## Aspects of Quantum Coherence

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## Abstract

In this work our aim is to study several aspects related to quantum coherence as understood to correspond with the non-classical behaviour that can be observed for certain particular states of a physical system. In particular we are interested in the possible mechanisms that result in dynamically induced transitions between quantum and classical regimes. The thesis is organized as follows:

The first chapter dubs as an introduction and serves to set out the basic philosophy underlying the questions addressed in this thesis. It also presents some elementary properties of states and state spaces in Quantum Theory including what we have chosen to define as classical and quantum behaviour.

In chapter 2 we study some of the aspects related to observing quantum behaviour and of the properties of our main definition of classicality (and quantumness). Here we also study some of the restrictions imposed on measurements by the existence of globally conserved quantities (Wigner-Araki-Yanase theorem) and their relationship to weak measurements coupled to postselection.

In the following chapter we review some of the basic tools used in the description of open quantum system dynamics that will be applied in other chapters.

In chapter 4 we review the basics of decoherence and analyse the importance of the choice of initial conditions when trying to study the dynamical emergence of classical behaviour within Quantum Theory.

Next we study the other direction of the transition and focus on how to obtain pure quantum states from states that originally were classically mixed.

Along the same lines, in chapter 6 we cover some topics related to the production of pure quantum states from measurements. We pay special attention to a model of the non-selective continuous monitoring of a system coupled to another unmonitored system.

Lastly we explore some of the possible similarities between the theory of phase transitions and the quantum-classical transition.

We must emphasize that all the work done in this thesis assumes that Quantum Theory is generally valid (at least within a broad enough range of energies). Thus, when we say that a state is “classical” we will mainly be referring to one of all the possible states contained in Quantum Theory, but that is susceptible to being interpreted as corresponding to “classical” behaviour. Similarly when we speak of creating a “quantum”, or “quantum coherent”, state we mean that the system has evolved to this state from one of the “classical” ones, but all of these still correspond to valid states within Quantum Theory.

In the opinion of the author the main original contributions that can be found in this thesis are the following:

- The recognition of the relationship between the Wigner-Araki-Yanase theorem and weak measurements coupled to postselection (sections 2.2 and 2.4);
- A mathematical proof of the possible ambiguities arising when two observers try to decide if a state corresponds to quantum or classical behaviour (section 2.6);
- The implications of initial correlations in decoherence models. In particular how the choice of certain (correlated) initial conditions can result in residual coherence and the production of pure quantum states in a model that otherwise results in ideal decoherence when (locally equivalent) uncorrelated initial conditions are used (section 4.2);
- Various results related to the production of quantum states from initially classical states (sections 5.2 to 5.4);
- The analysis of the inverse of a generalized depolarizing channel (section 5.7);
- The study of a model of the non-selective continuous monitoring, in the quantum Zeno limit, of a subsystem A interacting with an unmonitored subsystem B. In particular the absence of the purification of B, which has been previously predicted in the selective case, but the possibility of coherent dynamics for B (section 6.4);
- The identification of the loose equivalent of a broken symmetry and order parameter in the quantum-classical transition (section 7.2).