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arXiv:math-ph/0603022

Comment on "Stronger subadditivity of entropy" by Lieb and Seiringer math-ph/0412009

Mary Beth Ruskai

Published Mar 8, 2006• 0 citations
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Abstract

We show how recent results of Lieb and Seiringer [math-ph/0412009; Phys. Rev. A 71, 062329 (2005)] can be obtained from repeated use of the monotonicity of relative entropy under partial traces, and explain how to use their approach to obtain tighter bounds in many situations.
DOI
10.48550/arxiv.math-ph/0603022

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The strong subadditivity of entropy plays a key role in several areas of physics and mathematics. It states that the entropy $S[\ensuremath{\varrho}]=\ensuremath{-}\mathrm{Tr}(\ensuremath{\varrho}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varrho})$ of a density matrix ${\ensuremath{\varrho}}_{123}$ on the product of three Hilbert spaces satisfies $S[{\ensuremath{\varrho}}_{123}]\ensuremath{-}S[{\ensuremath{\varrho}}_{12}]\ensuremath{\leqslant}S[{\ensuremath{\varrho}}_{23}]\ensuremath{-}S[{\ensuremath{\varrho}}_{2}]$. We strengthen this to $S[{\ensuremath{\varrho}}_{123}]\ensuremath{-}S[{\ensuremath{\varrho}}_{12}]\ensuremath{\leqslant}{\ensuremath{\sum}}_{\ensuremath{\alpha}}{n}^{\ensuremath{\alpha}}(S[{\ensuremath{\varrho}}_{23}^{\ensuremath{\alpha}}]\ensuremath{-}S[{\ensuremath{\varrho}}_{2}^{\ensuremath{\alpha}}])$, where the ${n}^{\ensuremath{\alpha}}$ are weights and the ${\ensuremath{\varrho}}_{23}^{\ensuremath{\alpha}}$ are partitions of ${\ensuremath{\varrho}}_{23}$. Correspondingly, there is a strengthening of the theorem that the map $A\ensuremath{\mapsto}\mathrm{Tr}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[L+\mathrm{ln}\phantom{\rule{0.2em}{0ex}}A]$ is concave. As applications we prove some monotonicity and convexity properties of the Wehrl coherent state entropy and entropy inequalities for quantum gases.
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