2310.1007/23.1424-0661Annales Henri PoincaréA Journal of Theoretical and Mathematical PhysicsAnn. Henri Poincaré1424-06371424-0661Springer International PublishingChams00023-020-00891-889110.1007/s00023-020-00891-8Original PaperRelating Relative Entropy, Optimal Transport and Fisher Information: A Quantum HWI InequalityDattaNilanjana12RouzéCambyse3bgrid.5335.00000000121885934Statistical Laboratory, Centre for Mathematical SciencesUniversity of CambridgeCB 30WBCambridgeUKgrid.5335.00000000121885934DAMTP, Centre for Mathematical SciencesUniversity of CambridgeCB 30WACambridgeUKgrid.6936.a0000000123222966Department of MathematicsTechnische Universität München85748GarchingGermany
Quantum Markov semigroups characterize the time evolution of an important class of open quantum systems. Studying convergence properties of such a semigroup and determining concentration properties of its invariant state have been the focus of much research. Quantum versions of functional inequalities (like the modified logarithmic Sobolev and Poincaré inequalities) and the so-called transportation cost inequalities have proved to be essential for this purpose. Classical functional and transportation cost inequalities are seen to arise from a single geometric inequality, called the Ricci lower bound, via an inequality which interpolates between them. The latter is called the HWI inequality, where the letters I, W and H are, respectively, acronyms for the Fisher information (arising in the modified logarithmic Sobolev inequality), the so-called Wasserstein distance (arising in the transportation cost inequality) and the relative entropy (or Boltzmann H function) arising in both. Hence, classically, the above inequalities and the implications between them form a remarkable picture which relates elements from diverse mathematical fields, such as Riemannian geometry, information theory, optimal transport theory, Markov processes, concentration of measure and convexity theory. Here, we consider a quantum version of the Ricci lower bound introduced by Carlen and Maas and prove that it implies a quantum HWI inequality from which the quantum functional and transportation cost inequalities follow. Our results hence establish that the unifying picture of the classical setting carries over to the quantum one.
Munich Center for Quantum Science and TechnologyDFG cluster of excellence 2111publisher-imprint-nameBirkhäuservolume-issue-count12issue-article-count10issue-toc-levels0issue-pricelist-year2020issue-copyright-holderSpringer Nature Switzerland AGissue-copyright-year2020article-contains-esmNoarticle-numbering-styleContentOnlyarticle-registration-date-year2020article-registration-date-month1article-registration-date-day21article-toc-levels0toc-levels0volume-typeRegularjournal-productNonStandardArchiveJournalnumbering-styleContentOnlyarticle-grants-typeOpenChoicemetadata-grantOpenAccessabstract-grantOpenAccessbodypdf-grantOpenAccessbodyhtml-grantOpenAccessbibliography-grantOpenAccessesm-grantOpenAccessonline-firstfalsepdf-file-referenceBodyRef/PDF/23_2020_Article_891.pdfpdf-typeTypesettarget-typeOnlinePDFissue-online-date-year2020issue-online-date-month6issue-online-date-day27issue-print-date-year2020issue-print-date-month6issue-print-date-day27issue-typeRegulararticle-typeOriginalPaperjournal-subject-primaryPhysicsjournal-subject-secondaryTheoretical, Mathematical and Computational Physicsjournal-subject-secondaryDynamical Systems and Ergodic Theoryjournal-subject-secondaryQuantum Physicsjournal-subject-secondaryMathematical Methods in Physicsjournal-subject-secondaryClassical and Quantum Gravitation, Relativity Theoryjournal-subject-secondaryElementary Particles, Quantum Field Theoryjournal-subject-collectionPhysics and Astronomyopen-accesstrueIntroduction
Realistic physical systems that are relevant for quantum information processing are inherently open. They undergo unwanted but unavoidable interactions with the surrounding environment and are hence subject to noise and decoherence. Under the Markovian approximation, which is valid when the system is only weakly coupled to its environment, the resulting dissipative dynamics of the system is described by a quantum Markov semigroup (QMS), whose generator we denote by L\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {L}}}$$\end{document}. The analysis of quantum Markov semigroups is hence a key component of the theory of open quantum systems and quantum information. An important problem in the study of a QMS is the analysis of its convergence properties, in particular its mixing time, which is the time taken by any state evolving under the action of the QMS to come close to its invariant state.1
Functional and Transportation Cost Inequalities
Classically, given a measure μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document}, functional inequalities, e.g. the Poincaré inequality (usually denoted as PI(λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document})) [29] and the (modified) logarithmic Sobolev inequality (or log-Sobolev in short), denoted as MLSI(α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}) [22], constitute a powerful tool for deriving mixing times of a Markov semigroup with invariant measure μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document} and determining concentration properties of μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document}. They are also related to the so-called transportation cost inequalities denoted by TC1(c1)\documentclass[12pt]{minimal}
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\begin{document}$$_1(c_1)$$\end{document} and TC2(c2)\documentclass[12pt]{minimal}
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\begin{document}$$_2(c_2)$$\end{document}. Here α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}, c1\documentclass[12pt]{minimal}
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\begin{document}$$c_1$$\end{document} and c2\documentclass[12pt]{minimal}
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\begin{document}$$c_2$$\end{document} denote constants appearing in the respective inequalities. Consider a compact manifold M\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {M}}$$\end{document}, and let P(M)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {P}}}({\mathcal {M}})$$\end{document} be the set of probability measures on M\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {M}}$$\end{document}. Given a measure μ∈P(M)\documentclass[12pt]{minimal}
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\begin{document}$$\mu \in {{\mathcal {P}}}({\mathcal {M}})$$\end{document}, the inequality TC1(c1)\documentclass[12pt]{minimal}
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\begin{document}$$_1(c_1)$$\end{document} (resp. TC2(c2)\documentclass[12pt]{minimal}
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\begin{document}$$_2(c_2)$$\end{document}) provides an upper bound on the so-called Wasserstein distance W1\documentclass[12pt]{minimal}
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\begin{document}$$W_1$$\end{document} (resp. W2\documentclass[12pt]{minimal}
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\begin{document}$$W_2$$\end{document}), between any probability measure ν∈P(M)\documentclass[12pt]{minimal}
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\begin{document}$$\nu \in {{\mathcal {P}}}({\mathcal {M}})$$\end{document} and the given measure μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document}, in terms of the square root of the relative entropy of ν\documentclass[12pt]{minimal}
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\begin{document}$$\nu $$\end{document} with respect to μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document}. Since μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document} is fixed, this relative entropy is simply a functional of ν\documentclass[12pt]{minimal}
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\begin{document}$$\nu $$\end{document} and, due to its close links with the Boltzmann H-functional, is often denoted by the letter H in the literature. The notion of Wasserstein distances first appeared in the theory of optimal transport, which was initiated by Monge [21] and later analysed by Kantorovich [15]. In its original formulation by Monge, the problem of optimal transport concerns finding the optimal way, in the sense of minimal transportation cost, of moving a sand pile between two locations (see also [30]). In 1986, Marton [20] showed that transportation cost inequalities are also useful for deriving concentration of measure properties of the given measure μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document}.2
The classical inequalities discussed above can be shown to be obtainable from a single geometric inequality, involving a quantity called the Ricci curvature of the manifold M\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {M}}}$$\end{document}, and referred to as the Ricci lower bound. In fact, there is an inherent relation between the geometry of the manifold and a diffusion process (whose associated Markov semigroup has generator L\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}$$\end{document}, say) defined on it: the diffusion process can be used to explore the geometry of M\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {M}}}$$\end{document}, and conversely, the latter determines the mixing time of the diffusion process. Finding a quantum analogue of this appealing geometric inequality is hence a problem of fundamental interest and is considered in this paper. Before we present our results on this problem, we first need to explain the statement of the Ricci lower bound in the classical setting. In fact, it is instructive to start from the very definition of curvature which generalizes to the Ricci curvature for the case of a Riemannian manifold.
Ricci Curvature and Ricci Lower Bound (Classical Setting)
Given a surface S\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^3$$\end{document}, the Gauss curvatureκ\documentclass[12pt]{minimal}
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\begin{document}$$\kappa $$\end{document} of S\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {S}$$\end{document} is a measure of its local boundedness. More precisely, given a point x∈S\documentclass[12pt]{minimal}
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\begin{document}$$x\in \mathcal {S}$$\end{document} and any two mutually orthogonal unit tangent vectors u, v at x, the distance between two geodesics γu\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _v$$\end{document}, starting at x, with respective directions u and v, obeys the following Taylor expansions:dg(γu(t),γv(t))=2t1-κ(x)12t2+Ot→0(t3),t≥0,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} d_\mathrm{g}(\gamma _\mathbf {u}(t),\gamma _\mathbf {v}(t))=\sqrt{2}t\left( 1-\frac{\kappa (x)}{12}t^2+\mathcal {O}_{t\rightarrow 0}(t^3)\right) , \quad t \ge 0, \end{aligned}$$\end{document}where dg\documentclass[12pt]{minimal}
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\begin{document}$$d_\mathrm{g}$$\end{document} is the geodesic distance defined with respect to the metric g induced on S\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {S}$$\end{document} by the Euclidean metric. In the case when κ=0\documentclass[12pt]{minimal}
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\begin{document}$$\kappa =0$$\end{document} uniformly on the surface, the latter is flat, and we recover the Pythagoras theorem from Eq. (1.1). More generally, let x be a point on a d-dimensional compact Riemannian manifold M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document}, let u belong to the tangent space TxM\documentclass[12pt]{minimal}
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\begin{document}$$T_x\mathcal {M}$$\end{document} at the point x of M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document}, and complete the vector u into an orthonormal basis (u,v2,…,vd)\documentclass[12pt]{minimal}
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\begin{document}$$(u,v_2,\ldots ,v_d)$$\end{document} of TxM\documentclass[12pt]{minimal}
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\begin{document}$$T_x\mathcal {M}$$\end{document}. Then, the Ricci curvature of M\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {M}}$$\end{document}, evaluated at u, is the averaged Gauss curvature over orthogonal surfaces defined by all the geodesics starting at x with direction given by the unit vectors belonging to the vector subspace spanned by u and any other vector vi\documentclass[12pt]{minimal}
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\begin{document}$$i=2,\ldots ,d$$\end{document}. The expression for the Ricci curvature [30] is given in terms of the Laplace–Beltrami operator (denoted simply as Δ\documentclass[12pt]{minimal}
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\begin{document}$$\Delta $$\end{document}), and hence, the curvature is usually denoted as Ric(Δ)\documentclass[12pt]{minimal}
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\begin{document}$${\text {Ric}}(\Delta )$$\end{document}. Since Δ\documentclass[12pt]{minimal}
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\begin{document}$$\Delta $$\end{document} is the generator of the heat semigroup, the curvature provides a bridge between the geometry of the manifold and the evolution on it induced by heat diffusion. There is an important inequality, known as the Ricci lower bound, which is denoted by Ric(Δ)≥κ\documentclass[12pt]{minimal}
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\begin{document}$${\text {Ric}}(\Delta )\ge \kappa $$\end{document} [3], and is the property that the Ricci curvature is uniformly bounded below by a real parameter κ≥0\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \ge 0$$\end{document}. Intuitively, the inequality is related to concentration of the uniform measure on M\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {M}}$$\end{document}, which is known to be the unique invariant measure of heat diffusion. For example, in the case of the sphere, which has constant Ricci curvature given in terms of its radius, the Haar measure can be shown to concentrate around any great circle. One can relax the condition of uniformity of the measure in order to allow for the study of concentration of measure phenomena for different measures μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document}, invariant for other diffusions processes on M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document}. In this more general framework, the Ricci lower bound is denoted by Ric(L)≥κ\documentclass[12pt]{minimal}
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\begin{document}$${\text {Ric}}(\mathcal {L})\ge \kappa $$\end{document}, where L\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}$$\end{document} denotes the generator of the diffusion semigroup associated with μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document} (Fig. 1).
