Abstract
Let be a tensor product of Hilbert spaces and let be the closest separable state in the Hilbert-Schmidt norm to an entangled state Let denote the closest separable state to along the line segment from to where I is the identity matrix. Following A. O. Pittenger and M. H. Rubin [Linear Algebr. Appl. 346, 75 (2002)] a witness detecting the entanglement of can be constructed in terms of I, and If representations of and as convex combinations of separable projections are known, then the entanglement of 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, We illustrate our general approach with a two-parameter family of three-qubit bound entangled states for which 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 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 on the far face from Up to a normalization this coincides with the witness obtained by Gühne et al. for the particular example analyzed there.
- Received 8 August 2002
DOI:https://doi.org/10.1103/PhysRevA.67.012327
©2003 American Physical Society