√
1−
(
cosα
sin 
)2
= µ(a, b, α), (A.11)
where
µ(a, b, α) ≡
2
pi
(√
sin2 b− cos2 α−
√
sin2 a− cos2 α
)
, (A.12)
for cos2 α ≤ sin2 b and cos2 α ≤ sin2 a. We define
ν(, α) ≡ ξ
(pi
2
, , α
)
. (A.13)
From the definition of h43(θ), given by (A.3), and Equations (A.9) – (A.13), it is
145
Appendix A. Details of Numerical Work
straightforward to obtain that
[ d
dθ
h43(θ)
]
θ=pi/2
= −2
[
sin
(pi
6
)
+ sin
(pi
3
)]
+ ν
(
0,
pi
2
)
+ ν
(pi
6
,
pi
3
)
+ ν
(pi
6
,
2pi
3
)
+ν
(pi
3
,
pi
6
)
+ ν
(pi
3
,
5pi
6
)
+ µ
(
0,
pi
6
,
pi
2
)
− µ
(pi
6
,
pi
6
,
pi
3
)
+µ
(pi
6
,
pi
3
,
pi
3
)
− µ
(pi
6
,
pi
3
,
pi
2
)
+ µ
(pi
6
,
pi
3
,
2pi
3
)
−µ
(pi
3
,
pi
3
,
pi
6
)
+ µ
(pi
3
,
pi
2
,
pi
2
)
+ µ
(pi
3
,
pi
2
,
pi
6
)
−µ
(pi
3
,
pi
2
,
2pi
3
)
− µ
(pi
3
,
pi
2
,
pi
3
)
+ µ
(pi
3
,
pi
2
,
5pi
6
)
. (A.14)
We use (A.8), (A.12) and (A.13), and notice that ν
(
0, pi2
)
= 0 in order to evaluate
the previous expression. We obtain
[ d
dθ
h43(θ)
]
θ=pi/2
=
1
pi
[
6− 4
(√
3−
√
2
)]
= 1.5, (A.15)
as claimed.
146
Appendix B
Code for the Computer Program
In this appendix, we present the code of our Mathematica program used to com-
pute the correlations given by (A.3). The code is given in six parts. Each part is
given in a figure of this appendix.
From Figure B.1, we see that the integrals are computed with a precision of
6 and an accuracy of 7. The code precision gives the effective number of digits of
precision in the computed values, while the accuracy corresponds to the effective
number of digits to the right of the decimal point in the computed values. Thus,
considering the factor 2pi in (A.2) and the number of terms of the form (A.2)
in the expressions for the correlations (A.3) being of the order of 10, we expect
the obtained correlations to have a precision of the order of 10−5, which is good
enough for our claims in section 2.4. For example, it is easily observed from
the plots in Figure 2.6 that C3−0.038pi(θ) < C1(θ) for θ ∈
[
0.386pi, pi2
)
1 and that
C1(θ) − C3−0.038pi(θ) achieves values of the order of 10
−2 in this range. This can
also be seen from the numerical values output by the program.
The given code takes a computation time of a few seconds and outputs results
with a precision of the order of 10−5. One of the reasons for computing analyti-
1The value 0.386pi for the upper bound θwmax ≤ 0.386pi was not obtained just from observa-
tion of the plots. It was obtained by computing the correlations C3−0.038pi (θ) and C1(θ) numeri-
cally at several values of θ to guarantee the given precision. Similarly, the bound θsmax ≤ 0.345pi
was obtained by computing C2∆(θ) and −C1(θ) for several values of θ and ∆ to guarantee the
given precision.
147
Appendix B. Code for the Computer Program
cally the integrals with respect to ω in (A.1) is that, by doing this, the numerical
computations of our program take times of a few seconds, compared to times of
hours if the integrals with respect to ω are computed numerically. Furthermore,
by computing the integrals with respect to ω analytically, the achieved numerical
precision is much higher. In fact, we decided to constrain our numerical computa-
tions to colourings with azimuthal symmetry in order to have a high control over
the obtained precision. We expect that arbitrary colourings can be investigated
numerically with adequate numerical techniques and computer programs.
Figure B.1: Program code I. This code defines the function given by (A.2) and
the correlation function for colouring 2.
148
Figure B.2: Program code II. This code defines the correlation function for colour-
ing 3.
149
Appendix B. Code for the Computer Program
Figure B.3: Program code III. This code defines the correlation function for
colouring 4.
150
Figure B.4: Program code IV. This code defines the correlation function for
colouring 2∆.
151
Appendix B. Code for the Computer Program
Figure B.5: Program code V. This code defines the correlation function for colour-
ing 3δ.
