The inequalities of quantum information theory

Let ρ denote the density matrix of a quantum state having n parts 1, ..., n. For I⊆N={1, ..., n}, let ρ_I=Tr_NI/(ρ) denote the density matrix of the state comprising those parts i such that i∈I, and let S(ρ_I) denote the von Neumann (1927) entropy of the state ρ_I. The collection of ν=2ⁿ numbers {S(ρ_I)}_I⊆N may be regarded as a point, called the allocation of entropy for ρ, in the vector space R^ν. Let A_n denote the set of points in R^ν that are allocations of entropy for n-part quantum states. We show that A~_n~ (the topological closure of A_n) is a closed convex cone in R^ν. This implies that the approximate achievability of a point as an allocation of entropy is determined by the linear inequalities that it satisfies. Lieb and Ruskai (1973) have established a number of inequalities for multipartite quantum states (strong subadditivity and weak monotonicity). We give a finite set of instances of these inequalities that is complete (in the sense that any valid linear inequality for allocations of entropy can be deduced from them by taking positive linear combinations) and independent (in the sense that none of them can be deduced from the others by taking positive linear combinations). Let B_n denote the polyhedral cone in R^ν determined by these inequalities. We show that A~_n~=B_n for n≤3. The status of this equality is open for n≥4. We also consider a symmetric version of this situation, in which S(ρ_I) depends on I only through the number i=≠I of indexes in I and can thus be denoted S(ρ_i). In this case, we give for each n a finite complete and independent set of inequalities governing the symmetric allocations of entropy {S(ρ_i)}_0≤i≤n in Rⁿ⁺¹.