A Stronger Subadditivity of Entropy

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.