152
Figure B.6: Program code VI. This code computes the correlations. The first
three lines output the correlations plotted in Figure 2.5. The next three lines
output the correlations plotted in Figure 2.6. The last two lines output the
correlations plotted in Figure 2.7.
153

Appendix C
Details of the Primed PBT
Protocol
We provide specific details of the primed PBT protocol described in section 3.5,
which achieves success probability qj and satisfies Equation (3.8) for the corre-
sponding primed states,
η′j = γ
′
j,i =
I
2n
. (C.1)
We use the following identity satisfied for any quantum state ρ of dimension
2n:
I
2n
≡
1
4n
4n∑
l=1
VlρV
†
l ; (C.2)
where we define the set of unitary operations {Vl}4
n
l=1 ≡ {σ0, σ1, σ2, σ3}
⊗n, σ0 is
the identity acting on C2, and {σi}3i=1 are the Pauli matrices.
The following, primed, PBT protocol achieves success probability qj and sat-
isfies (C.1).
Firstly, consider the stage previous to the implementation of PBT in which the
resource state is prepared and distributed to Alice and Bob. An ancilla a′ with
Hilbert space Ha′ of dimension 4n is prepared in the state |φ〉 ≡ 12n
∑4n
l=1|µl〉,
where {|µl〉}4
n
l=1 is an orthonormal basis of Ha′ . Consider the resource state
155
Appendix C. Details of the Primed PBT Protocol
|ξ〉AB for the PBT protocol defined by (3.3) – (3.5). In the primed protocol,
the controlled unitary
∑4n
l=1(|µl〉〈µl|)a′
N⊗
i=1
(Vl)Bi is applied on |φ〉a′|ξ〉AB in order
to prepare the new resource state
|ξ′〉a′AB ≡
1
2n
4n∑
l=1
N⊗
i=1
(Vl)Bi |µl〉a′ |ξ〉AB, (C.3)
where (Vl)Bi acts on HBi only. Recall that B ≡ B1B2 · · ·BN . The system a
′A is
sent to Alice and the system B is sent to Bob. In this protocol, the initial state
of the system Bj is η′j ≡ Tra′AB˜j(|ξ
′〉〈ξ′|)a′AB. From the definitions of η′j, |ξ
′〉 and
ηj (Equation (3.31)), the identity (C.2), and the fact that ηj is independent of
|ψ〉, it is straightforward to obtain that
η′j =
1
4n
4n∑
l=1
VlηjV
†
l =
I
2n
, (C.4)
as claimed.
Now consider the implementation of the primed PBT protocol. Alice applies
the unitary operation Waa′ ≡
∑4n
l=1(V
†
l )a ⊗ (|µl〉〈µl|)a′ on the system aa
′, as the
notation suggests. The global state transforms into
Waa′|ψ〉a|ξ〉a′AB =
1
2n
4n∑
l=1
(V †l )a
N⊗
i=1
(Vl)Bi |ψ〉a|µl〉a′|ξ〉AB. (C.5)
Then, Alice applies her operations corresponding to the PBT protocol defined
by (3.3) – (3.5) on the system aA only. With probability qj, Alice obtains outcome
k = j 6= 0. Due to the linearity of unitary evolution, it is not difficult to obtain
that, in this case, the global state transforms into
|G′j〉a′aAB =
1
2n
4n∑
l=1
N⊗
i=1
i 6=j
(Vl)Bi |µl〉a′ |ψ〉Bj |Rj〉aAB˜j . (C.6)
156
Thus, we see that the state |ψ〉 is teleported to port Bj, as required. This pro-
tocol works because, as we can see from (C.5), the operations on the system
B commute with the operations performed by Alice on aA, which is necessary
for satisfaction of the no-signalling principle. Therefore, this protocol is equiva-
lent to the following: conditioned on a′ being in the state |µl〉a′ , if an outcome
k = j 6= 0 is obtained, the state V †l |ψ〉 is teleported to the port Bj; then, Bob
applies
N⊗
i=1
(Vl)Bi , after which, the state of the system Bj transforms into |ψ〉, as
desired.
The state of the system Bj, after an outcome k = i /∈ {0, j} is obtained,
is γ′j,i ≡ Tra′aAB˜j(|G
′
i〉〈G
′
i|)a′aAB. From the definitions of γ
′
j,i and γj,i (Equa-
tion (3.32)), the identity (C.2), and the fact that γj,i is independent of |ψ〉, it can
easily be obtained that
γ′j,i =
1
4n
4n∑
l=1
Vlγj,iV
†
l =
I
2n
, (C.7)
as claimed.