The Gauss curvature
More recently, Sturm [27, 28] and Lott–Villani [18] showed that Ric(L)≥κ\documentclass[12pt]{minimal}
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\begin{document}$${\text {Ric}}(\mathcal {L})\ge \kappa $$\end{document} can be viewed as a (refined) convexity property (called the κ\documentclass[12pt]{minimal}
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\begin{document}$$\kappa $$\end{document}-displacement convexity) of H along geodesics on the Riemannian manifold obtained by endowing the set of probability measures P(M)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {P}(\mathcal {M})$$\end{document} on M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document} with the Wasserstein distance W2\documentclass[12pt]{minimal}
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\begin{document}$$W_2$$\end{document} [31]. This discovery led to a more robust notion of a Ricci lower bound, which does not explicitly depend on the expression of the Ricci curvature, and hence can be extended to more general metric spaces. Starting from this convexity property, one can then construct a diffusion semigroup for which H decreases the most along the direction of evolution induced by the semigroup. In this case, the path on the Riemannian manifold (P(M),W2)\documentclass[12pt]{minimal}
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\begin{document}$$({{\mathcal {P}}}({{\mathcal {M}}}), W_2)$$\end{document}, which corresponds to the actual evolution under the diffusion, is said to be gradient flow for H. It is a striking fact that this diffusion coincides with the one whose generator appears in the Bakry–Émery condition (see [10, 14]).
In [23], the authors introduced the so-called HWI(κ)\documentclass[12pt]{minimal}
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\begin{document}$$(\kappa )$$\end{document}-interpolation inequality, using which they reproved the so-called Bakry–Émery theorem, which states that for κ>0\documentclass[12pt]{minimal}
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\begin{document}$$\kappa >0$$\end{document}, Ric(L)≥κ\documentclass[12pt]{minimal}
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\begin{document}$${\text {Ric}}(\mathcal {L})\ge \kappa $$\end{document} implies MLSI(α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}) (for diffusions on Rn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^n$$\end{document} with associated generator L\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}$$\end{document}). The letters W, I and H are, respectively, acronyms for the Wasserstein distance W2\documentclass[12pt]{minimal}
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\begin{document}$$W_2$$\end{document} (appearing in TC2(c2)\documentclass[12pt]{minimal}
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\begin{document}$$_2(c_2)$$\end{document}), the Fisher information (which arises in MLSI(α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document})) and the relative entropy (also called the Boltzmann H-functional, as mentioned above) which appears in both these inequalities. They also showed that MLSI(α\documentclass[12pt]{minimal}
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\begin{document}$$_2(c_2)$$\end{document}. The term interpolation here comes from the fact that in the case κ=0\documentclass[12pt]{minimal}
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\begin{document}$$_2$$\end{document}(c) together with HWI(0) gives back MLSI(α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}).
In [11, 12, 19], a modified version of the Ricci lower bound was defined for Markov processes on finite sets, which led to the unification of the previously discussed functional and concentration inequalities in this discrete framework. In particular, it was proved in [11] that one can recover the Poincaré and modified log-Sobolev inequalities from the Ricci lower bound, provided the diameter of P(M)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {P}}}({{\mathcal {M}}})$$\end{document}, with respect to the Wasserstein distance, W2\documentclass[12pt]{minimal}
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\begin{document}$$W_2$$\end{document}, is bounded.
Ricci Lower Bound (Quantum Setting)
In the case of a quantum system with a finite-dimensional Hilbert space H\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {P}(\mathcal {M})$$\end{document} is replaced by the set D(H)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {D}}}({{\mathcal {H}}})$$\end{document} of quantum states (i.e. density matrices) on H\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {H}}}$$\end{document}. Then, in analogy with the classical case, starting with a primitive QMS with generator L\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {L}}}$$\end{document}, Carlen and Maas [6, 7] defined a quantum Wasserstein distance W2,L\documentclass[12pt]{minimal}
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\begin{document}$$W_{2,\mathcal {L}}$$\end{document} which renders D(H)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {D}}}({{\mathcal {H}}})$$\end{document} with a Riemannian structure, and for which the master equation associated with the QMS is gradient flow for the quantum relative entropy.
In [7], the authors proved that a quantum MLSI(α)\documentclass[12pt]{minimal}
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\begin{document}$$(\alpha )$$\end{document}, first introduced in [16], holds provided the quantum relative entropy (between a state on a geodesic on this manifold and the invariant state of the QMS) satisfies a quantum analogue of the κ\documentclass[12pt]{minimal}
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\begin{document}$$\alpha =\kappa >0$$\end{document}. This is denoted below by Ric(L)≥κ\documentclass[12pt]{minimal}
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\begin{document}$$(\mathcal {L})\ge \kappa $$\end{document} in analogy with the classical case, with L\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}$$\end{document} being the generator of the QMS.
The quantum versions of the Ricci lower bound, the HWI inequality, and the functional and transportation cost inequalities, all fit into a unifying picture which is analogous to the classical setting. It is given in Fig. 2.
Chain of quantum functional- and Talagrand inequalities and related concentrations for a primitive semigroup (Λt)t≥0\documentclass[12pt]{minimal}
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\begin{document}$$\Rightarrow $$\end{document} PI(λ)\documentclass[12pt]{minimal}
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\begin{document}$$(\lambda )$$\end{document} was proved in [16]. Here, “Exp.” refers to the notion of exponential concentration, whereas “Gauss.” refers to the stronger notion of Gaussian concentration. The implications MLSI(α\documentclass[12pt]{minimal}
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\begin{document}$$\Rightarrow $$\end{document}PI(λ\documentclass[12pt]{minimal}
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\begin{document}$$\Rightarrow $$\end{document}Gauss. were proved in [24]
Our Contribution:
In this paper, we analyse the quantum version of the Ricci lower bound introduced by Carlen and Maas [7] and derive various implications of it in Theorem 3. Moreover, we show that Ric(L)≥κ\documentclass[12pt]{minimal}
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\begin{document}$${\text {Ric}}(\mathcal {L})\ge \kappa $$\end{document} implies a quantum version of the celebrated HWI(κ)\documentclass[12pt]{minimal}
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\begin{document}$${\text {HWI}}(\kappa )$$\end{document} inequality which interpolates between the modified logarithmic Sobolev inequality and the transportation cost inequality (Theorem 5). We show that, in the case of κ>0\documentclass[12pt]{minimal}
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\begin{document}$${\text {HWI}}(\kappa )\Rightarrow {\text {MLSI}}(\kappa )$$\end{document} (Corollary 2), recovering the result of [7]. On the other hand, in Corollary 3, we establish that in the case when κ∈R\documentclass[12pt]{minimal}
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\begin{document}$$\kappa = 0$$\end{document}, we show that, under the assumption of boundedness of the diameter D of the set of states with respect to the quantum Wasserstein distance W2,L\documentclass[12pt]{minimal}
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\begin{document}$$c_1$$\end{document} (Theorem 6). Moreover, in the case of a unital QMS (i.e. one which has the completely mixed state as its unique invariant state), we show that it also implies MLSI(c2D-2)\documentclass[12pt]{minimal}
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\begin{document}$$c_2$$\end{document} (Theorem 7). We hence extend the results of [11] to the quantum regime.
Layout of the Paper
In Sect. 2, we introduce the necessary notations and definitions, including quantum Markov semigroups, the quantum Wasserstein distance and quantum functional inequalities. The quantum version of κ\documentclass[12pt]{minimal}
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\begin{document}$${\text {TC}}_2(c_2)$$\end{document} from it. In Sect. 5, we show that in the case in which κ=0\documentclass[12pt]{minimal}
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Notations and PreliminariesOperators, States and Entropic Quantities
In this paper, we denote by (H,⟨.|.⟩)\documentclass[12pt]{minimal}
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\begin{document}$$\mathop {{\hbox {Tr}}}\nolimits (\mathbb {I})=d$$\end{document}. Let P(H)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {P}({{\mathcal {H}}})$$\end{document} be the cone of positive semi-definite operators on H\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {H}}}$$\end{document} and P+(H)⊂P(H)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {P}_{+}({{\mathcal {H}}}) \subset \mathcal {P}({{\mathcal {H}}})$$\end{document} the set of (strictly) positive operators. Further, let D(H):={ρ∈P(H)∣Trρ=1}\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {D}}}({{\mathcal {H}}}):=\lbrace \rho \in \mathcal {P}({{\mathcal {H}}})\mid \mathop {{\hbox {Tr}}}\nolimits \rho =1\rbrace $$\end{document} denote the set of density operators (or states) on H\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {H}}}$$\end{document}, and D+(H):=D(H)∩P+(H)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {D}}}_+({{\mathcal {H}}}):={{\mathcal {D}}}({{\mathcal {H}}})\cap \mathcal {P}_+({{\mathcal {H}}})$$\end{document} denote the subset of faithful states. We denote the support of an operator A by supp(A)\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm {supp}}(A)$$\end{document}. Let I∈P(H)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {I}\in \mathcal {P}({{\mathcal {H}}})$$\end{document} be the identity operator on H\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {H}}}$$\end{document}, and id:B(H)↦B(H)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{id}:{{\mathcal {B}}}({{\mathcal {H}}})\mapsto {{\mathcal {B}}}({{\mathcal {H}}})$$\end{document} the identity map on operators on H\documentclass[12pt]{minimal}
