The inequalities of quantum information theory

Let rho denote the density matrix of a quantum state having n parts 1,...,n. For I subset of or equal to N = {1,...,n},let rho(I) = Tr-N\I(rho) denote the density matrix of the state comprising those parts i such that i is an element of I, and let S(rho(I)) denote the von Neumann entropy of the state rho(I). The collection of nu = 2(n) numbers {S(rho(I))}(Isubset of or equal toN) may be regarded as a point, called the allocation of entropy for rho, in the vector space R-nu. Let A(n)denote the set of points in R-nu that are allocations of entropy for n-part quantum states. We show that (A(n)) over bar (the topological closure of A(n)) is a closed convex cone in R-nu. 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 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-nu determined by these inequalities. We show that (A(n)) over bar = B-n for n less than or equal to 3. The status of this equality is open for n greater than or equal to 4. We also consider a symmetric version of this situation, in which S(rho(I)) depends on I only through the number i = #I of indexes in I and can thus be denoted S(rho(i)). In this case, we give for each n a finite complete and independent set of inequalities governing the symmetric allocations of entropy {S(rho(i))}(0<i<n) in Rn+1.