Geometry of entanglement witnesses and local detection of entanglement

Let H-[N]=H-1([d)]circle times.circle timesH(n)([d)] be a tensor product of Hilbert spaces and let tau(0) be the closest separable state in the Hilbert-Schmidt norm to an entangled state rho(0). Let tau(0) denote the closest separable state to rho(0) along the line segment from I/N to rho(0) where I is the identity matrix. Following A. O. Pittenger and M. H. Rubin [Linear Algebr. Appl. 346, 75 (2002)] a witness W-0 detecting the entanglement of rho(0) can be constructed in terms of I, tau(0), and tau(0). If representations of tau(0) and tau(0) as convex combinations of separable projections are known, then the entanglement of rho(0) can be detected by local measurements. Guhne [Phys. Rev. A 66, 062305 (2002)] obtain the minimum number of measurement settings required for a class of two-qubit states. We use our geometric approach to generalize their result to the corresponding two-qudit case when d is prime and obtain the minimum number of measurement settings. In those particular bipartite cases, tau(0)=tau(0). We illustrate our general approach with a two-parameter family of three-qubit bound entangled states for which tau(0)not equaltau(0) and we show that our approach works for n qubits. We elaborated earlier [A. O. Pittenger, Linear Algebr. App. 359, 235 (2003)] on the role of a "far face" of the separable states relative to a bound entangled state rho(0) constructed from an orthogonal unextendible product base. In this paper the geometric approach leads to an entanglement witness expressible in terms of a constant times I and a separable density mu(0) on the far face from rho(0). Up to a normalization this coincides with the witness obtained by Guhne for the particular example analyzed there.