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:1404.5999 · Journal of Mathematical Physics

Bounds on the concavity of quantum entropy

Isaac H. Kim, Mary Beth Ruskai

Published Sep 1, 2014• 10 citations
arXivHTMLPDF

Abstract

We give new upper and lower bounds on the concavity of quantum entropy. Comparisons are given with other results in the literature.
DOI
10.1063/1.4895757

Related papers

Bounds on the Concavity of Quantum Entropy

Isaac H. Kim, Mary Beth Ruskai

We give new upper and lower bounds on the concavity of quantum entropy. Comparisons are given with other results in the literature.

•Jan 1, 2014•8 citations

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

Bounds on general entropy measures

Dominic William Berry, Barry C. Sanders

Abstract. We show how to determine the maximum and minimum possible values of one measure of entropy for a given value of another measure of entropy. These maximum and minimum values are obtained for two standard forms of probability distribution (or quantum state) independent of the entropy measures, provided the entropy measures satisfy a concavity/convexity relation. These results may be applied to entropies for classical probability distributions, entropies of mixed quantum states and measures of entanglement for pure states. PACS numbers: 03.67.–a

•Jan 1, 2004•18 citations

Bounds on general entropy measures

Dominic William Berry, Barry C. Sanders

We show how to determine the maximum and minimum possible values of one measure of entropy for a given value of another measure of entropy. These maximum and minimum values are obtained for two standard forms of probability distribution (or quantum state) independent of the entropy measures, provided the entropy measures satisfy a concavity/convexity relation. These results may be applied to entropies for classical probability distributions, entropies of mixed quantum states and measures of entanglement for pure states.

arXiv:quant-ph/0305059•Journal of Physics A Mathematical and General•Nov 26, 2003•30 citations

On upper bound for the quantum entropy

Waldemar Hebisch, Robert Olkiewicz, Bogusław Zegarliński

Linear Algebra and its Applications•May 1, 2001•2 citations

Proof of the strong subadditivity of quantum-mechanical entropy

Élliott H. Lieb, Mary Beth Ruskai

We prove several theorems about quantum-mechanical entropy, in particular, that it is strongly subadditive.

Journal of Mathematical Physics•Dec 1, 1973•876 citations
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.
Abstract. We show how to determine the maximum and minimum possible values of one measure of entropy for a given value of another measure of entropy. These maximum and minimum values are obtained for two standard forms of probability distribution (or quantum state) independent of the entropy measures, provided the entropy measures satisfy a concavity/convexity relation. These results may be applied to entropies for classical probability distributions, entropies of mixed quantum states and measures of entanglement for pure states. PACS numbers: 03.67.–a
We show how to determine the maximum and minimum possible values of one measure of entropy for a given value of another measure of entropy. These maximum and minimum values are obtained for two standard forms of probability distribution (or quantum state) independent of the entropy measures, provided the entropy measures satisfy a concavity/convexity relation. These results may be applied to entropies for classical probability distributions, entropies of mixed quantum states and measures of entanglement for pure states.