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 12] <= S[rho(23])-S[rho(2)]. We strengthen this to S[rho(123)]-S rho(12)] <= Sigma(alpha)n(alpha) (S[rho(alpha)(23)]-S[rho(alpha)(2)]), where the n alpha are weights and the rho(alpha)(23) 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 coherent state entropy and entropy inequalities for quantum gases.