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arXiv:2410.02686

Optimal continuity bound for the von Neumann entropy under energy constraints

Becker, S., Nilanjana Datta, Michael G. Jabbour, Максим Евгеньевич Широков

Published Oct 3, 2024• 1 citation
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Abstract

Using techniques proposed in [Sason, IEEE Trans. Inf. Th. 59, 7118 (2013)] and [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)], and based on the results from the latter, we construct a globally optimal continuity bound for the von Neumann entropy. This bound applies to any state under energy constraints imposed by arbitrary Hamiltonians that satisfy the Gibbs hypothesis. This completely solves the problem of finding an optimal continuity bound for the von Neumann entropy in this setting, previously known only for pairs of states that are sufficiently close to each other. Our main technical result, a globally optimal semicontinuity bound for the von Neumann entropy under general energy constraints, leads to this continuity bound. To prove it, we also derive an optimal Fano-type inequality for random variables with a countably infinite alphabet and a general constraint, as well as optimal semicontinuity and continuity bounds for the Shannon entropy in the same setting. In doing so, we improve the results derived in [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)].

DOI
10.48550/arxiv.2410.02686

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Using techniques proposed in [Sason, IEEE Trans. Inf. Th. 59, 7118 (2013)] and [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)], and based on the results from the latter, we construct a globally optimal continuity bound for the von Neumann entropy. This bound applies to any state under energy constraints imposed by arbitrary Hamiltonians that satisfy the Gibbs hypothesis. This completely solves the problem of finding an optimal continuity bound for the von Neumann entropy in this setting, previously known only for pairs of states that are sufficiently close to each other. Our main technical result, a globally optimal semicontinuity bound for the von Neumann entropy under general energy constraints, leads to this continuity bound. To prove it, we also derive an optimal Fano-type inequality for random variables with a countably infinite alphabet and a general constraint, as well as optimal semicontinuity and continuity bounds for the Shannon entropy in the same setting. In doing so, we improve the results derived in [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)].
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We consider families of tight upper bounds on the difference S(ρ)−S(σ)S(\rho)-S(\sigma)S(ρ)−S(σ) with a rank/energy constraint imposed on the state ρ\rhoρ, which are valid provided that the state ρ\rhoρ partially majorizes the state σ\sigmaσ and is close to the state σ\sigmaσ with respect to the trace norm. The upper bounds within these families depend on the parameter mmm of partial majorization. The upper bounds corresponding to m=0m=0m=0 coincide with the (unconditional) optimal semicontinuity bounds for the von Neumann entropy with a rank/energy constraint obtained in [1] and [2]. State-dependent improvements of these semicontinuity bounds are proposed and analysed numerically. The notion of ε\varepsilonε-sufficient majorization dimension of the set of states with bounded energy is introduced and analysed. Classical versions of the above results formulated in terms of probability distributions and Shannon entropy are also considered. Bibliography: 20 titles.
We present two results derived from the sharp log-Sobolev inequality for the uniform measure on a complete graph which concern the von Neumann entropy and the Entanglement of Formation of a state of finite and infinite-dimensional quantum systems. The first result is a sharp Lipschitz lower semicontinuity bound for the von Neumann entropy at any mixed state ρρρ with uniform positive spectrum (i.e. a state proportional to a projector) w.r.t. the fidelity deficit: the inequality  S(ρ)−S(σ)≤Cρ(1−F(ρ,σ)) \,S(ρ)-S(σ)\leq C_ρ(1-F(ρ,σ))\,S(ρ)−S(σ)≤Cρ​(1−F(ρ,σ)) valid for any state σσσ, where CρC_ρCρ​ is a constant depending on the rank of ρρρ. The second result is a sharp Lipschitz lower semicontinuity bound for the Entanglement of Formation at any pure state ρρρ with uniform positive spectrum of marginal states w.r.t. the fidelity deficit: the inequality  EF(ρ)−EF(σ)≤Cρ(1−Trρσ) \,E_F(ρ)-E_F(σ)\leq C_ρ(1-\mathrm{Tr}ρσ)\,EF​(ρ)−EF​(σ)≤Cρ​(1−Trρσ) valid for any state σσσ, where CρC_ρCρ​ is a constant depending on the Schmidt rank of ρρρ. In both cases the optimal constant CρC_ρCρ​ is equal to the optimal constant KdK_{d}Kd​ in the log-Sobolev inequality for the complete graph with ddd vertices: in the first case d=rankρd=\mathrm{rank}ρd=rankρ, in the second one d=rankρA=rankρBd=\mathrm{rank}ρ_A=\mathrm{rank}ρ_Bd=rankρA​=rankρB​. The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.
Uniform continuity bounds on entropies are generally expressed in terms of a single distance measure between probability distributions or quantum states, typically, the total variation- or trace distance. However, if an additional distance measure is known, the continuity bounds can be significantly strengthened. Here, we prove a tight uniform continuity bound for the Shannon entropy in terms of both the local- and total variation distances, sharpening an inequality in (Sason, 2013). We also obtain a uniform continuity bound for the von Neumann entropy in terms of both the operator norm- and trace distances. We then apply our results to compute upper bounds on channel capacities. We first refine the concept of approximate degradable channels by introducing(ε,ν)(\varepsilon ,\nu)(ε,ν)–degradable channels. These are$\varepsilon –closeindiamondnormand–close in diamond norm and–closeindiamondnormand\nu $–close in completely bounded spectral norm to their complementary channel when composed with a degrading channel. This leads to improved upper bounds to the quantum- and private classical capacities of such channels. Moreover, these bounds can be further improved by considering certain unstabilized versions of the above norms. We show that upper bounds on the latter can be efficiently expressed as semidefinite programs. As an application, we obtain a new upper bound on the quantum capacity of the qubit depolarizing channel.
Given an arbitrary quantum state (σ), we obtain an explicit construction of a state ρε*(σ) [respectively, ρ*,ε(σ)] which has the maximum (respectively, minimum) entropy among all states which lie in a specified neighborhood (ε-ball) of σ. Computing the entropy of these states leads to a local strengthening of the continuity bound of the von Neumann entropy, i.e., the Audenaert-Fannes inequality. Our bound is local in the sense that it depends on the spectrum of σ. The states ρε*(σ) and ρ*,ε(σ) depend only on the geometry of the ε-ball and are in fact optimizers for a larger class of entropies. These include the Rényi entropy and the minimum- and maximum-entropies, providing explicit formulas for certain smoothed quantities. This allows us to obtain local continuity bounds for these quantities as well. In obtaining this bound, we first derive a more general result which may be of independent interest, namely, a necessary and sufficient condition under which a state maximizes a concave and Gâteaux-differentiable function in an ε-ball around a given state σ. Examples of such a function include the von Neumann entropy and the conditional entropy of bipartite states. Our proofs employ tools from the theory of convex optimization under non-differentiable constraints, in particular Fermat’s rule, and majorization theory.