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Continuity of entropies via integral representations

Mario Berta, Ludovico Lami, Marco Tomamichel

TL;DR

This work introduces a dimension-independent semi-continuity bound for the quantum relative entropy via Frenkel's integral representation, establishing that $D(\rho\|\omega) - D(\sigma\|\omega) \le \varepsilon \log(M-1) + h_2(\varepsilon)$ under $\tfrac{1}{2}\|\rho-\sigma\|_1 \le \varepsilon$ and $\rho \le M\omega$. The bound, which is tight, unifies and extends classical continuity results to the fully quantum setting and serves as a versatile tool for a suite of applications: it yields improved Fannes--Audenaert and conditional-entropy bounds (including Wilde's conjecture in the equal-mMarginals case), enhanced continuity for entanglement cost, refined bounds for approximate degradability in quantum channels, and general results for filtered entropies and infinite-dimensional transformation rates. Collectively, these results advance quantitative control over how small perturbations in quantum states or channels affect key information measures, with broad implications for quantum information processing and entanglement theory. The work also highlights open questions, notably the full resolution of Wilde's conjecture and tight continuity for quantum mutual information and squashed entanglement, pointing to future directions for extending the integral-representation framework.

Abstract

We show that Frenkel's integral representation of the quantum relative entropy provides a natural framework to derive continuity bounds for quantum information measures. Our main general result is a dimension-independent semi-continuity relation for the quantum relative entropy with respect to the first argument. Using it, we obtain a number of results: (1) a tight continuity relation for the conditional entropy in the case where the two states have equal marginals on the conditioning system, resolving a conjecture by Wilde in this special case; (2) a stronger version of the Fannes-Audenaert inequality on quantum entropy; (3) better estimates on the quantum capacity of approximately degradable channels; (4) an improved continuity relation for the entanglement cost; (5) general upper bounds on asymptotic transformation rates in infinite-dimensional entanglement theory; and (6) a proof of a conjecture due to Christandl, Ferrara, and Lancien on the continuity of 'filtered' relative entropy distances.

Continuity of entropies via integral representations

TL;DR

This work introduces a dimension-independent semi-continuity bound for the quantum relative entropy via Frenkel's integral representation, establishing that under and . The bound, which is tight, unifies and extends classical continuity results to the fully quantum setting and serves as a versatile tool for a suite of applications: it yields improved Fannes--Audenaert and conditional-entropy bounds (including Wilde's conjecture in the equal-mMarginals case), enhanced continuity for entanglement cost, refined bounds for approximate degradability in quantum channels, and general results for filtered entropies and infinite-dimensional transformation rates. Collectively, these results advance quantitative control over how small perturbations in quantum states or channels affect key information measures, with broad implications for quantum information processing and entanglement theory. The work also highlights open questions, notably the full resolution of Wilde's conjecture and tight continuity for quantum mutual information and squashed entanglement, pointing to future directions for extending the integral-representation framework.

Abstract

We show that Frenkel's integral representation of the quantum relative entropy provides a natural framework to derive continuity bounds for quantum information measures. Our main general result is a dimension-independent semi-continuity relation for the quantum relative entropy with respect to the first argument. Using it, we obtain a number of results: (1) a tight continuity relation for the conditional entropy in the case where the two states have equal marginals on the conditioning system, resolving a conjecture by Wilde in this special case; (2) a stronger version of the Fannes-Audenaert inequality on quantum entropy; (3) better estimates on the quantum capacity of approximately degradable channels; (4) an improved continuity relation for the entanglement cost; (5) general upper bounds on asymptotic transformation rates in infinite-dimensional entanglement theory; and (6) a proof of a conjecture due to Christandl, Ferrara, and Lancien on the continuity of 'filtered' relative entropy distances.
Paper Structure (10 sections, 10 theorems, 92 equations)

This paper contains 10 sections, 10 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. A semi-continuity relation for the relative entropy
  3. Applications
  4. Improved Fannes--Audenaert inequality
  5. Improved continuity of conditional entropy
  6. Improved continuity of entanglement cost
  7. Approximate degradability
  8. Improved continuity of filtered entropies
  9. Upper bound on transformation rates in infinite dimensions
  10. Conclusion

Key Result

Theorem 1

Let $\rho,\sigma,\omega$ be quantum states on the same separable Hilbert space such that for some $\varepsilon \in [0,1]$ and $M \geq 1$. Then, we have that Moreover, for every such $\varepsilon$ and $M$, there exists a triple of states $\rho, \sigma$ and $\omega$ obeying semicontinuity_relent_thm_conditions_states for which the inequality semicontinuity_relent is tight.

Theorems & Definitions (25)

  • Theorem 1
  • Remark 2
  • proof : Proof of Theorem \ref{['semicontinuity_relent_thm']}
  • Corollary 3
  • Remark 4
  • Theorem 5
  • proof : Proof of Theorem \ref{['equal_marginals_thm']}
  • Conjecture 6: Mark2020
  • Lemma 7
  • proof : Proof of Lemma \ref{['lem:output-entropy']}
  • ...and 15 more