Convexity and the separability problem of quantum mechanical density matrices

AbstractA finite-dimensional quantum mechanical system is modelled by a density ρ, a trace one, positive semi-definite matrix on a suitable tensor product space H[N]. For the system to demonstrate experimentally certain non-classical behavior, ρ cannot be in S, a closed convex set of densities whose extreme points have a specificed tensor product form. Two mathematical problems in the quantum computing literature arise from this context: 1.the determination whether a given ρ is in S, and2.a measure of the “entanglement” of such a ρ in terms of its distance from S. In this paper we describe these two problems in detail for a linear algebra audience, discuss some recent results from the quantum computing literature, and prove some new results. We emphasize the roles of densities ρ as both operators on the Hilbert space H[N] and also as points in a real Hilbert space M. We are able to compute the nearest separable densities τ0 to ρ0 in particular classes of inseparable densities and we use the Euclidean distance between the two in M to quantify the entanglement of ρ0. We also show the role of τ0 in the construction of separating hyperplanes, so-called entanglement witnesses in the quantum computing literature