  Strong subadditivity of 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 prove 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.
