From Quasi-Entropy to Various Quantum Information Quantities

Abstract. We study a class of ergodic quantum Markov semigroups on finitedimensional unital C * -algebras. These semigroups have a unique stationary state σ, and we are concerned with those that satisfy a quantum detailed balance condition with respect to σ. We show that the evolution on the set of states that is given by such a quantum Markov semigroup is gradient flow for the relative entropy with respect to σ in a particular Riemannian metric on the set of states. This metric is a non-commutative analog of the 2-Wasserstein metric, and in several interesting cases we are able to show, in analogy with work of Otto on gradient flows with respect to the classical 2-Wasserstein metric, that the relative entropy is strictly and uniformly convex with respect to the Riemannian metric introduced here. As a consequence, we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum Markov semigroups with detailed balance.

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“…is jointly convex on S + × A; see [35,36]. Note that if ρ and A are scalars, the right-hand side reduces to A 2 /ρ.…”

Section: Geodesic Convexity and Relaxation To Equilibriummentioning

confidence: 99%

Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance

Cárlen

Maas

2017

Journal of Functional Analysis

127

250

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“…However, the right hand sides of the above equations coincide as the sequence {K α (φ (A n ) ||φ(X ))} n∈N is convergent, see (5). So, we deduced that…”

Section: Claim 7 the Map φ Is A Homeomorphismmentioning

confidence: 99%

“…Important properties of these notions (both the more restricted one given in Definition 1 as well as the just mentioned more general one) were investigated in several further papers. Without presuming to be exhaustive here we refer only to the works [3], [5], [6], [14], [16].…”

Section: Introductionmentioning

confidence: 99%

Maps on positive definite operators preserving the quantum $$\chi _\alpha ^2$$ χ α 2 -divergence

et al. 2017

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ABSTRACT. We describe the structure of all bijective maps on the cone of positive definite operators acting on a finite and at least twodimensional complex Hilbert space which preserve the quantum χ 2 α -divergence for some α ∈ [0, 1]. We prove that any such transformation is necessarily implemented by either a unitary or an antiunitary operator. Similar results concerning maps on the cone of positive semidefinite operators as well as on the set of all density operators are also derived.

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“…In the important case of finite dimensional space H and the generalized inverse P −1 , numerous properties of the quantum f -divergence have been obtained in the recent papers [33][34][35][36] and the references therein. We omit the details.…”

Section: Classical Quantum F -Divergencementioning

confidence: 99%