Superadditivity of Quantum Relative Entropy for General States

Capel, Ángela; Lucia, Ángelo; Pérez-Garcı́a, David

doi:10.1109/tit.2017.2772800

Cited by 20 publications

(17 citation statements)

References 23 publications

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“…For that, we show in increasing order of difficulty several results of quasi-factorization, classifying them into two classes depending on the possible overlap of the subregions where the relative entropy is conditioned and the number of such subregions. Moreover, we remark that one of them is equivalent to a result that was proven in [9].…”

Section: Introductionmentioning

confidence: 63%

Quantum conditional relative entropy and quasi-factorization of the relative entropy

Capel¹,

Lucia²,

Pérez-Garcı́a³

2018

J. Phys. A: Math. Theor.

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The existence of a positive log-Sobolev constant implies a bound on the mixing time of a quantum dissipative evolution under the Markov approximation. For classical spin systems, such constant was proven to exist, under the assumption of a mixing condition in the Gibbs measure associated to their dynamics, via a quasi-factorization of the entropy in terms of the conditional entropy in some sub-σ-algebras.In this work we analyze analogous quasi-factorization results in the quantum case. For that, we define the quantum conditional relative entropy and prove several quasifactorization results for it. As an illustration of their potential, we use one of them to obtain a positive log-Sobolev constant for the heat-bath dynamics with product fixed point.arXiv:1804.09525v2 [quant-ph]

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Section: Introductionmentioning

confidence: 63%

Quantum conditional relative entropy and quasi-factorization of the relative entropy

Capel¹,

Lucia²,

Pérez-Garcı́a³

2018

J. Phys. A: Math. Theor.

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“…Nevertheless, let us recall here some situations for which we already know that inequality (37) holds. First, if σ Λ is a tensor product, this inequality holds with f = 1 [9], as a consequence of strong subadditivity. Moreover, for a more general σ Λ , if A and B are separated enough, we have seen in Step 2 that it also holds with f = 1, due to the structure of quantum Markov chain of σ Λ .…”

Section: Proofmentioning

confidence: 99%

On the modified logarithmic Sobolev inequality for the heat-bath dynamics for 1D systems

Bardet

Capel

Lucia

et al. 2021

Journal of Mathematical Physics

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The mixing time of Markovian dissipative evolutions of open quantum manybody systems can be bounded using optimal constants of certain quantum functional inequalities, such as the modified logarithmic Sobolev constant. For classical spin systems, the positivity of such constants follows from a mixing condition for the Gibbs measure, via quasi-factorization results for the entropy. Inspired by the classical case, we present a strategy to derive the positivity of the modified logarithmic Sobolev constant associated to the dynamics of certain quantum systems from some clustering conditions on the Gibbs state of a local, commuting Hamiltonian. In particular we show that for the heat-bath dynamics for 1D systems, the modified logarithmic Sobolev constant is positive under the assumptions of a mixing condition on the Gibbs state and a strong quasi-factorization of the relative entropy.

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“…This concludes the proof. Lemma 10 (Superadditivity of relative entropy [42]). Let H AB = H A ⊗ H B be a bipartite Hilbert space, and let ρ AB , σ A , and σ B be density operators.…”

Section: Appendix B: Technical Lemmasmentioning

confidence: 99%

Relative entropy and catalytic relative majorization

Rethinasamy

Wilde

2020

Phys. Rev. Research

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Given two pairs of quantum states, a fundamental question in the resource theory of asymmetric distinguishability is whether there exists a quantum channel converting one pair to the other. In this work, we reframe this question in such a way that a catalyst can be used to help perform the transformation, with the only constraint on the catalyst being that its reduced state is returned unchanged, so that it can be used again to assist a future transformation. What we find here, for the special case in which the states in a given pair are commuting, and thus quasiclassical, is that this catalytic transformation can be performed if and only if the relative entropy of one pair of states is larger than that of the other pair. This result endows the relative entropy with a fundamental operational meaning that goes beyond its traditional interpretation in the setting of independent and identical resources. Our finding thus has an immediate application and interpretation in the resource theory of asymmetric distinguishability, and we expect it to find application in other domains.

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Superadditivity of Quantum Relative Entropy for General States

Cited by 20 publications

References 23 publications

Quantum conditional relative entropy and quasi-factorization of the relative entropy

Quantum conditional relative entropy and quasi-factorization of the relative entropy

On the modified logarithmic Sobolev inequality for the heat-bath dynamics for 1D systems

Relative entropy and catalytic relative majorization

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