Coauthors: Noah Linden, Ben Ibinson

This talk will be mainly based on a paper with N Linden. Pippenger has initiated the generalization of the programme to find all the "laws of information theory" to quantum entropy. The standard inequalities derive from strong subadditivity (SSA). SSA of the von Neumann entropy, proved in 1973 by Lieb and Ruskai, is a cornerstone of quantum coding theory. All other known inequalities for entropies of quantum systems may be derived from it. Here we prove a new inequality for the von Neumann entropy which we show is independent of strong subadditivity: it is an inequality which is true for any four party quantum state, provided that it satisfies three linear relations (constraints) on the entropies of certain reduced states. In the talk I will also discuss the possibility of finding an unconstrained inequality (work with N Linden and B Ibinson).