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[̺] = −Tr ( ̺ ln̺ ) of a density matrix ̺123 on the product of three Hilbert spaces satisfies S[̺123] − S[̺23] ≤ S[̺12] − S[̺2]. We strengthen this to S[̺123]−S[̺12] ≤ ∑ α n ( S[̺ 23 ]−S[̺α 2 ] ) , where the n are weights and the ̺ 23 are partitions of ̺23. Correspondingly, there is a strengthening of the theorem that the map A 7→ Tr exp[L + lnA] is concave. As applications we prove some monotonicity and convexity properties of the Wehrl entropy and entropy inequalities for quantum gases.