A Stronger Subadditivity of Entropy

Abstract

The strong subadditivity of entropy plays a key role in several areas of physics and mathematics. It states that the entropy S[ρ]= - Tr (ρ\ln ρ) of a density matrix ρ_{123} on the product of three Hilbert spaces satisfies S[ρ_{123}] - S[ρ_{23}] \leq S[ρ_{12}]- S[ρ_2]. We strengthen this to S[ρ_{123}] - S[ρ_{12}] \leq \sum_αn^α(S[ρ_{23}^α] - S[ρ_2^α]), where the n^αare weights and the ρ_{23}^αare partitions of ρ_{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.

LaTeX2e, 24 pages