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\begin{document}$$p,q\ge 1$$\end{document}, the p-Schatten norm of an operator A∈B(H)\documentclass[12pt]{minimal}
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\begin{document}$$A\in {{\mathcal {B}}}({{\mathcal {H}}})$$\end{document} is denoted by ‖A‖p:=(Tr|A|p)1/p\documentclass[12pt]{minimal}
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\begin{document}$$\Vert A\Vert _p:=(\mathop {{\hbox {Tr}}}\nolimits |A|^p)^{1/p}$$\end{document}, and the p→q\documentclass[12pt]{minimal}
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\begin{document}$$p\rightarrow q$$\end{document}-norm of a superoperator Λ:B(H)→B(H)\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda :{{\mathcal {B}}}({{\mathcal {H}}})\rightarrow {{\mathcal {B}}}({{\mathcal {H}}})$$\end{document} by ‖Λ‖p→q\documentclass[12pt]{minimal}
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\begin{document}$$\Vert \Lambda \Vert _{p\rightarrow q}$$\end{document}. Such a linear map is said to be unital if Λ(I)=I\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda (\mathbb {I})=\mathbb {I}$$\end{document}. Given two states ρ,σ∈D(H)\documentclass[12pt]{minimal}
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\begin{document}$$\rho ,\sigma \in {{\mathcal {D}}}({{\mathcal {H}}})$$\end{document}, the quantum relative entropy between ρ\documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document} and σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document} is defined as:D(ρ‖σ):=Tr(ρ(logρ-logσ))ifsupp(ρ)⊆supp(σ),+∞else.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {D}(\rho \Vert \sigma ):= \left\{ \begin{array}{ll} \mathop {{\hbox {Tr}}}\nolimits (\rho (\log \rho -\log \sigma )) &{}\quad {\text {if}}\,{\text {supp}}(\rho )\subseteq \mathop {\hbox {supp}}\nolimits (\sigma ),\\ +\infty &{}\quad {\text {else}}. \end{array} \right. \end{aligned}$$\end{document}
Quantum Markov Semigroups and the Detailed Balance Condition
In the Heisenberg picture, a quantum Markov semigroup (QMS) on a finite-dimensional Hilbert space H\documentclass[12pt]{minimal}
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\begin{document}$$\left( \Lambda _t\right) _{t \ge 0}$$\end{document} of linear, completely positive, unital maps on B(H)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {B}}}({\mathcal {H}})$$\end{document} satisfying the following properties
The parameter t plays the role of time. For each quantum Markov semigroup, there exists an operator L\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}$$\end{document} called the generator, or Lindbladian, of the semigroup, such thatddtΛt=Λt∘L=L∘Λt.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Lambda _t=\Lambda _t\circ \mathcal {L}=\mathcal {L}\circ \Lambda _t. \end{aligned}$$\end{document}In the Schrödinger picture, the dual of Λt\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda _t$$\end{document} is written Λ∗t\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda _{*t}$$\end{document}, for any t≥0\documentclass[12pt]{minimal}
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\begin{document}$$t\ge 0$$\end{document}. Similarly, we denote by L∗\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {L}}}_*$$\end{document} the dual of L\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {L}}}$$\end{document}. The QMS is said to be primitive (or ergodic) if there exists a unique invariant state σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}, i.e. such that Λ∗t(σ)=σ\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda _{*t}(\sigma )=\sigma $$\end{document}. Such a QMS is said to satisfy the detailed balance condition if the following holds:Tr(σL(X)∗Y)=Tr(σX∗L(Y)),X,Y∈B(H).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathop {{\hbox {Tr}}}\nolimits (\sigma \mathcal {L}(X)^*Y)=\mathop {{\hbox {Tr}}}\nolimits ( \sigma X^*\mathcal {L}(Y)), \quad X,Y\in {{\mathcal {B}}}({{\mathcal {H}}}). \end{aligned}$$\end{document}In the context of quantum logarithmic Sobolev inequalities (introduced later), the quantum Fisher information of ρ\documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document} with respect to the state σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}, first defined in [25], is particularly useful:Iσ(ρ):=-Tr(L∗(ρ)(logρ-logσ)),ρ∈D+(H)+∞,otherwise.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\text {I}}_\sigma (\rho ):=\left\{ \begin{array}{ll} -\mathop {{\hbox {Tr}}}\nolimits (\mathcal {L}_*(\rho )(\log \rho -\log \sigma ) ), &{}\quad \rho \in {{\mathcal {D}}}_+({{\mathcal {H}}})\\ +\infty ,&{}\quad \text {otherwise}. \end{array}\right. \end{aligned}$$\end{document}This quantity is also referred to as entropy production and denoted by EPσ\documentclass[12pt]{minimal}
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\begin{document}$${\text {EP}}_\sigma $$\end{document} in the literature. We will use both notations in what follows. The following theorem provides a structure for the generators of primitive QMS satisfying the detailed balance condition:
Theorem 1
([1, 7]). Let σ∈D+(H)\documentclass[12pt]{minimal}
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\begin{document}$$\sigma \in {{\mathcal {D}}}_+({{\mathcal {H}}})$$\end{document}, and let (Λt)t≥0\documentclass[12pt]{minimal}
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\begin{document}$$(\Lambda _t)_{t\ge 0}$$\end{document} be a quantum Markov semigroup on B(H)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {B}}}({{\mathcal {H}}})$$\end{document}. Suppose that the generator L\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}$$\end{document} of (Λt)t≥0\documentclass[12pt]{minimal}
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\begin{document}$$(\Lambda _t)_{t\ge 0}$$\end{document} satisfies the detailed balance condition with respect to a full-rank invariant state σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}. Then there exists an index set J\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {J}$$\end{document} of cardinality |J|≤d2-1\documentclass[12pt]{minimal}
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\begin{document}$$|\mathcal {J}|\le d^2-1$$\end{document}, where d=dim(H)\documentclass[12pt]{minimal}
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\begin{document}$$d=\dim ({{\mathcal {H}}})$$\end{document}, such that L\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}$$\end{document} takes the following form for any f∈B(H)\documentclass[12pt]{minimal}
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\begin{document}$$f\in {{\mathcal {B}}}({{\mathcal {H}}})$$\end{document}:L(f)=∑j∈Jcje-ωj/2L~j∗[f,L~j]+eωj/2[L~j,f]L~j∗\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathcal {L}(f)&=\sum _{j\in \mathcal {J}}c_j\left( \mathrm {e}^{-\omega _j/2}{\tilde{L}}_j^*[f,{\tilde{L}}_j]+\mathrm {e}^{\omega _j/2}[{\tilde{L}}_j,f]{\tilde{L}}_j^*\right) \end{aligned}$$\end{document}where ωj∈R\documentclass[12pt]{minimal}
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\begin{document}$$c_j>0$$\end{document} for all j∈J\documentclass[12pt]{minimal}
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\begin{document}$$\{{\tilde{L}}_j\}_{j\in \mathcal {J}}$$\end{document} is a set of operators in B(H)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {B}}}({{\mathcal {H}}})$$\end{document} with the properties:
In their celebrated paper [3] (see also [2]), Bakry and Emery found an elegant criterion which implies the logarithmic Sobolev inequality in the setting of diffusions. In this case of Markov semigroups defined on a Riemannian manifold M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document}, this criterion, called the Ricci lower bound, which is a special case of the Bakry–Emery condition, was shown later on to be equivalent to the so-called κ\documentclass[12pt]{minimal}
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\begin{document}$$\kappa $$\end{document}-displacement convexity of the relative entropy along geodesics in the Wasserstein space of probability measures on M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document} in [26]. This notion of κ\documentclass[12pt]{minimal}
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\begin{document}$$\kappa $$\end{document}-displacement convexity was extended to the framework of (necessarily non-diffusive) finite Markov chains by Maas in [19]. Carlen and Maas generalized this notion to the quantum regime in [7] and proved that it implies the modified logarithmic Sobolev inequality as well as the contractivity of the Wasserstein metric under the flow associated with the underlying quantum semigroup (Λt)t≥0\documentclass[12pt]{minimal}
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\begin{document}$$(\Lambda _t)_{t\ge 0}$$\end{document}. In their previous article [6], the same authors had already studied this quantum extension of the notion of κ\documentclass[12pt]{minimal}
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\begin{document}$$\kappa $$\end{document}-displacement convexity in the particular case of the fermionic Fokker–Planck equation. In this section, we provide a systematic analysis of the κ\documentclass[12pt]{minimal}
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\begin{document}$$({{\mathcal {D}}}_+({{\mathcal {H}}}),g_{\mathcal {L}})$$\end{document}.
Geodesic Equations
Similarly to Theorem 2.4 of [12], Carlen and Maas provided in [7] the set of faithful states D+(H)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {D}}}_+({{\mathcal {H}}})$$\end{document} with a Riemannian structure with associated Riemannian distance given by W2,L\documentclass[12pt]{minimal}
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\begin{document}$$W_{2,\mathcal {L}}$$\end{document}. Therefore, the local existence and uniqueness of constant speed geodesics is guaranteed by standard Riemannian geometry. We first recall that a constant speed geodesic (γ(s),U(s))s∈[0,1]\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} through Eq. (2.6), satisfies a Euler–Lagrange equation that we derive in Theorem 2. This result is a direct generalization of Theorem 5.3 in [6]. We start by recalling the abstract framework. Let (V,⟨.,.⟩)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}\subset \mathcal {W}_z$$\end{document} be a relatively open subset. Let D:M→B(W)\documentclass[12pt]{minimal}
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\begin{document}$$D:\mathcal {M}\rightarrow {{\mathcal {B}}}(\mathcal {W})$$\end{document} be a smooth function such that D(x) is self-adjoint and invertible for all x∈M\documentclass[12pt]{minimal}
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\begin{document}$$x\in \mathcal {M}$$\end{document}. We shall write C(x):=D(x)-1\documentclass[12pt]{minimal}
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\begin{document}$$C(x):=D(x)^{-1}$$\end{document}. Consider the Lagrangian L:W×M→R\documentclass[12pt]{minimal}
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\begin{document}$$L:\mathcal {W}\times \mathcal {M}\rightarrow \mathbb {R}$$\end{document} defined by L(p,x)=⟨C(x)p,p⟩\documentclass[12pt]{minimal}
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\begin{document}$$L(p,x)=\langle C(x) p,p\rangle $$\end{document} and the associated minimization problem:infu(.)∈C1([0,1],M)∫01L(u′(t),u(t))dt:u(0)=u0,u(1)=u1,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \inf _{u(.)\in C^1([0,1],\mathcal {M})}\left( \int _0^1 L(u'(t),u(t))dt:~~ u(0)=u_0,~ u(1)=u_1\right) , \end{aligned}$$\end{document}where u0,u1∈M\documentclass[12pt]{minimal}
