Distinguishability of States and von Neumann Entropy

Abstract

Consider an ensemble of pure quantum states |ψ_j>, j=1,...,n taken with prior probabilities p_j respectively. We show that it is possible to increase all of the pairwise overlaps |<ψ_i|ψ_j>| i.e. make each constituent pair of the states more parallel (while keeping the prior probabilities the same), in such a way that the von Neumann entropy S is increased, and dually, make all pairs more orthogonal while decreasing S. We show that this phenomenon cannot occur for ensembles in two dimensions but that it is a feature of almost all ensembles of three states in three dimensions. It is known that the von Neumann entropy characterises the classical and quantum information capacities of the ensemble and we argue that information capacity in turn, is a manifestation of the distinguishability of the signal states. Hence our result shows that the notion of distinguishability within an ensemble is a global property that cannot be reduced to considering distinguishability of each constituent pair of states.