Geometry of entanglement witnesses and local detection of entanglement

Let $H^{\left[N\right]}{=H}^{\left[d_1\right]}\otimes \cdots \otimes H^{\left[d_n\right]}$ 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 ${\tilde{\tau }}_0$ 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 ${\tilde{\tau }}_0.$ If representations of ${\tau }_0$ and ${\tilde{\tau }}_0$ as convex combinations of separable projections are known, then the entanglement of ${\rho }_0$ can be detected by local measurements. Gühne et al. [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, ${\tau }_0={\tilde{\tau }}_0.$ We illustrate our general approach with a two-parameter family of three-qubit bound entangled states for which ${\tau }_0\ne {\tilde{\tau }}_0$ 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 Gühne et al. for the particular example analyzed there.