A Stronger Subadditivity of Entropy

TL;DR

This paper strengthens the strong subadditivity of entropy, providing a more general inequality and exploring its implications for quantum entropy properties and related inequalities.

Contribution

It introduces a stronger form of the strong subadditivity of entropy and extends related concavity theorems, with applications to quantum entropy inequalities.

Findings

Strengthened the strong subadditivity inequality for entropy.

Proved new monotonicity and convexity properties of Wehrl entropy.

Derived entropy inequalities for quantum gases.

Abstract

The strong subadditivity of entropy plays a key role in several areas of physics and mathematics. It states that the entropy S[\rho]= - Tr (\rho \ln \rho) of a density matrix \rho_{123} on the product of three Hilbert spaces satisfies S[\rho_{123}] - S[\rho_{23}] \leq S[\rho_{12}]- S[\rho_2]. We strengthen this to S[\rho_{123}] - S[\rho_{12}] \leq \sum_\alpha n^\alpha (S[\rho_{23}^\alpha ] - S[\rho_2^\alpha ]), where the n^\alpha are weights and the \rho_{23}^\alpha are partitions of \rho_{23}. Correspondingly, there is a strengthening of the theorem that the map A -> Tr \exp[L + \ln A] is concave. As applications we prove some monotonicity and convexity properties of the Wehrl entropy and entropy inequalities for quantum gases.