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\begin{document}$$u_0, u_1\in \mathcal {M}$$\end{document} are given boundary values. Then, the Euler–Lagrange equations are equivalent to the following system of equations:u′(t)-D(u(t))v(t)=0,v′(t)+12⟨∂xD(u(t))v(t),v(t)⟩=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{aligned}&u'(t)-D(u(t))v(t)=0,\\&v'(t)+\frac{1}{2}\langle \partial _x D(u(t))v(t),v(t)\rangle =0. \end{aligned} \right. \end{aligned}$$\end{document}Here, we apply this abstract result to the case where V=Bsa(H)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {V}={{\mathcal {B}}}_{sa}({{\mathcal {H}}})$$\end{document}, with inner product ⟨.,.⟩\documentclass[12pt]{minimal}
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\begin{document}$$\langle .,. \rangle $$\end{document} the usual Hilbert–Schmidt inner product, W={A∈V:Tr(A)=0}\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {W}=\{A\in \mathcal {V}:~ \mathop {{\hbox {Tr}}}\nolimits (A)=0\}$$\end{document}, z:=I/dim(H)\documentclass[12pt]{minimal}
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\begin{document}$$z:=\mathbb {I}/\dim ({{\mathcal {H}}})$$\end{document}, and M=D+(H)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}={{\mathcal {D}}}_+({{\mathcal {H}}})$$\end{document}. Indeed, any density operator ρ\documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document} can be written as ρ=I/dimH+K\documentclass[12pt]{minimal}
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\begin{document}$$\rho =\mathbb {I}/\dim {{{\mathcal {H}}}}+K$$\end{document}, for some self-adjoint and traceless operator K. For any ρ∈D+(H)\documentclass[12pt]{minimal}
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\begin{document}$$\rho \in {{\mathcal {D}}}_+({{\mathcal {H}}})$$\end{document}, we already proved in Lemma 3 that Dω→(ρ):U↦-div([ρ]ω→∇U)\documentclass[12pt]{minimal}
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\begin{document}$$D_{\vec {\omega }}(\rho ): U\mapsto -{\text {div}}([\rho ]_{\vec {\omega }}\nabla U)$$\end{document} is invertible and self-adjoint. Now we use the following identity (see [6] p. 21):ddt(ρ+tA)αt=0=∫01∫0αρα-β(1-s)I+sρAρβ(1-s)I+sρdβds\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left. \frac{\mathrm{d}}{\mathrm{d}t} (\rho +t A)^{\alpha }\right| _{t=0}=\int _0^1 \int _0^\alpha \frac{\rho ^{\alpha -\beta }}{(1-s)\mathbb {I}+s\rho } A \frac{\rho ^\beta }{(1-s)\mathbb {I}+s \rho }\mathrm{d}\beta \mathrm{d}s \end{aligned}$$\end{document}for any 0<α<1\documentclass[12pt]{minimal}
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\begin{document}$$0<\alpha <1$$\end{document}, ρ∈D+(H)\documentclass[12pt]{minimal}
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\begin{document}$$\rho \in {{\mathcal {D}}}_+({{\mathcal {H}}})$$\end{document} and A∈W\documentclass[12pt]{minimal}
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\begin{document}$$A \in \mathcal {W}$$\end{document}. Hence for all A,U∈W\documentclass[12pt]{minimal}
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\begin{document}$$A,U\in \mathcal {W}$$\end{document},ddtt=0⟨Dω→(ρ+tA)[U],U⟩=ddtt=0∑j∈Jcj⟨∂jU,[ρ+tA]ωj∂jU⟩=ddtt=0∑j∈Jcj⟨∂jU,∫01eωj(1/2-α)(ρ+tA)α∂jU(ρ+tA)1-α⟩dα=⟨A,∇U.ρ∇U⟩,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left. \frac{\mathrm{d}}{\mathrm{d}t}\right| _{t=0} \langle D_{\vec {\omega }}(\rho +tA)[U],U\rangle&=\left. \frac{\mathrm{d}}{\mathrm{d}t}\right| _{t=0} \sum _{j\in {{\mathcal {J}}}} c_j \langle \partial _j U,[\rho +tA]_{\omega _j} \partial _j U\rangle \nonumber \\&=\left. \frac{\mathrm{d}}{\mathrm{d}t}\right| _{t=0}\sum _{j\in {{\mathcal {J}}}} c_j \langle \partial _j U,\int _0^1\mathrm {e}^{\omega _j(1/2-\alpha )} (\rho +t A)^\alpha \nonumber \\&\quad \partial _j U(\rho + tA)^{1-\alpha }\rangle \mathrm{d}\alpha \nonumber \\&=\langle A,\nabla U._\rho \nabla U\rangle , \end{aligned}$$\end{document}where for two vectors V→1,V→2\documentclass[12pt]{minimal}
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\begin{document}$$\vec {V}_1, \vec {V}_2$$\end{document} in ⨁jB(H)\documentclass[12pt]{minimal}
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\begin{document}$$\bigoplus _j {{\mathcal {B}}}({{\mathcal {H}}})$$\end{document},V→1.ρV→2:=∑j∈Jcj∫01∫01eωj(1/2-α)(χj(V→1,V→2∗,ρ,α,s)+χj(V→1∗,V→2,ρ,1-α,s))dαds,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \vec {V}_1._\rho \vec {V}_2&:= \sum _{j\in {{\mathcal {J}}}}c_j \int _0^1\int _0^1\mathrm {e}^{\omega _j(1/2-\alpha )}( \chi _j(\vec {V}_1,\vec {V}_2^*,\rho ,\alpha ,s)\nonumber \\&\quad +\chi _j(\vec {V}_1^*,\vec {V}_2,\rho ,1-\alpha ,s)) \mathrm{d}\alpha \mathrm{d}s, \end{aligned}$$\end{document}whereχj(V→1,V→2,ρ,α,s):=∫0αρβ(1-s)I+sρ(V1)jρ1-α(V2)jρα-β(1-s)I+sρdβ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \chi _j(\vec {V}_1,\vec {V}_2,\rho ,\alpha ,s):= \int _0^\alpha \frac{\rho ^\beta }{(1-s)\mathbb {I}+s\rho }(V_1)_j~\rho ^{1-\alpha }~(V_2)_j\frac{\rho ^{\alpha -\beta }}{(1-s)\mathbb {I}+s\rho }\mathrm{d}\beta . \end{aligned}$$\end{document}Therefore, in our context the Euler–Lagrange equations (3.1) reduce to the following:
Theorem 2
The geodesic equations in the Riemannian manifold (D+(H),W2,L)\documentclass[12pt]{minimal}
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\begin{document}$$({{\mathcal {D}}}_+({{\mathcal {H}}}),W_{2,\mathcal {L}})$$\end{document} are given byddsγ(s)+div([γ(s)]ω→∇U(s))=0,[2mm]dU(s)ds+12∇U(s).γ(s)∇U(s)=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}s}\gamma (s)+{\text {div}}([\gamma (s)]_{\vec {\omega }} \nabla U(s))=0,\\[2mm]&\frac{\mathrm{d}U(s)}{\mathrm{d}s}+\frac{1}{2} \nabla U(s)._{\gamma (s)} \nabla U(s)=0. \end{aligned} \right. \end{aligned}$$\end{document}
Different Formulations of Quantum κ\documentclass[12pt]{minimal}
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\begin{document}$$\kappa $$\end{document}-Displacement Convexity
In analogy with [12], we say that a primitive quantum Markov semigroup (Λt)t≥0\documentclass[12pt]{minimal}
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\begin{document}$$(\Lambda _t)_{t\ge 0}$$\end{document} with associated invariant state σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document} and generator L\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}$$\end{document} of the form of Eq. (2.3) has Ricci curvature bounded from below by a constant κ∈R\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \in \mathbb {R}$$\end{document} if the following inequality holds:
where (γ(s),U(s))s∈(-ε,ε)\documentclass[12pt]{minimal}
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\begin{document}$$(\gamma (s),U(s))_{s\in (-\varepsilon ,\varepsilon )}$$\end{document} is the unique solution to the geodesic equation (3.5) such that D+(H)∋ρ:=γ(0)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {D}}}_+({{\mathcal {H}}})\ni \rho :=\gamma (0)$$\end{document} and U(0)=U\documentclass[12pt]{minimal}
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\begin{document}$$U(0)=U$$\end{document}. We also refer to the above inequality as the quantum Ricci lower bound. Theorem 2 is useful to derive an expression for the second derivative of the relative entropy D(γ(s)‖σ)\documentclass[12pt]{minimal}
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\begin{document}$$D(\gamma (s)\Vert \sigma )$$\end{document} with respect to s, where (γ(s))s∈(-ε,ε)\documentclass[12pt]{minimal}
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\begin{document}$$(\gamma (s))_{s\in (-\varepsilon ,\varepsilon )}$$\end{document} is a constant speed geodesic with associated tangent vector ∇U(s)\documentclass[12pt]{minimal}
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\begin{document}$$\nabla U(s)$$\end{document} at each s. We already know from the gradient flow equation (2.12) thatddsD(γ(s)‖σ)=-gL,γ(s)(γ˙(s),L∗(γ(s)))=∑j∈Jcj⟨∂jU(s),[γ(s)]ωj∂j(logγ(s)-logσ)⟩=∑j∈Jcj⟨∂jU(s),[γ(s)]ωj(L~jlog(e-ωj/2γ(s))-log(eωj/2γ(s))L~j)⟩,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}s} D(\gamma (s)\Vert \sigma )\\&\quad =-g_{\mathcal {L},\gamma (s)}(\dot{\gamma }(s),\mathcal {L}_*(\gamma (s)))\\&\quad = \sum _{j\in {{\mathcal {J}}}} c_j \langle \partial _j U(s),[\gamma (s)]_{\omega _j}\partial _j(\log \gamma (s)-\log \sigma )\rangle \\&\quad =\sum _{j\in {{\mathcal {J}}}} c_j \langle \partial _j U(s),[\gamma (s)]_{\omega _j} ({\tilde{L}}_j\log (\mathrm {e}^{-\omega _j/2}\gamma (s))-\log (\mathrm {e}^{\omega _j/2}\gamma (s)){\tilde{L}}_j)\rangle , \end{aligned}$$\end{document}where the second line comes from Theorem 5.10 in [7], and the last identity comes from Lemma 5.9 of [7]. Now by identity (5.6) of the same paper,[γ(s)]ωj(L~jlog(e-ωj/2γ(s))-log(eωj/2γ(s))L~j)=e-ωj/2L~jγ(s)-eωj/2γ(s)L~j,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}{}[\gamma (s)]_{\omega _j} ({\tilde{L}}_j\log (\mathrm {e}^{-\omega _j/2}\gamma (s))-\log (\mathrm {e}^{\omega _j/2}\gamma (s)){\tilde{L}}_j)=\mathrm {e}^{-\omega _j/2}{\tilde{L}}_j\gamma (s)-\mathrm {e}^{\omega _j/2}\gamma (s){\tilde{L}}_j, \end{aligned}$$\end{document}so that we finally getddsD(γ(s)‖σ)=∑j∈Jcj⟨∂jU(s),e-ωj/2L~jγ(s)-eωj/2γ(s)L~j⟩.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}s}D(\gamma (s)\Vert \sigma )=\sum _{j\in {{\mathcal {J}}}} c_j \langle \partial _j U(s),\mathrm {e}^{-\omega _j/2}{\tilde{L}}_j\gamma (s)-\mathrm {e}^{\omega _j/2}\gamma (s){\tilde{L}}_j\rangle . \end{aligned}$$\end{document}Differentiating once more, we get:d2ds2D(γ(s)‖σ)s=0=∑j∈Jcj⟨∂jddsU(s)s=0,e-ωj/2L~jρ-eωj/2ρL~j⟩+⟨∂jU,e-ωj/2L~jγ˙(0)-eωj/2γ˙(0)L~j⟩.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left. \frac{\mathrm{d}^2}{\mathrm{d}s^2}D(\gamma (s)\Vert \sigma )\right| _{s=0}=&\sum _{j\in {{\mathcal {J}}}} c_j ~\left\{ \langle \partial _j \left. \frac{\mathrm{d}}{\mathrm{d}s}U(s)\right| _{s=0},\mathrm {e}^{-\omega _j/2} {\tilde{L}}_j\rho -\mathrm {e}^{\omega _j/2}\rho {\tilde{L}}_j\rangle \right. \nonumber \\&\left. +\langle \partial _j U,\mathrm {e}^{-\omega _j/2}{\tilde{L}}_j \dot{\gamma }(0)-\mathrm {e}^{\omega _j/2}\dot{\gamma }(0) {\tilde{L}}_j\rangle \right\} . \end{aligned}$$\end{document}We first take care of the second line of Eq. (3.6). Using Theorem 2 as well as Eq. (2.3), we find⟨∂jU,e-ωj/2L~jγ˙(0)-eωj/2γ˙(0)L~j⟩=-⟨∂jU,e-ωj/2L~jdiv([ρ]ω→∇U)-eωj/2div([ρ]ω→∇U)L~j⟩=-⟨∂jU,e-ωj/2L~j∑k∈Jck[[ρ]ωk∂kU,L~k∗]-eωj/2∑k∈Jck[[ρ]ωk∂kU,L~k∗]L~j⟩=∑k∈Jcke-ωj/2⟨∂k(L~j∗∂jU),[ρ]ωk∂kU⟩-eωj/2⟨∂k(∂jUL~j∗),[ρ]ωk∂kU⟩=∑k∈Jck⟨∂ke-ωj/2L~j∗∂jU-eωj/2∂jUL~j∗,[ρ]ωk∂kU⟩.