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[ϱ]=−Tr(ϱ ln ϱ) of a density matrix ϱ₁₂₃ on the product of three Hilbert spaces satisfies S[ϱ₁₂₃]−S[ϱ₁₂]⩽S[ϱ₂₃]−S[ϱ₂]. We strengthen this to S[ϱ₁₂₃]−S[ϱ₁₂]⩽∑_αn^α(S[ϱ₂₃^α]−S[ϱ₂^α]), where the n^α are weights and the ϱ₂₃^α are partitions of ϱ₂₃. 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.