If Alice obtains the outcome k = 0, the final global state is
|G′0〉a′aAB =
1
2n
4n∑
l=1
N⊗
i=1
(Vl)Bi|µl〉a′|F
(ψl)〉aAB, (C.8)
where ψl refers to dependence on the state V
†
l |ψ〉. In this case, the final state of
the system Bj is ω
′(ψ)
j ≡ Tra′aAB˜j (|G
′
0〉〈G
′
0|)a′aAB. From the previous definition
of ω′(ψ)j and that one of ω
(ψ)
j , given by (3.33), it is straightforward to obtain the
expression (3.36):
ω′(ψ)j =
1
4n
4n∑
l=1
Vlω
(ψl)
j V
†
l . (C.9)
157

Appendix D
Bound for Nonlocal Strategies in
the IC-2 Game
Recall the nonlocal strategies in the IC-2 game presented in section 4.5.3. We
show the bound (4.60):
Q ≤ Q′, (D.1)
where Q′ ≡ 14
(
1+ 3√n
)
and Q = 1n
∑n−1
k=0 P (y
0
k = x
0
k, y
1
k = x
1
k) is the success proba-
bility. The proof is an extension of the one given in [108] for the IC-1 game. It re-
quires several steps. Firstly, we define the quantity E~x,k ≡ (−1)x
0
k+x
1
k〈ψ|Aˆ~xBˆk|ψ〉
in terms of the Hermitian operators:
Aˆ~x ≡
1∑
r=0
1∑
s=0
(−1)r+s|ν~xr,s〉〈ν
~x
r,s|,
Bˆk ≡
1∑
t=0
1∑
u=0
(−1)t+u|wkt,u〉〈w
k
t,u|,
which act on HA and HB, respectively. We write the state |ψ〉 in the basis
{|ν~xr,s〉|w
k
t,u〉}, we use that y
j
k = a
j
k ⊕ b
j
k, for j = 0, 1, and we use the fact that ~x is
159
Appendix D. Bound for Nonlocal Strategies in the IC-2 Game
a random variable of 4n possible values to obtain that
1
n
n−1∑
k=0
[
P
(
y0k = x
0
k, y
1
k = x
1
k
)
+ P
(
y0k 6= x
0
k, y
1
k 6= x
1
k
)]
=
1
2
(
1 +
1
n4n
n−1∑
k=0
∑
~x
E~x,k
)
.
(D.2)
Then, we show that
1
2
(
1 +
1
n4n
n−1∑
k=0
∑
~x
E~x,k
)
≤
1
2
(
1 +
1
√
n
)
, (D.3)
which from (D.2) implies
1
n
n−1∑
k=0
[
P
(
y0k = x
0
k, y
1
k = x
1
k
)
+ P
(
y0k 6= x
0
k, y
1
k 6= x
1
k
)]
≤
1
2
(
1 +
1
√
n
)
. (D.4)
We follow a similar procedure to obtain
1
n
n−1∑
k=0
[
P
(
y0k = x
0
k, y
1
k = x
1
k
)
+ P
(
y0k = x
0
k, y
1
k 6= x
1
k
)]
≤
1
2
(
1 +
1
√
n
)
, (D.5)
1
n
n−1∑
k=0
[
P
(
y0k = x
0
k, y
1
k = x
1
k
)
+ P
(
y0k 6= x
0
k, y
1
k = x
1
k
)]
≤
1
2
(
1 +
1
√
n
)
. (D.6)
Adding (D.4) – (D.6), using normalization of probabilities and arranging terms
we have that
1
n
n−1∑
k=0
P
(
y0k = x
0
k, y
1
k = x
1
k
)
≤
1
4
(
1 +
3
√
n
)
,
as claimed in (D.1).
We show Equation (D.2). Writing |ψ〉 in the basis {|ν~xr,s〉|w
k
t,u〉}, we have
|ψ〉 =
∑
r,s,t,u∈{0,1}
C~x,kr,s,t,u|ν
~x
r,s〉|w
k
t,u〉.
160
From the definition of E~x,k, it follows that
E~x,k = (−1)
x0k+x
1
k
∑
r,s,t,u∈{0,1}
(−1)r+s+t+u|C~x,kr,s,t,u|
2.
= (−1)x
0
k+x
1
k
∑
a0k,a
1
k,b
0
k,b
1
k∈{0,1}
(−1)a
0
k⊕b
0
k(−1)a
1
k⊕b
1
kP (a0k, a
1
k, b
0
k, b
1
k|~x)
= (−1)x
0
k+x
1
k
∑
y0k,y
1
k∈{0,1}
(−1)y
0
k+y
1
kP (y0k, y
1
k|~x)
= P
(
y0k = x
0
k, y
1
k = x
1
k|~x
)
+ P
(
y0k 6= x
0
k, y
1
k 6= x
1
k|~x
)
−P
(
y0k = x
0
k, y
1
k 6= x
1
k|~x
)
− P
(
y0k 6= x
0
k, y
1
k = x
1
k|~x
)
.