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\langle \partial _j U,\mathrm {e}^{-\omega _j/2}{\tilde{L}}_j \dot{\gamma }(0)-\mathrm {e}^{\omega _j/2}\dot{\gamma }(0) {\tilde{L}}_j\rangle \\&\quad = -\langle \partial _j U,\mathrm {e}^{-\omega _j/2}{\tilde{L}}_j {\text {div}}([\rho ]_{\vec {\omega }} \nabla U)-\mathrm {e}^{\omega _j/2}{\text {div}}([\rho ]_{\vec {\omega }} \nabla U){\tilde{L}}_j\rangle \\&\quad = -\langle \partial _j U,\mathrm {e}^{-\omega _j/2}{\tilde{L}}_j\sum _{k\in {{\mathcal {J}}}} c_k[[\rho ]_{\omega _k} \partial _k U, {\tilde{L}}_k^* ]-\mathrm {e}^{\omega _j/2} \sum _{k\in {{\mathcal {J}}}} c_k [[\rho ]_{\omega _k} \partial _k U,{\tilde{L}}_k^* ]{\tilde{L}}_j\rangle \\&\quad =\sum _{k\in {{\mathcal {J}}}} c_k \left( \mathrm {e}^{-\omega _j/2}\langle \partial _k ({\tilde{L}}_j^* \partial _j U),[\rho ]_{\omega _k}\partial _k U\rangle -\mathrm {e}^{\omega _j/2}\langle \partial _k ( \partial _j U {\tilde{L}}_j^*),[\rho ]_{\omega _k}\partial _k U\rangle \right) \\&\quad = \sum _{k\in {{\mathcal {J}}}} c_k \langle \partial _k \left( \mathrm {e}^{-\omega _j/2} {\tilde{L}}_j^* \partial _j U-\mathrm {e}^{\omega _j/2} \partial _j U {\tilde{L}}_j^*\right) ,[\rho ]_{\omega _k}\partial _k U\rangle . \end{aligned}$$\end{document}Hence by (2.3),∑j∈Jcj⟨∂jU,e-ωj/2L~jγ˙(0)-eωj/2γ˙(0)L~j⟩=-∑kck⟨∂kL(U),[ρ]ωk∂kU⟩=-⟨∇L(U),∇U⟩L,ρ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sum _{j\in {{\mathcal {J}}}} c_j \langle \partial _j U,\mathrm {e}^{-\omega _j/2}{\tilde{L}}_j \dot{\gamma }(0)-\mathrm {e}^{\omega _j/2}\dot{\gamma }(0) {\tilde{L}}_j\rangle&= -\sum _k c_k \langle \partial _k \mathcal {L}(U),[\rho ]_{\omega _k}\partial _k U\rangle \nonumber \\&=-\langle \nabla \mathcal {L}(U),\nabla U\rangle _{\mathcal {L},\rho }. \end{aligned}$$\end{document}By (3.5), the first line of (3.6) is equal to12∑j∈Jcj⟨∂j(∇U.ρ∇U),eωj/2ρL~j-e-ωj/2L~jρ⟩=12∑j∈Jcj⟨∇U.ρ∇U,[L~j∗,ρL~j]eωj/2-e-ωj/2[L~j∗,L~jρ]⟩=12⟨∇U.ρ∇U,L∗(ρ)⟩,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\frac{1}{2} \sum _{j\in {{\mathcal {J}}}} c_j \langle \partial _j (\nabla U._{\rho }\nabla U),\mathrm {e}^{\omega _j/2} \rho {\tilde{L}}_j-\mathrm {e}^{-\omega _j/2} {\tilde{L}}_j\rho \rangle \nonumber \\&\quad =\frac{1}{2} \sum _{j\in {{\mathcal {J}}}} c_j \langle \nabla U._\rho \nabla U,[{\tilde{L}}_j^*,\rho {\tilde{L}}_j]\mathrm {e}^{\omega _j/2}-\mathrm {e}^{-\omega _j/2} [{\tilde{L}}_j^*,{\tilde{L}}_j\rho ]\rangle \nonumber \\&\quad = \frac{1}{2} \langle \nabla U._{\rho } \nabla U, \mathcal {L}_*(\rho )\rangle , \end{aligned}$$\end{document}where we used that, replacing L~j\documentclass[12pt]{minimal}
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\begin{document}$${\tilde{L}}_j$$\end{document} by L~j∗\documentclass[12pt]{minimal}
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\begin{document}$${\tilde{L}}_j^*$$\end{document} so that ωj→-ωj\documentclass[12pt]{minimal}
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\begin{document}$$\omega _j\rightarrow -\omega _j$$\end{document} and cj→cj\documentclass[12pt]{minimal}
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\begin{document}$$c_j\rightarrow c_j$$\end{document},L∗(ρ)=∑j∈Jcjeωj/2[L~j∗ρ,L~j]+e-ωj/2[L~j,ρL~j∗]=∑j∈Jcje-ωj/2[L~jρ,L~j∗]+eωj/2[L~j∗,ρL~j],\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathcal {L}_*(\rho )&=\sum _{j\in {{\mathcal {J}}}} c_j \left( \mathrm {e}^{\omega _j/2} [{\tilde{L}}_j^*\rho ,{\tilde{L}}_j]+\mathrm {e}^{-\omega _j/2}[{\tilde{L}}_j,\rho {\tilde{L}}_j^*] \right) \\&= \sum _{j\in {{\mathcal {J}}}} c_j \left( \mathrm {e}^{-\omega _j/2} [{\tilde{L}}_j\rho ,{\tilde{L}}_j^*]+\mathrm {e}^{\omega _j/2} [{\tilde{L}}_j^*,\rho {\tilde{L}}_j]\right) , \end{aligned}$$\end{document}Hence, using (3.7) and (3.8), (3.6) reduces tod2ds2D(γ(s)‖σ)s=0=12⟨∇U.ρ∇U,L∗(ρ)⟩-⟨∇L(U),∇U⟩L,ρ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left. \frac{\mathrm{d}^2}{\mathrm{d}s^2}D(\gamma (s)\Vert \sigma )\right| _{s=0}&=\frac{1}{2} \langle \nabla U._{\rho }\nabla U,\mathcal {L}_*(\rho )\rangle -\langle \nabla \mathcal {L}(U),\nabla U\rangle _{\mathcal {L},\rho }. \end{aligned}$$\end{document}One can compare this expression with the one derived in Proposition 4.3 of [12]. To make this analogy more clear, we denote the quantity on the right-hand side of Eq. (3.9) by B(ρ,U)\documentclass[12pt]{minimal}
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\begin{document}$$B(\rho ,U)$$\end{document} so thatd2ds2D(γ(s)‖σ)s=0=B(ρ,U).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left. \frac{\mathrm{d}^2}{\mathrm{d}s^2}D(\gamma (s)\Vert \sigma )\right| _{s=0}= B(\rho ,U). \end{aligned}$$\end{document}The following lemma extends Lemma 4.6 of [12] to the quantum regime, as well as part of the proof of Proposition 5.11 of [6], and is proven to be useful in what follows:
Let L\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}$$\end{document} be the generator of an ergodic QMS (Λt)t≥0\documentclass[12pt]{minimal}
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\begin{document}$$(\Lambda _t)_{t\ge 0}$$\end{document}, with unique invariant state σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}, of the form of Eq. 2.3). Then, for κ∈R\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \in \mathbb {R}$$\end{document}, the following are equivalent:
The proof is inspired by the one of Theorem 4.5 of [12]. That (i)⇔(ii)\documentclass[12pt]{minimal}
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\begin{document}$$(i)\Leftrightarrow (ii)$$\end{document} follows from Eq. (3.10). We use Lemma 5 to show that (ii)⇒(iii)\documentclass[12pt]{minimal}
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\begin{document}$$(ii)\Rightarrow (iii)$$\end{document}: Take a smooth path (γ(s),U(s))s∈[0,1]\documentclass[12pt]{minimal}
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\begin{document}$$(\gamma (s),U(s))_{s\in [0,1]}$$\end{document} such that γ(0)=ω\documentclass[12pt]{minimal}
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\begin{document}$$\gamma (0)=\omega $$\end{document}, γ(1)=ρ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma (1)=\rho $$\end{document} and∫01‖γ˙(s)‖gL,γ(s)2ds≤W2,L(ρ,ω)2+ε.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \int _0^1 \Vert \dot{\gamma }(s)\Vert _{g_{\mathcal {L},\gamma (s)}}^2\mathrm{d}s\le W_{2,\mathcal {L}}(\rho ,\omega )^2+\varepsilon . \end{aligned}$$\end{document}With the notations of Lemma 5,12∂te2κst‖∂sγ(s,t)‖gL,γ(s,t)2+∂se2κstD(γ(s,t)‖σ)≤2κte2κstD(γ(s,t)‖σ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \frac{1}{2}\partial _t \left( \mathrm {e}^{2\kappa s t}\Vert \partial _s \gamma (s,t) \Vert _{g_{\mathcal {L},\gamma (s,t)}}^2 \right) +\partial _s \left( \mathrm {e}^{2\kappa s t}D(\gamma (s,t)\Vert \sigma ) \right) \le 2\kappa t\mathrm {e}^{2\kappa s t }D(\gamma (s,t)\Vert \sigma ). \end{aligned}$$\end{document}Integrating with respect to t∈[0,h]\documentclass[12pt]{minimal}
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\begin{document}$$t\in [0,h]$$\end{document}, for some h>0\documentclass[12pt]{minimal}
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\begin{document}$$h>0$$\end{document}, and s∈[0,1]\documentclass[12pt]{minimal}
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\begin{document}$$s\in [0,1]$$\end{document},12∫01e2κsh‖∂sγ(s,h)‖gL,γ(s,h)2-‖∂sγ(s,0)‖gL,γ(s,0)2ds\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\frac{1}{2}\int _0^1 \left( \mathrm {e}^{2\kappa s h}\Vert \partial _s \gamma (s,h) \Vert _{g_{\mathcal {L},\gamma (s,h)}}^2 -\Vert \partial _s \gamma (s,0)\Vert _{g_{\mathcal {L},\gamma (s,0)}}^2 \right) \mathrm{d}s \end{aligned}$$\end{document}+∫0he2κtD(γ(1,t)‖σ)-D(γ(0,t)‖σ)dt≤2κ∫01ds∫0hdtte2κstD(γ(s,t)‖σ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\qquad +\int _0^h \left( \mathrm {e}^{2\kappa t} D(\gamma (1,t)\Vert \sigma )- D(\gamma (0,t)\Vert \sigma ) \right) \mathrm{d}t\nonumber \\&\quad \le 2\kappa \int _0^1\mathrm{d}s\int _0^h \mathrm{d}t~ t\,\mathrm {e}^{2\kappa st} D(\gamma (s,t)\Vert \sigma ). \end{aligned}$$\end{document}The following inequality, for which a classical equivalent is given in the proof of Theorem 4.5 of [12], can be derived similarly to Lemma 5.1 of [9]:m(κh)W2,L(ρh,ω)2≤∫01e2κsh‖∂sγ(s,h)‖gL,γ(s,h)2ds,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} m(\kappa h)W_{2,\mathcal {L}}(\rho _h,\omega )^2\le \int _0^1 \mathrm {e}^{2\kappa s h} \Vert \partial _s \gamma (s,h) \Vert _{g_{\mathcal {L},\gamma (s,h)}}^2 \mathrm{d}s, \end{aligned}$$\end{document}where m(x):=xex/sinh(x)\documentclass[12pt]{minimal}
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\begin{document}$$m(x):= x\mathrm {e}^x/\sinh (x)$$\end{document}. Indeed, define f:s↦e2κsh\documentclass[12pt]{minimal}
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\begin{document}$$f:s\mapsto \mathrm {e}^{2\kappa s h}$$\end{document} and denote Lf:=∫011f(s)ds\documentclass[12pt]{minimal}
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\begin{document}$$L_{f}:=\int _0^1 \frac{1}{f(s)}\mathrm{d}s$$\end{document}. Then, let g:[0,1]↦[0,1]\documentclass[12pt]{minimal}
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\begin{document}$$g:[0,1]\mapsto [0,1]$$\end{document} be the smooth increasing map defined as g(s)=Lf-1∫0s1f(u)du\documentclass[12pt]{minimal}
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\begin{document}$$g(s)=L_{f}^{-1}\int _0^s\frac{1}{f(u)}\mathrm{d}u$$\end{document} and denote its inverse k such that k′(g(s))=Lff(s).\documentclass[12pt]{minimal}
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\begin{document}$$k'(g(s))=L_{f}f(s).$$\end{document} Then define the reparametrized curve (γ(k(r),h),k′(r)U(k(r),h))r∈[0,1]\documentclass[12pt]{minimal}
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\begin{document}$$(\gamma (k(r),h ),\,k'(r)\,U(k(r),h) )_{r\in [0,1]}$$\end{document} which satisfies the continuity equation:∂rγ(k(r),h)=k′(r)∂1γ(k(r),h)=-k′(r)div([γ(k(r),h)]ω→∇U(k(r),h)),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \partial _r \gamma (k(r),h)&=k'(r) \partial _1 \gamma (k(r),h)\\&=-k'(r) {\text {div}}([\gamma (k(r),h)]_{\vec {\omega }}\nabla U(k(r),h) ), \end{aligned}$$\end{document}where we used Eq. (3.11) in order to establish the second line. This curve satisfies γ(k(0),h)=ω\documentclass[12pt]{minimal}
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\begin{document}$$\gamma (k(0),h)=\omega $$\end{document} and γ(k(1),h)=ρh\documentclass[12pt]{minimal}
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\begin{document}$$\gamma (k(1),h)=\rho _h$$\end{document}, so thatW2,L(ρh,ω)2≤∫01‖∂rγ(k(r),h)‖gL,γ(k(r),h)2dr=∫01k′(r)2‖∇U(k(r),h)‖L,γ(k(r),h)2dr=∫01k′(g(s))‖∇U(s,h)‖L,γ(s,h)2ds=Lf∫01f(s)‖∂sγ(s,h)‖gL,γ(s,h)2ds,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} W_{2,\mathcal {L}}(\rho _h,\omega )^2&\le \int _0^1 \Vert \partial _r \gamma (k(r),h)\Vert ^2_{g_{\mathcal {L},\gamma (k(r),h)}}\mathrm{d}r\\&= \int _0^1 k'(r)^2 \Vert \nabla U(k(r),h)\Vert ^2_{\mathcal {L},\gamma (k(r),h)}\mathrm{d}r\\&= \int _0^1 k'(g(s)) \Vert \nabla U(s,h)\Vert _{\mathcal {L},\gamma (s,h)}^2\mathrm{d}s\\&= L_f \int _0^1 f(s)\Vert \partial _s \gamma (s,h)\Vert _{g_{\mathcal {L},\gamma (s,h)}}^2\mathrm{d}s, \end{aligned}$$\end{document}which directly leads to (3.20). This inequality, together with (3.17), impliesm(hκ)2W2,L(ρh,ω)2-12W2,L(ρ,ω)2-ε+∫0he2κtdtD(ρh‖σ)-hD(ω‖σ)≤12∫01e2κsh‖∂sγ(s,t)‖gL,γ(s,t)2ds-12∫01‖γ˙(s)‖gL,γ(s)2ds+∫0he2κtD(ρt‖σ)dt-hD(ω‖σ)≤2κ∫01∫0hte2κstD(γ(s,t)‖σ)dtds.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\frac{m(h\kappa )}{2}W_{2,\mathcal {L}}(\rho _h,\omega )^2-\frac{1}{2}W_{2,\mathcal {L}}(\rho ,\omega )^2-\varepsilon +\int _0^h\mathrm {e}^{2\kappa t} \mathrm{d}t~D(\rho _h\Vert \sigma )-hD(\omega \Vert \sigma )\\&\quad \le \frac{1}{2}\int _0^1 \mathrm {e}^{2\kappa s h}\Vert \partial _s \gamma (s,t)\Vert _{g_{\mathcal {L},\gamma (s,t)}}^2 \mathrm{d}s-\frac{1}{2}\int _0^1 \Vert \dot{\gamma }(s)\Vert _{g_{\mathcal {L},\gamma (s)}}^2\mathrm{d}s\\&\qquad +\int _0^h \mathrm {e}^{2\kappa t} D(\rho _t\Vert \sigma )~\mathrm{d}t-hD(\omega \Vert \sigma )\\&\quad \le 2\kappa \int _0^1\int _0^h ~t\mathrm {e}^{2\kappa st }D(\gamma (s,t)\Vert \sigma )\,\mathrm{d}t\,\mathrm{d}s. \end{aligned}$$\end{document}where, in the first inequality, we also used the monotonicity of the relative entropy so that D(ρh‖σ)=D(ρh‖Λ∗hσ)≤D(ρt‖Λ∗tσ)=D(ρt‖σ)\documentclass[12pt]{minimal}
