The Inequalities of Quantum Information Theory

Given an n-part quantum state, we consider the 2ⁿ substates obtained by restricting attention to a subset of the parts. The entropies (in the sense of von Neumann) of these substates may be regarded as a point, called the allocation of entropy, in a (2ⁿ)-dimensional real vector space. We show that the topological closure of the set of allocations of entropy form a convex cone. We show that a set of inequalities due to Lieb and Ruskai characterize this cone when n is at most 3. We also consider the symmetric situation in which the entropy depends only on the number of parts in the substate. In this case, the topological closure of the set of allocations of entropy (in (n+1)-dimensional space) again form a convex cone, and we give inequalities characterizing this cone for all n.