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 ${\tau }_0$ 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. 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, ${\tau }_0={\tilde{\tau }}_0$. 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 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.

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