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\begin{document}$$D(\rho _h\Vert \sigma )=D(\rho _h\Vert \Lambda _{*h}\sigma )\le D(\rho _t\Vert \Lambda _{*t}\sigma )=D(\rho _t\Vert \sigma )$$\end{document}, and in the second one that for all t>0\documentclass[12pt]{minimal}
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\begin{document}$$t>0$$\end{document}, γ(1,t)=ρt\documentclass[12pt]{minimal}
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\begin{document}$$\gamma (1,t)=\rho _t$$\end{document}, γ(0,t)=ω\documentclass[12pt]{minimal}
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\begin{document}$$\gamma (0,t)=\omega $$\end{document}, as well as (3.19). Since for all s∈[0,1]\documentclass[12pt]{minimal}
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\begin{document}$$s\in [0,1]$$\end{document}, t↦D(γ(s,t)‖σ)\documentclass[12pt]{minimal}
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\begin{document}$$t\mapsto D(\gamma (s,t)\Vert \sigma )$$\end{document} is bounded,limh→01h∫01∫0hte2κstD(γ(s,t)‖σ)dtds=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \lim _{h\rightarrow 0} \frac{1}{h}\int _0^1\int _0^h ~t\mathrm {e}^{2\kappa st} D(\gamma (s,t)\Vert \sigma )\mathrm{d}t~\mathrm{d}s=0. \end{aligned}$$\end{document}Moreover,limh→01h∫0he2κtdtD(ρh‖σ)-hD(ω‖σ)=D(ρ‖σ)-D(ω‖σ)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \lim _{h\rightarrow 0}\frac{1}{h}\left( \int _0^h \mathrm {e}^{2\kappa t}\mathrm{d}t~D(\rho _h\Vert \sigma )-hD(\omega \Vert \sigma )\right) =D(\rho \Vert \sigma )-D(\omega \Vert \sigma ) \end{aligned}$$\end{document}Since ε>0\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon >0$$\end{document} is arbitrary, we arrive atddhh=0+m(κh)2W2,L(ρh,ω)2+D(ρ‖σ)-D(ω‖σ)≤0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left. \frac{\mathrm{d}}{\mathrm{d}h}\right| _{h=0^+}\left( \frac{m(\kappa h)}{2}W_{2,\mathcal {L}}(\rho _h,\omega )^2\right) +D(\rho \Vert \sigma )-D(\omega \Vert \sigma )\le 0. \end{aligned}$$\end{document}The result for t=0\documentclass[12pt]{minimal}
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\begin{document}$$t=0$$\end{document} follows from the fact that the first term in the left-hand side above is equal to κ2W2,L(ρh,ω)2+12ddhh=0+W2,L(ρh,ω)2\documentclass[12pt]{minimal}
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\begin{document}$$\frac{\kappa }{2} W_{2,\mathcal {L}}(\rho _h,\omega )^2+\frac{1}{2}\left. \frac{\mathrm{d}}{\mathrm{d}h}\right| _{h=0^+}W_{2,\mathcal {L}}(\rho _h,\omega )^2$$\end{document}. The case t≥0\documentclass[12pt]{minimal}
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\begin{document}$$t\ge 0$$\end{document} directly follows from the case t=0\documentclass[12pt]{minimal}
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\begin{document}$$t=0$$\end{document}.
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\begin{document}$$(iii)\Rightarrow (iv)$$\end{document} follows from Theorem 3.3 of [9] together with the fact that (D(H),W2,L)\documentclass[12pt]{minimal}
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\begin{document}$$({{\mathcal {D}}}({{\mathcal {H}}}),W_{2,\mathcal {L}})$$\end{document} is complete (cf. Proposition 2).
(iv)⇒(v)\documentclass[12pt]{minimal}
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\begin{document}$$(iv)\Rightarrow (v)$$\end{document} follows directly from Theorem 3.2 of [9].
(v)⇒(i)\documentclass[12pt]{minimal}
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\begin{document}$$(v)\Rightarrow (i)$$\end{document} can easily be proved as follows: let 0<ε<ε′\documentclass[12pt]{minimal}
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\begin{document}$$0<\varepsilon <\varepsilon '$$\end{document}, and without loss of generality, let γ:(-ε′,ε′)→D+(H)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma :(-\varepsilon ',\varepsilon ')\rightarrow {{\mathcal {D}}}_+({{\mathcal {H}}})$$\end{document} be speed 1 geodesic, and that γ(0)=ρ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma (0)=\rho $$\end{document}. Then, construct the following constant speed geodesic γ~:[0,1]→D+(H)\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{\gamma }:[0,1]\rightarrow {{\mathcal {D}}}_+({{\mathcal {H}}})$$\end{document} as follows: for any s∈[0,1]\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{\gamma }(s):= \gamma (2\varepsilon s-\varepsilon )$$\end{document}. It then follows that W2,L(γ~(0),γ~(1))=2ε\documentclass[12pt]{minimal}
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\begin{document}$$W_{2,\mathcal {L}}(\tilde{\gamma }(0),\tilde{\gamma }(1))=2\varepsilon $$\end{document}. Moreover, by applying (3.16) to γ~\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{\gamma }$$\end{document}, we find, after a suitable rearrangement of the terms:D(γ(ε)‖σ)-2D(ρ‖σ)+D(γ(-ε)‖σ)ε2≥κ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \frac{D(\gamma (\varepsilon )\Vert \sigma )-2D(\rho \Vert \sigma )+D(\gamma (-\varepsilon )\Vert \sigma )}{\varepsilon ^2}\ge \kappa . \end{aligned}$$\end{document}The result follows after taking the limit ε→0\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon \rightarrow 0$$\end{document}. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Other Equivalent Formulations of Displacement Convexity
Here, we provide other characterizations of the Ricci curvature lower bound in terms of some contraction properties of the Wasserstein metric along the semigroup (Pt)t≥0\documentclass[12pt]{minimal}
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\begin{document}$$(\mathcal {P}_t)_{t\ge 0}$$\end{document}. In the next theorem, the characterization of displacement convexity in terms of gradient estimates can be interpreted as a non-commutative version of Bakry–Émery’s original gradient bound (see Theorem 4.7.2 of [4]): in particular they showed that the Ricci curvature lower bound is equivalent to the following pointwise inequality for smooth enough functions:Γ(Pt(f),Pt(f))≤e-2κtPt(Γ(f,f)),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varGamma (P_t(f),P_t(f))\le \mathrm {e}^{-2\kappa t}P_t(\varGamma (f,f))\,, \end{aligned}$$\end{document}where Γ\documentclass[12pt]{minimal}
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\begin{document}$$\varGamma $$\end{document} stands for the carré du champ operator:Γ(f,g):=∇f.∇g.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varGamma (f,g):=\nabla f.\nabla g\,. \end{aligned}$$\end{document}
In the commutative diffusive setting, the Ricci curvature lower bound is also known to be equivalent to the contraction of the Wasserstein distance along the semigroup (Pt)t≥0\documentclass[12pt]{minimal}
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\begin{document}$$(P_t)_{t\ge 0}$$\end{document} (see Theorem 9.7.2 of [4]):W2(Pt∗(ν),Pt∗(ν′))≤e-κtW2(ν,ν′).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} W_2(P_{t*}(\nu ),\,P_{t*}(\nu '))\le \mathrm {e}^{-\kappa t}W_2(\nu ,\nu ')\,. \end{aligned}$$\end{document}This still holds true in the non-commutative, finite-dimensional setting:
Proposition 4
For any κ∈R\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \in \mathbb {R}$$\end{document}, Ric(L)≥κ\documentclass[12pt]{minimal}
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\begin{document}$${\text {Ric}}(\mathcal {L})\ge \kappa $$\end{document} is equivalent to the contraction of the Wasserstein distance along the flow generated by (Λt)t≥0\documentclass[12pt]{minimal}
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\begin{document}$$(\Lambda _t)_{t\ge 0}$$\end{document}: for any ρ,ω∈D+(H)\documentclass[12pt]{minimal}
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\begin{document}$$\rho ,\omega \in {{\mathcal {D}}}_+({{\mathcal {H}}})$$\end{document}W2,L(Λt∗(ρ),Λt∗(ω))≤e-κtW2,L(ρ,ω).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} W_{2,{{\mathcal {L}}}}(\Lambda _{t*}(\rho ),\Lambda _{t*}(\omega ))\le \mathrm {e}^{-\kappa t}W_{2,{{\mathcal {L}}}}(\rho ,\omega )\,. \end{aligned}$$\end{document}
Proof
The direct implication follows from Proposition 3.1 of [9] and Theorem 3(iii). The reverse implication is proved as in inequality (2.12) of [9], using the smooth Riemannian structure provided by (D+(H),W2,L)\documentclass[12pt]{minimal}
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\begin{document}$$({{\mathcal {D}}}_+({{\mathcal {H}}}),W_{2,{{\mathcal {L}}}})$$\end{document} in the finite-dimensional case. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
In the commutative diffusive setting, the contraction of (3.21) is actually known to be equivalent to its “square root” version, usually referred to as the strong gradient bound:Γ(Pt(f),Pt(f))≤e-κtPt(Γ(f,f)).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sqrt{ \varGamma (P_t(f),P_t(f))}\le \mathrm {e}^{-\kappa t}P_t(\sqrt{\varGamma (f,f)})\,. \end{aligned}$$\end{document}The proof of (3.24)⇒\documentclass[12pt]{minimal}
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\begin{document}$$\Rightarrow $$\end{document}(3.21) follows by a simple use of Jensen’s inequality, the converse being the content of Theorem 3.3.18 of [4]. The advantage of this formulation arises from the fact that some canonical semigroups (e.g. the quantum Ornstein–Uhlenbeck semigroup on Rn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^n$$\end{document}) saturate the inequality, or equivalently:L,∇=κ∇.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left[ L,\,\nabla \right] =\kappa \nabla \,. \end{aligned}$$\end{document}Therefore, the Ricci lower bound is equivalent to comparing the commutation of a semigroup with the gradient to the one of a canonical semigroup. A similar reasoning recently lead [13] to formulate a Bakry–Émery condition for birth and death processes on N\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {N}$$\end{document} in terms of a comparison to the Poisson process. Going back to our non-commutative setting, [7] showed that the quantum Ornstein–Uhlenbeck semigroup, as well as its fermionic version on the Clifford algebra, does satisfy such a commutation relation. They used this fact to derive the modified logarithmic Sobolev constant for these QMS via the contraction (3.23). In the next proposition, we recall their argument:
Proposition 5
Assume that the following equalities hold: there exists κ∈R\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \in \mathbb {R}$$\end{document} such that, for any j∈J\documentclass[12pt]{minimal}
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\begin{document}$$j\in {{\mathcal {J}}}$$\end{document} and any t≥0\documentclass[12pt]{minimal}
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\begin{document}$$t\ge 0$$\end{document},∂j∘Λt=e-κtΛt∘∂j.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \partial _{j}\circ \Lambda _t= \mathrm {e}^{-\kappa t} \Lambda _t\circ \partial _{j}\,. \end{aligned}$$\end{document}Then, Ric(L)≥κ\documentclass[12pt]{minimal}
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\begin{document}$${\text {Ric}}(\mathcal {L})\ge \kappa $$\end{document} holds.
