dorsal/arxiv
View SchemaInequalities for Quantum Entropy: A Review with Conditions for Equality
| Authors | Mary Beth Ruskai |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0205064 |
| URL | https://arxiv.org/abs/quant-ph/0205064 |
| DOI | 10.1063/1.1497701 |
| Journal | J. Math. Phys. 43, 4358-4375 (2002); erratum 46, 019901 (2005). |
Abstract
This paper presents self-contained proofs of the strong subadditivity inequality for quantum entropy and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein's inequality and one of Lieb's less well-known concave trace functions, allows one to obtain conditions for equality. Using the fact that the Holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy, we show that equality can be attained for that bound only when the states in the ensemble commute. The paper concludes with an Appendix giving a short description of Epstein's elegant proof of the relevant concavity theorem of Lieb.
{
"annotation_id": "45b6dd09-1a3b-42ca-9859-ff924b1284c6",
"date_created": "2026-03-02T18:01:52.883000Z",
"date_modified": "2026-03-02T18:01:52.883000Z",
"file_hash": "b5e19c3b3c692c3b72420775fc20dc565dcf56cb99393aeaa875723f98500d8c",
"private": false,
"record": {
"abstract": "This paper presents self-contained proofs of the strong subadditivity\ninequality for quantum entropy and some related inequalities for the quantum\nrelative entropy, most notably its convexity and its monotonicity under\nstochastic maps. Moreover, the approach presented here, which is based on\nKlein\u0027s inequality and one of Lieb\u0027s less well-known concave trace functions,\nallows one to obtain conditions for equality. Using the fact that the Holevo\nbound on the accessible information in a quantum ensemble can be obtained as a\nconsequence of the monotonicity of relative entropy, we show that equality can\nbe attained for that bound only when the states in the ensemble commute. The\npaper concludes with an Appendix giving a short description of Epstein\u0027s\nelegant proof of the relevant concavity theorem of Lieb.",
"arxiv_id": "quant-ph/0205064",
"authors": [
"Mary Beth Ruskai"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"doi": "10.1063/1.1497701",
"journal_ref": "J. Math. Phys. 43, 4358-4375 (2002); erratum 46, 019901 (2005).",
"title": "Inequalities for Quantum Entropy: A Review with Conditions for Equality",
"url": "https://arxiv.org/abs/quant-ph/0205064"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "1b435263-0023-4ec3-8daf-be7d5f645896",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}