The geometry of entanglement witnesses and local detection of entanglement

Abstract

Let $H^{[ N]}=H^{[ d_{1}]}\otimes ... \otimes H^{[ d_{n}]}$ be a tensor product of Hilbert spaces and let $τ_{0}$ be the closest separable state in the Hilbert-Schmidt norm to an entangled state $ρ_{0}$. Let $\tildeτ_{0}$ denote the closest separable state to $ρ_{0}$ along the line segment from $I/N$ to $ρ_{0}$ where $I$ is the identity matrix. Following [pitrubmat] a witness $W_{0}$ detecting the entanglement of $ρ_{0}$ can be constructed in terms of $I, τ_{0}$ and $\tildeτ_{0}$. If representations of $τ_{0}$ and $\tildeτ_{0}$ as convex combinations of separable projections are known, then the entanglement of $ρ_{0}$ can be detected by local measurements. Gühne \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, $τ_{0}=\tildeτ_{0}$. We illustrate our general approach with a two parameter family of three qubit bound entangled states for which $τ_{0} \neq \tildeτ_{0}$ 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 $ρ_{0}$ 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 $μ_{0}$ on the far face from $ρ_{0}$. Up to a normalization this coincides with the witness obtained in [bruss1] for the particular example analyzed there.