Proof
From Proposition 4, it is enough to prove that (3.23) holds. Assume that (γ(s))s∈[0,1]\documentclass[12pt]{minimal}
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\begin{document}$$(\gamma (s))_{s\in [0,1]}$$\end{document} is a minimal geodesic relating ρ\documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document} to σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document} and denote by (A(s))s∈[0,1]\documentclass[12pt]{minimal}
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\begin{document}$$(\mathbf {A}(s))_{s\in [0,1]}$$\end{document} the unique solution of the continuity equationγ˙(s)=divA(s).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \dot{\gamma }(s)={\text {div}}\mathbf {A}(s)\,. \end{aligned}$$\end{document}By duality, Eq. (3.25) implies that Λt∗γ˙(s)=e-κtdivΛ→t∗A(s)\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda _{t*}\dot{\gamma }(s)=\mathrm {e}^{-\kappa t}{\text {div}}\vec {\Lambda }_{t*}\mathbf {A}(s)$$\end{document}, where Λ→t∗A:=(Λt∗Aj)j∈J\documentclass[12pt]{minimal}
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\begin{document}$$\vec {\Lambda }_{t*}\mathbf {A}:=(\Lambda _{t*}A_j)_{j\in {{\mathcal {J}}}}$$\end{document}. Then, denoting γ(s,t):=Λt∗(γ(s))\documentclass[12pt]{minimal}
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\begin{document}$$\gamma (s,t):=\Lambda _{t*}(\gamma (s))$$\end{document},gL,γ(s,t)(γ˙(s,t),γ˙(s,t))=e-2κt∑j∈J⟨P∗t(Aj(s)),[γ(s,t)]ωj-1(Pt∗(Aj(s)))⟩≤e-2κt∑j∈J⟨Aj(s),[γ(s)]ωj-1Aj(s))⟩=e-2κtgL,γ(s)(γ˙(s),γ˙(s)),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} g_{{{\mathcal {L}}},\gamma (s,t)}(\dot{\gamma }(s,t),\dot{\gamma }(s,t))&=\mathrm {e}^{-2\kappa t}\sum _{j\in {{\mathcal {J}}}}\,\langle \mathcal {P}_{*t}(A_j(s)),\,[\gamma (s,t)]^{-1}_{\omega _j}(\mathcal {P}_{t*}(A_j(s)))\rangle \\&\le \mathrm {e}^{-2\kappa t}\sum _{j\in {{\mathcal {J}}}}\,\langle A_j(s),\,[\gamma (s)]^{-1}_{\omega _j}A_j(s))\rangle \\&=\mathrm {e}^{-2\kappa t} g_{{{\mathcal {L}}},\gamma (s)}(\dot{\gamma }(s),\dot{\gamma }(s))\,, \end{aligned}$$\end{document}where the inequality arises from the property of monotonicity of Fisher information metrics (cf. [17]). The result follows after taking the integral over the geodesic path. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Example: The Quantum Depolarizing Semigroup
In this section, we derive a Ricci curvature lower bound on perhaps the simplest possible QMS: the depolarizing semigroup: define (Λtdep)t≥0\documentclass[12pt]{minimal}
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\begin{document}$$(\Lambda _t^\mathrm{dep})_{t\ge 0}$$\end{document} on B(Cd)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Lambda ^\mathrm{dep}_t(X)=\mathrm {e}^{-t}X\,+(1-\mathrm {e}^{-t})\frac{1}{d}\mathop {{\hbox {Tr}}}\nolimits (X)\,\mathbb {I}\,. \end{aligned}$$\end{document}
In [7] it was proved that, in the case when κ>0\documentclass[12pt]{minimal}
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\begin{document}$$\kappa =\alpha _1$$\end{document}. This is, for example, the case of the classical and quantum Ornstein–Uhlenbeck processes. Here, we study the case of κ∈R\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \in \mathbb {R}$$\end{document}. In [12], the authors proved that, in the classical discrete framework, Ric(L)≥κ\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \in \mathbb {R}$$\end{document} implies an HWI-like inequality (see Theorem 7.3). Here, we provide a quantum generalization of their result.
Theorem 5
Assume that Ric(L)≥κ\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}$$\end{document} satisfies the following inequality
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\begin{document}$$\kappa >0$$\end{document}, we recover the result of [7]:
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\begin{document}$$(\kappa )$$\end{document} still implies a modified log-Sobolev inequality under the further condition that a transportation cost inequality holds. This is a direct quantum generalization of Theorem 7.8 of [12] (see also Corollary 3.1 of [23])
The proof is identical to the one of Corollary 3.1 of [23]. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Similarly, we can show that Ric(L)≥κ\documentclass[12pt]{minimal}
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\begin{document}$$\kappa \in \mathbb {R}$$\end{document} implies MLSI as long as MLSI+TC2\documentclass[12pt]{minimal}
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\begin{document}$$_2$$\end{document} holds.
The result follows directly from the convexity of the quantum relative entropy (cf. (v) of Theorem 3):0≤D(γ(1/2)‖σ)≤12D(ρ‖σ)+12D(ω‖σ)-κ8W2,L(ρ,ω)2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} 0\le D(\gamma (1/2)\Vert \sigma )\le \frac{1}{2}D(\rho \Vert \sigma )+\frac{1}{2} D(\omega \Vert \sigma )-\frac{\kappa }{8} W_{2,\mathcal {L}}(\rho ,\omega )^2. \end{aligned}$$\end{document}for a given constant speed geodesic (γ(s))s∈[0,1]\documentclass[12pt]{minimal}
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\begin{document}$$(\gamma (s))_{s\in [0,1]}$$\end{document} relating ρ\documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document} and ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document}. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
From Ricci Lower Bound to the Poincaré Inequality
In this section, we show that Ric(L)≥0\documentclass[12pt]{minimal}
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\begin{document}$${\text {Ric}}(\mathcal {L})\ge 0$$\end{document} together with a condition of finiteness of the diameter of D(H)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {D}}}({{\mathcal {H}}})$$\end{document} with respect to the distance W2,L\documentclass[12pt]{minimal}
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\begin{document}$$W_{2,\mathcal {L}}$$\end{document} implies the Poincaré inequality, hence extending Proposition 5.9 of [11] to our non-commutative setting. Throughout this section, we fix (Pt)t≥0\documentclass[12pt]{minimal}
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\begin{document}$$(\mathcal {P}_t)_{t\ge 0}$$\end{document} to be a primitive QMS on B(H)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {B}}}({{\mathcal {H}}})$$\end{document}, H\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {H}}}$$\end{document} finite dimensional, with unique invariant state σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document} and associated generator L\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}$$\end{document}, satisfying σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}-DBC. The next result is a non-commutative extension of the fourth equivalent statement in Theorem 4.7.2 of [4]:
Let f∈Bsa(H)\documentclass[12pt]{minimal}
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\begin{document}$$f\in {{\mathcal {B}}}_{sa}({{\mathcal {H}}})$$\end{document} be an eigenvector of L\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}$$\end{document} with associated eigenvalue opposite to the spectral gap λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document} of L\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}$$\end{document}. Without loss of generality, ‖f‖∞=1\documentclass[12pt]{minimal}
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\begin{document}$$\Vert f\Vert _\infty =1$$\end{document}, and by primitivity of (Λt)t≥0\documentclass[12pt]{minimal}
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\begin{document}$$(\Lambda _t)_{t\ge 0} $$\end{document}, Tr(σf)=0\documentclass[12pt]{minimal}
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\begin{document}$$\mathop {{\hbox {Tr}}}\nolimits (\sigma f)=0$$\end{document}. Now, note that Λt(f)=e-λtf\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda _t(f)=\mathrm {e}^{-\lambda t}f$$\end{document}. Therefore, the reverse Poincaré inequality (5.1) in the case when κ=0\documentclass[12pt]{minimal}
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\begin{document}$$\kappa =0$$\end{document} implies that for any ρ∈D+(H)\documentclass[12pt]{minimal}
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\begin{document}$$\rho \in {{\mathcal {D}}}_+({{\mathcal {H}}})$$\end{document},‖∇f‖L,ρ2≤e2λt2t‖f‖∞2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Vert \nabla f\Vert _{\mathcal {L},\rho }^2\le \frac{\mathrm {e}^{2\lambda t}}{2t}\Vert f\Vert _\infty ^2. \end{aligned}$$\end{document}Optimizing over t and using ‖f‖∞=1\documentclass[12pt]{minimal}
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\begin{document}$$\Vert f\Vert _\infty =1$$\end{document}, we find‖∇f‖L,ρ2≤eλ‖f‖∞2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Vert \nabla f\Vert _{\mathcal {L},\rho }^2\le \mathrm {e}\lambda \Vert f\Vert _\infty ^2. \end{aligned}$$\end{document}Given the following spectral decomposition of f=∑μμPμ\documentclass[12pt]{minimal}
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\begin{document}$$f=\sum _\mu \mu P_\mu $$\end{document}, since Tr(σf)=0\documentclass[12pt]{minimal}
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\begin{document}$$\mathop {{\hbox {Tr}}}\nolimits (\sigma f)=0$$\end{document}, the minimum and maximum eigenvalues of f, respectively, denoted by μmin\documentclass[12pt]{minimal}
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\begin{document}$$\mu _{\min }$$\end{document} and μmax\documentclass[12pt]{minimal}
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\begin{document}$$\mu _{\max }$$\end{document}, obey μmin<0<μmax\documentclass[12pt]{minimal}
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\begin{document}$$\mu _{\min }<0<\mu _{\max }$$\end{document}. Since we assumed ‖f‖∞=1\documentclass[12pt]{minimal}
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\begin{document}$$\Vert f\Vert _\infty =1$$\end{document}, this implies that given a path (γ(s),U(s))s∈[0,1]\documentclass[12pt]{minimal}
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\begin{document}$$(\gamma (s),U(s))_{s\in [0,1]}$$\end{document} in D(H)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {D}}}({{\mathcal {H}}})$$\end{document} joining the states γ(0)=PμmaxTr(Pμmax)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma (0)=\frac{P_{\mu _{\max }}}{\mathop {{\hbox {Tr}}}\nolimits (P_{\mu _{\max }})}$$\end{document} and γ(1)=PμminTr(Pμmin)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma (1)= \frac{P_{\mu _{\min }}}{\mathop {{\hbox {Tr}}}\nolimits (P_{\mu _{\min }} )}$$\end{document} such that ∫01‖γ˙(s)‖gL,γ(s)2ds≤W2,L(γ(0),γ(1))2+ε\documentclass[12pt]{minimal}
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\begin{document}$$\int _{0}^1 \Vert \dot{\gamma }(s)\Vert _{g_{\mathcal {L},\gamma (s)}}^2 \mathrm{d}s\le W_{2,\mathcal {L}}(\gamma (0),\gamma (1))^2+\varepsilon $$\end{document},1≤|μmax-μmin|=TrfPμminTr(Pμmin)-PμmaxTr(Pμmax)=Trf∫01γ˙(s)ds=∫01∑j∈Jcj⟨∂jf,[γ(s)]ωj∂jU(s)⟩ds≤(D2+ε)∫01‖∇f‖L,γ(s)2ds1/2≤(D2+ε)λe,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} 1\le |\mu _{\max } - \mu _{\min }|&=\left| \mathop {{\hbox {Tr}}}\nolimits f\left( \frac{P_{\mu _{\min }}}{\mathop {{\hbox {Tr}}}\nolimits (P_{\mu _{\min }} )} - \frac{P_{\mu _{\max }}}{\mathop {{\hbox {Tr}}}\nolimits (P_{\mu _{\max }} )} \right) \right| \\&= \left| \mathop {{\hbox {Tr}}}\nolimits \left( f\int _0^1 \dot{\gamma }(s) \mathrm{d}s\right) \right| \\&=\left| \int _0^1 \sum _{j\in {{\mathcal {J}}}} c_j \langle \partial _j f,[\gamma (s)]_{\omega _j}\partial _j U(s)\rangle \mathrm{d}s \right| \\&\quad \le \sqrt{( D^2+\varepsilon )}\left( \int _0^1 \Vert \nabla f\Vert ^2_{\mathcal {L},\gamma (s)}\mathrm{d}s\right) ^{1/2}\le \sqrt{(D^2+\varepsilon ) \lambda \mathrm {e}}, \end{aligned}$$\end{document}where in the last line we used the Cauchy–Schwarz inequality with respect to the inner product ∑j∈Jcj⟨.,∫01[γ(s)]ωjds.⟩\documentclass[12pt]{minimal}
