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Optimal continuity bound for the von Neumann entropy under energy constraints

S. Becker, N. Datta, M. G. Jabbour, M. E. Shirokov

TL;DR

The paper addresses the problem of obtaining a globally optimal, ε-uniform modulus of continuity for the von Neumann entropy under energy constraints defined by Hamiltonians satisfying the Gibbs hypothesis. It develops a semicontinuity bound for $S(\rho)-S(\sigma)$ that is tight for all $\varepsilon\in[0,1]$, and derives a matching energy-constrained continuity bound, both expressed through Gibbs-based quantities $F_H(E)$, $F^+_H(E)$, and $Z_H(E)$ and the entropy-maximizing constructions. The approach hinges on a classical-quantum reduction: first proving a Fano-type inequality for random variables with a countably infinite alphabet under a general constraint and constrained Shannon entropy bounds, then lifting these results to the quantum setting via passive-state diagonalisation. By establishing tightness and an exact identity linking $F_H^+$ and $F_H$, the work resolves the problem of optimal continuity bounds for the von Neumann entropy in infinite dimensions under energy restrictions, extending prior results that were limited to nearby states. The findings provide rigorous tools for information processing in infinite-dimensional quantum systems under energy constraints and advance the theoretical understanding of entropy continuity in quantum information theory.

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)].

Optimal continuity bound for the von Neumann entropy under energy constraints

TL;DR

The paper addresses the problem of obtaining a globally optimal, ε-uniform modulus of continuity for the von Neumann entropy under energy constraints defined by Hamiltonians satisfying the Gibbs hypothesis. It develops a semicontinuity bound for that is tight for all , and derives a matching energy-constrained continuity bound, both expressed through Gibbs-based quantities , , and and the entropy-maximizing constructions. The approach hinges on a classical-quantum reduction: first proving a Fano-type inequality for random variables with a countably infinite alphabet under a general constraint and constrained Shannon entropy bounds, then lifting these results to the quantum setting via passive-state diagonalisation. By establishing tightness and an exact identity linking and , the work resolves the problem of optimal continuity bounds for the von Neumann entropy in infinite dimensions under energy restrictions, extending prior results that were limited to nearby states. The findings provide rigorous tools for information processing in infinite-dimensional quantum systems under energy constraints and advance the theoretical understanding of entropy continuity in quantum information theory.

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)].
Paper Structure (8 sections, 6 theorems, 66 equations)

This paper contains 8 sections, 6 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Some auxiliary results and a proof of Theorem \ref{['SCB']}
  5. Preliminary results: Classical entropy bounds
  6. Proof of Theorem \ref{['theo:FanoG']}
  7. Proof of Theorem \ref{['theo:classical']}
  8. Proof of Theorem \ref{['SCB']}

Key Result

Theorem 1

Let $H$ be a positive operator on $\mathcal{H}$ satisfying conditions (H-cond) and (star) and $E>0$. (A) If $\rho$ is a state in $\mathfrak{S}(\mathcal{H})$ such that $\mathrm{Tr} H\rho\leq E$ and $\varepsilon\in(0,1]$ is arbitrary, then for any state $\sigma$ in $\mathfrak{S}(\mathcal{H})$ such that $\,\frac{1}{2}\|\rho-\sigma\|_1\leq \varepsilon$, where the l.h.s. may be equal to $-\infty$. Her

Theorems & Definitions (15)

  • Theorem 1: Semicontinuity bound for the von Neumann entropy of energy-constrained states
  • Remark 1
  • Remark 2
  • Corollary 1: Continuity bound for the von Neumann entropy of energy-constrained states
  • Corollary 2
  • Definition 1: Entropies: Discrete random variables
  • Theorem 2: A Fano-type inequality for random variables with a countably infinite alphabet and a general constraint
  • Remark 3
  • Remark 4
  • proof : Proof of Remark \ref{['rem:tightClass-1']}
  • ...and 5 more