Using normalization of probabilities and arranging terms, we obtain
P
(
y0k = x
0
k, y
1
k = x
1
k|~x
)
+ P
(
y0k 6= x
0
k, y
1
k 6= x
1
k|~x
)
=
1
2
(1 + E~x,k).
Since each possible value of ~x occurs with probability 4−n, multiplying the pre-
vious equation by n−14−n and summing over all possible values of ~x and k, we
obtain Equation (D.2).
Now we show Equation (D.3). We define the normalized states:
|A〉 =
1
√
4n
∑
~x
Aˆ~x|ψ〉 ⊗ |~x〉,
|Bk〉 =
1
√
4n
∑
~x
(−1)x
0
k+x
1
kBˆk|ψ〉 ⊗ |~x〉,
where the set of states |~x〉 is orthonormal. We have that 〈A|Bk〉 = Ek, where
Ek ≡ 14n
∑
~xE~x,k. Since the states |Bk〉 satisfy 〈Bk′|Bk〉 = δk′,k,
∑
k|Bk〉〈Bk| sat-
isfies
(∑
k|Bk〉〈Bk|
)2
=
∑
k|Bk〉〈Bk| and
(∑
k|Bk〉〈Bk|
)†
=
∑
k|Bk〉〈Bk|, hence,
161
Appendix D. Bound for Nonlocal Strategies in the IC-2 Game
∑
k|Bk〉〈Bk| is a projector. Thus, we obtain
n−1∑
k=0
E2k =
n−1∑
k=0
〈A|Bk〉
2
= 〈A|
(
n−1∑
k=0
|Bk〉〈Bk|
)
|A〉
≤ 1. (D.7)
Since the root mean square is not smaller than the average value, we have
1
n
n−1∑
k=0
Ek ≤
√
√
√
√ 1
n
n−1∑
k=0
E2k
≤
1
√
n
, (D.8)
where in the second line we have used (D.7). Equation (D.3) follows from (D.8)
and the definition of Ek.
We adopt a similar procedure to show Equation (D.5). We define E0~x,k ≡
(−1)x
0
k〈ψ|Aˆ0~xBˆ
0
k|ψ〉 in terms of the Hermitian operators:
Aˆ0~x ≡
1∑
r=0
1∑
s=0
(−1)r|ν~xr,s〉〈ν
~x
r,s|,
Bˆ0k ≡
1∑
t=0
1∑
u=0
(−1)t|wkt,u〉〈w
k
t,u|.
It follows that
E0~x,k = (−1)
x0k
∑
a0k,a
1
k,b
0
k,b
1
k∈{0,1}
(−1)a
0
k⊕b
0
kP (a0k, a
1
k, b
0
k, b
1
k|~x)
= (−1)x
0
k
1∑
y0k=0
(−1)y
0
kP (y0k|~x)
= P
(
y0k = x
0
k|~x
)
− P
(
y0k 6= x
0
k|~x
)
.
162
From the previous equation, it is straightforward to obtain that
1
n
n−1∑
k=0
[
P
(
y0k = x
0
k, y
1
k = x
1
k
)
+ P
(
y0k = x
0
k, y
1
k 6= x
1
k
)]
=
1
2
(
1 +
1
n4n
n−1∑
k=0
∑
~x
E0~x,k
)
.
(D.9)
We define the normalized states:
|A0〉 =
1
√
4n
∑
~x
Aˆ0~x|ψ〉 ⊗ |~x〉,
|B0k〉 =
1
√
4n
∑
~x
(−1)x
0
kBˆ0k|ψ〉 ⊗ |~x〉.
We have that 〈A0|B0k〉 = E
0
k , where E
0
k ≡
1
4n
∑
~xE
0
~x,k. Similar to (D.8), we obtain
1
n
n−1∑
k=0
E0k ≤
1
√
n
. (D.10)
From the definition of E0k and Equations (D.9) and (D.10), we obtain (D.5).
In a similar way, Equation (D.6) can be shown using E1~x,k ≡ (−1)
x1k〈ψ|Aˆ1~xBˆ
1
k|ψ〉
in terms of the Hermitian operators:
Aˆ1~x ≡
1∑
r=0
1∑
s=0
(−1)s|ν~xr,s〉〈ν
~x
r,s|,
Bˆ1k ≡
1∑
t=0
1∑
u=0
(−1)u|wkt,u〉〈w
k
t,u|,
and noticing that 〈A1|B1k〉 = E
1
k , for E
1
k ≡
1
4n
∑
~xE
1
~x,k and the states
|A1〉 =
1
√
4n
∑
~x
Aˆ1~x|ψ〉 ⊗ |~x〉,
|B1k〉 =
1
√
4n
∑
~x
(−1)x
1
kBˆ1k|ψ〉 ⊗ |~x〉.
163

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