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 $S\left[\varrho \right]=-\mathrm{Tr}\left(\varrho \ln \varrho \right)$ of a density matrix ${\varrho }_{123}$ on the product of three Hilbert spaces satisfies $S\left[{\varrho }_{123}\right]-S\left[{\varrho }_{12}\right]⩽S\left[{\varrho }_{23}\right]-S\left[{\varrho }_2\right]$. We strengthen this to $S\left[{\varrho }_{123}\right]-S\left[{\varrho }_{12}\right]⩽{\sum }_{\alpha }{n}^{\alpha }(S\left[{\varrho }_{23}^{\alpha }\right]-S\left[{\varrho }_2^{\alpha }\right])$, where the $n^{\alpha }$ are weights and the ${\varrho }_{23}^{\alpha }$ are partitions of ${\varrho }_{23}$. Correspondingly, there is a strengthening of the theorem that the map $A\mapsto \mathrm{Tr}\exp [L+\ln A]$ is concave. As applications we prove some monotonicity and convexity properties of the Wehrl coherent state entropy and entropy inequalities for quantum gases.