The relative entropy of two n-party quantum states is an important quantity
exhibiting, for example, the extent to which the two states are different. The
relative entropy of the states formed by reducing two n-party to a smaller
number m of parties is always less than or equal to the relative entropy of
the two original n-party states. This is the monotonicity of relative entropy.
Using techniques from convex geometry, we prove that monotonicity under
restrictions is the only general inequality satisfied by relative entropies. In
doing so we make a connection to secret sharing schemes with general access
structures.
A suprising outcome is that the structure of allowed relative entropy values
of subsets of multiparty states is much simpler than the structure of allowed
entropy values. And the structure of allowed relative entropy values (unlike
that of entropies) is the same for classical probability distributions and
quantum states.Comment: 15 pages, 3 embedded eps figure
