The geometry of entanglement witnesses and local detection of entanglement

Let $H^{[ N]}=H^{[ d_{1}]}\otimes ... \otimes H^{[ d_{n}]}$ 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 [pitrubmat] 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. G\"{u}hne \textit{et. al.} in [bruss1] 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 $\tau_{0} \neq \tilde{\tau}_{0}$ and we show our approach works for $n$ qubits. In [pitt] we elaborated 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 in [bruss1] for the particular example analyzed there.