Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2504.08098 · Sbornik Mathematics

Semicontinuity bounds for the von Neumann entropy and partial majorization

Максим Евгеньевич Широков

Published Jan 1, 2026• 0 citations
arXivHTMLPDF

Abstract

We consider families of tight upper bounds on the difference $S(\rho)-S(\sigma)$ with a rank/energy constraint imposed on the state $\rho$, which are valid provided that the state $\rho$ partially majorizes the state $\sigma$ and is close to the state $\sigma$ with respect to the trace norm. The upper bounds within these families depend on the parameter $m$ of partial majorization. The upper bounds corresponding to $m=0$ 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 $\varepsilon$-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.

DOI
10.4213/sm10371e

Related papers

Partial majorization and Schur concave functions on the sets of quantum and classical states

M. E. Shirokov

We construct for a Schur concave function $f$ on the set of quantum states a tight upper bound on the difference $f(ρ)-f(σ)$ for a quantum state $ρ$ with finite $f(ρ)$ and any quantum state $σ$ $m$-partially majorized by the state $ρ$ in the sense described in [1]. We also obtain a tight upper bound on this difference under the additional condition $\frac{1}{2}\|ρ-σ\|_1\leq\varepsilon$ and find simple sufficient conditions for vanishing this bound with $\,\min\{\varepsilon,1/m\}\to0\,$. The obtained results are applied to the von Neumann entropy. The concept of $\varepsilon$-sufficient majorization rank of a quantum state with finite entropy is introduced and a tight upper bound on this quantity is derived and applied to the Gibbs states of a quantum oscillator. We also show how the obtained results can be reformulated for Schur concave functions on the set of probability distributions with a finite or countable set of outcomes.

quant-pharXiv:2604.13033•Lobachevskii Journal of Mathematics•Apr 15, 2026•0 citations

Optimal continuity bound for the von Neumann entropy under energy constraints

Becker, S., Nilanjana Datta, Michael G. Jabbour +1 more

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)].

arXiv:2410.02686•Oct 3, 2024•1 citation

How to improve the semicontinuity bounds in [Lett. Math. Phys., 113, 121 (2023)]

Максим Евгеньевич Широков

We show how to improve the semicontinuity bounds in [1] by optimizing the proof of the basic technical lemma. In this optimization we apply the modified version of the trick used in the resent article [2]. The most important applications are the semicontinuity bound for the von Neumann entropy with the energy constraint and the semicontinuity bounds for the entanglement of formation with the rank/energy constraint.

arXiv:2409.10546•Sep 1, 2024•0 citations

Tightening Continuity Bounds for Entropies and Bounds on Quantum Capacities

Michael G. Jabbour, Nilanjana Datta

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$(\varepsilon ,\nu)$–degradable channels. These are$\varepsilon $–close in diamond norm and$\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.

arXiv:2310.17329•IEEE Journal on Selected Areas in Information Theory•Jan 1, 2024•3 citations

Log-Sobolev inequality, von Neumann entropy and Entanglement of Formation

A. S. Holevo, M. E. Shirokov

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 $ρ$ with uniform positive spectrum (i.e. a state proportional to a projector) w.r.t. the fidelity deficit: the inequality $\,S(ρ)-S(σ)\leq C_ρ(1-F(ρ,σ))\,$ valid for any state $σ$, where $C_ρ$ is a constant depending on the rank of $ρ$. The second result is a sharp Lipschitz lower semicontinuity bound for the Entanglement of Formation at any pure state $ρ$ with uniform positive spectrum of marginal states w.r.t. the fidelity deficit: the inequality $\,E_F(ρ)-E_F(σ)\leq C_ρ(1-\mathrm{Tr}ρσ)\,$ valid for any state $σ$, where $C_ρ$ is a constant depending on the Schmidt rank of $ρ$. In both cases the optimal constant $C_ρ$ is equal to the optimal constant $K_{d}$ in the log-Sobolev inequality for the complete graph with $d$ vertices: in the first case $d=\mathrm{rank}ρ$, in the second one $d=\mathrm{rank}ρ_A=\mathrm{rank}ρ_B$. The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.

quant-pharXiv:2609.12667•Sep 14, 2026•0 citations

Tight uniform continuity bound for a family of entropies

Eric P. Hanson, Nilanjana Datta

We prove a tight uniform continuity bound for a family of entropies which includes the von Neumann entropy, the Tsallis entropy and the $α$-Rényi entropy, $S_α$, for $α\in (0,1)$. We establish necessary and sufficient conditions for equality in the continuity bound and prove that these conditions are the same for every member of the family. Our result builds on recent work in which we constructed a state which was majorized by every state in a neighbourhood ($\varepsilon$-ball) of a given state, and thus was the minimal state in majorization order in the $\varepsilon$-ball. This minimal state satisfies a particular semigroup property, which we exploit to prove our bound.

arXiv:1707.04249•Jul 13, 2017•8 citations
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 construct for a Schur concave function fff on the set of quantum states a tight upper bound on the difference f(ρ)−f(σ)f(ρ)-f(σ)f(ρ)−f(σ) for a quantum state ρρρ with finite f(ρ)f(ρ)f(ρ) and any quantum state σσσ mmm-partially majorized by the state ρρρ in the sense described in [1]. We also obtain a tight upper bound on this difference under the additional condition 12∥ρ−σ∥1≤ε\frac{1}{2}\|ρ-σ\|_1\leq\varepsilon21​∥ρ−σ∥1​≤ε and find simple sufficient conditions for vanishing this bound with  min⁡{ε,1/m}→0 \,\min\{\varepsilon,1/m\}\to0\,min{ε,1/m}→0. The obtained results are applied to the von Neumann entropy. The concept of ε\varepsilonε-sufficient majorization rank of a quantum state with finite entropy is introduced and a tight upper bound on this quantity is derived and applied to the Gibbs states of a quantum oscillator. We also show how the obtained results can be reformulated for Schur concave functions on the set of probability distributions with a finite or countable set of outcomes.
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)].
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.
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.
We prove a tight uniform continuity bound for a family of entropies which includes the von Neumann entropy, the Tsallis entropy and the ααα-Rényi entropy, SαS_αSα​, for α∈(0,1)α\in (0,1)α∈(0,1). We establish necessary and sufficient conditions for equality in the continuity bound and prove that these conditions are the same for every member of the family. Our result builds on recent work in which we constructed a state which was majorized by every state in a neighbourhood (ε\varepsilonε-ball) of a given state, and thus was the minimal state in majorization order in the ε\varepsilonε-ball. This minimal state satisfies a particular semigroup property, which we exploit to prove our bound.