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\begin{document}$$\sum _{j\in {{\mathcal {J}}}}c_j\langle .~,\int _0^1[\gamma (s)]_{\omega _j}\mathrm{d}s~.\rangle $$\end{document}, and the result directlyfollows. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
From Ricci Lower Bound to Modified Log-Sobolev Inequality
In [11], a modified logarithmic Sobolev inequality was proved to hold under the conditions that Ric(L)≥0\documentclass[12pt]{minimal}
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\begin{document}$${\text {Ric}}(\mathcal {L})\ge 0$$\end{document} and that the diameter of the underlying space, in terms of the modified Wasserstein distance, is bounded. Here, we extend their results to the quantum regime under the further assumption that the semigroup (Λt)t≥0\documentclass[12pt]{minimal}
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\begin{document}$$(\Lambda _t)_{t\ge 0}$$\end{document} is unital, leaving the study of the general case to later. The idea of the proof is to get a non-tight logarithmic Sobolev inequality from HWI(0)\documentclass[12pt]{minimal}
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\begin{document}$${\text {HWI}}(0)$$\end{document} and then to tighten it using ideas borrowed from [5]. In what follows, we denote by d the dimension of H\documentclass[12pt]{minimal}
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Given two states ρ,ω∈D(H)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sum _{i\in {{\mathcal {A}}}} q(i,j)&=\mu _j\mathop {{\hbox {Tr}}}\nolimits (Q_j)\\ \sum _{j\in {{\mathcal {B}}}}q(i,j)&=\lambda _i \mathop {{\hbox {Tr}}}\nolimits (P_i). \end{aligned}$$\end{document}The set of couplings between ρ\documentclass[12pt]{minimal}
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\begin{document}$$\Pi (\rho ,\omega )$$\end{document}. In analogy with the classical literature (see, for example, [12]), given an ergodic semigroup (Λt)t≥0\documentclass[12pt]{minimal}
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\begin{document}$$(\Lambda _t)_{t\ge 0}$$\end{document} with associated generator L\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}$$\end{document}, the coupling Wasserstein distance of order two between ρ\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} is defined as follows:W2,L,c(ρ,ω)2:=infq∈Π(ρ,σ)∑i∈A,j∈Bq(i,j)W2,L(ρi,ωj)2,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} W_{2,\mathcal {L},c}(\rho ,\omega )^2:=\inf _{q\in \Pi (\rho ,\sigma )} \sum _{i\in {{\mathcal {A}}},~j\in {{\mathcal {B}}}} q(i,j) W_{2,\mathcal {L}} (\rho _i,\omega _j)^2, \end{aligned}$$\end{document}whereρi:=PiTrPi,ωj:=QjTr(Qj),i∈A,j∈B.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \rho _i:= \frac{P_i}{\mathop {{\hbox {Tr}}}\nolimits P_i},~~~~~~~~~~\omega _j:=\frac{Q_j}{\mathop {{\hbox {Tr}}}\nolimits (Q_j)},~~~~~~~~~i\in {{\mathcal {A}}},~j\in {{\mathcal {B}}}. \end{aligned}$$\end{document}The following result is a quantum generalization of Proposition 2.14 of [12]:
Proposition 8
Let (Λt)t≥0\documentclass[12pt]{minimal}
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\begin{document}$$(\Lambda _t)_{t\ge 0}$$\end{document} be a primitive QMS, with unique invariant state σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document} and associated generator L\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}$$\end{document}, satisfying the detailed balance condition. Then, for any ρ,ω∈D+(H)\documentclass[12pt]{minimal}
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\begin{document}$$\rho ,\omega \in {{\mathcal {D}}}_+({{\mathcal {H}}})$$\end{document},W2,L(ρ,ω)≤W2,L,c(ρ,ω).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} W_{2,\mathcal {L}}(\rho ,\omega )\le W_{2,\mathcal {L},c}(\rho ,\omega ). \end{aligned}$$\end{document}
In what follows, we restrict our analysis to the case of a primitive QMS (Λt)t≥0\documentclass[12pt]{minimal}
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\begin{document}$$(\Lambda _t)_{t\ge 0}$$\end{document} with unique invariant state I/d\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {I}/d$$\end{document} that satisfies the detailed balance condition. In order to prove the main result of this section, we need the following two lemmas that are extensions of Lemmas 6.2 and 6.3 of [11]:
This is a direct rewriting of Lemma 2.5 of [5]. □\documentclass[12pt]{minimal}
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Theorem 7
Let (Λt)t≥0\documentclass[12pt]{minimal}
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\begin{document}$$(\Lambda _t)_{t\ge 0}$$\end{document} be a primitive semigroup with unique invariant state I/d\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {I}/d$$\end{document} and associated generator L\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}$$\end{document}. Assume that Ric(L)≥0\documentclass[12pt]{minimal}
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\begin{document}$${\text {Ric}}(\mathcal {L})\ge 0$$\end{document} and that DiamL(D(H))≤D\documentclass[12pt]{minimal}
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\begin{document}$${\text {Diam}}_{\mathcal {L}}({{\mathcal {D}}}({{\mathcal {H}}}))\le D$$\end{document}. Then MLSI(cD-2)\documentclass[12pt]{minimal}
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\begin{document}$${\text {MLSI}}(cD^{-2})$$\end{document} holds, for some universal constant c.
In this paper, we prove that a classical picture, relating various inequalities which are useful in the analysis of Markov semigroups, carries over to the quantum setting. Classically, a key element of this picture is a geometric inequality called the Ricci lower bound. Functional and transportation cost inequalities, which play an important role in the study of mixing times of a primitive Markov semigroup and concentration properties of its invariant measure, can be obtained from this geometric inequality. The connection between them is provided by an interpolating inequality called the HWI inequality. In this paper, we analyse a quantum version of the Ricci lower bound (due to Carlen and Maas [7]) and show that it implies a quantum HWI inequality, from which quantum versions of the functional and transportation cost inequalities (which are relevant for the analysis of quantum Markov semigroups) follow.
Acknowledgements
Open Access funding provided by Projekt DEAL. CR acknowledges support by the DFG cluster of excellence 2111 (Munich Center for Quantum Science and Technology). The authors would like to thank Ivan Bardet for helpful discussions.
ReferencesAlickiROn the detailed balance condition for non-Hamiltonian systems19761022492581976RpMP...10..249A5732260363.6011410.1016/0034-4877(76)90046-XBakryDBernardPL’hypercontractivité et son utilisation en théorie des semigroupes1992BerlinSpringer1114Bakry, D., Émery, M.: Diffusions hypercontractives. In: Séminaire de Probabilités XIX 1983/84, pp. 177–206. Springer (1985)BakryDGentilILedouxM2014BerlinSpringer1376.6000210.1007/978-3-319-00227-9BartheFKolesnikovAVMass transport and variants of the logarithmic sobolev inequality200818492197924389061170.4603110.1007/s12220-008-9039-6CarlenEAMaasJAn analog of the 2-Wasserstein metric in non-commutative probability under which the Fermionic Fokker–Planck equation is gradient flow for the entropy201433138879262014CMaPh.331..887C32480531297.3524110.1007/s00220-014-2124-8CarlenEAMaasJGradient flow and entropy inequalities for quantum Markov semigroups with detailed balance201727351810186936667291386.4605710.1016/j.jfa.2017.05.003Carlen, E.A., Maas, J.: Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems (2018). arXiv preprint arXiv:1811.04572DaneriSSavaréGEulerian calculus for the displacement convexity in the Wasserstein distance20084031104112224528821166.5801110.1137/08071346XErbarMThe heat equation on manifolds as a gradient flow in the Wasserstein space20104611232010AnIHP..46....1E26417671215.3501610.1214/08-AIHP30602Erbar, M., Fathi, M.: Poincaré, modified logarithmic Sobolev and isoperimetric inequalities for Markov chains with non-negative Ricci curvature (2016). arXiv preprint arXiv:1612.00514ErbarMMaasJRicci curvature of finite markov chains via convexity of the entropy20122063997103829894491256.5302810.1007/s00205-012-0554-zJohnsonOA discrete log-Sobolev inequality under a Bakry–Émery type condition2017534195219701390.3907710.1214/16-AIHP77811JordanRKinderlehrerDOttoFThe variational formulation of the Fokker–Planck equation199829111716171710915.3512010.1137/S0036141096303359KantorovichLOn the translocation of masses194213319920196190061.09705KastoryanoMJTemmeKQuantum logarithmic Sobolev inequalities and rapid mixing20135450522022013JMP....54e2202K30989231379.8102110.1063/1.4804995LesniewskiARuskaiMBMonotone Riemannian metrics and relative entropy on noncommutative probability spaces19994011570257241999JMP....40.5702L17223340968.8100610.1063/1.533053LottJVillaniCRicci curvature for metric-measure spaces via optimal transport200916990399124806191178.5303810.4007/annals.2009.169.903MaasJGradient flows of the entropy for finite Markov chains201126182250229228245781237.6005810.1016/j.jfa.2011.06.009MartonKA simple proof of the blowing-up lemma (corresp.)19863234454461986ptlc.book.....M8382130594.9400310.1109/TIT.1986.1057176Monge, G.: Mémoire sur la théorie des déblais et des remblais. Hist. l’Acad. R. Sci. Paris, pp. 666 – 704 (1781)OlkiewiczRZegarlinskiBHypercontractivity in noncommutative Lp spaces1999161124628516702300923.4606610.1006/jfan.1998.3342OttoFVillaniCGeneralization of an inequality by Talagrand and links with the logarithmic Sobolev inequality2000173236140017606200985.5801910.1006/jfan.1999.3557Rouzé, C., Datta, N.: Concentration of Quantum States from Quantum Functional and Talagrand Inequalities (2017). arXiv preprint arXiv:1704.02400SpohnHEntropy production for quantum dynamical semigroups1978195122712301978JMP....19.1227S4755400428.4702210.1063/1.523789SturmK-TTransport inequalities, gradient estimates, entropy, and Ricci curvature2005LVIII92394021428791078.53028SturmK-TOn the geometry of metric measure spaces200619616513122372061105.5303510.1007/s11511-006-0002-8SturmK-TOn the geometry of metric measure spaces. II2006196113317722372071106.5303210.1007/s11511-006-0003-7TemmeKKastoryanoMJRuskaiMWolfMMVerstraeteFThe χ\documentclass[12pt]{minimal}
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\begin{document}$$\chi $$\end{document}2-divergence and mixing times of quantum Markov processes201051121222012010JMP....51l2201T27791701314.81124VillaniC2008BerlinSpringer1156.53003von RenesseM-KSturmK-TTransport inequalities, gradient estimates, entropy and Ricci curvature200558792394021428791078.5302810.1002/cpa.20060Wolf, M.M.: Quantum Channels and Operations: Guided tour. Lecture notes, vol. 5 (2012). http://www-m5.ma.tum.de/foswiki/pubM. Accessed 13 Mar 2019
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