Checking the optimality of entanglement witnesses: an application to structural physical approximations

Abstract

In 2008, the conjecture that structural physical approximations to optimal entanglement witnesses are separable states (in general unnormalized) was posed. In an attempt to disprove it, in [K.-C. Ha and S.-H. Kye, Separable states with unique decompositions, arXiv:1210.1088v3], Ha and Kye proposed a decomposable entanglement witness whose SPA is entangled and argued that it is optimal. In this note, which is based on a comment to the latter work [R. Augusiak et al., Comment on "Separable states with unique decompositions", arXiv:1304.2040v1], we show, both analytically and numerically, that this entanglement witness is not optimal, and as such it is not a counterexample to the conjecture. To this end, we make use of a method for checking optimality of entanglement witnesses developed already in [M. Lewenstein et al., Phys. Rev. A 62, 052310 (2000)], however, hardly exploited so far in the literature.

10 pages, one figure, paper based on the comment arXiv:1304.2040v1

References in corpus (9)

• Entanglement detection
• Structural approximations to positive maps and entanglement breaking channels
• Generic local distinguishability and completely entangled subspaces
• Entanglement witnesses arising from exposed positive linear maps
• Separable states with unique decompositions
• Exposed positive maps: a sufficient condition
• The structural physical approximations and optimal entanglement witnesses
• Disproving the conjecture on structural physical approximation to optimal decomposable entanglement witnesses
• Extremal entanglement witnesses

Cited by in corpus (8)

• Free versus Bound Entanglement: Machine learning tackling a NP-hard problem
• Separable states with unique decompositions
• Comparing bound entanglement of bell diagonal pairs of qutrits and ququarts
• Almost complete solution for the NP-hard separability problem of Bell diagonal qutrits
• On witnessing arbitrary bipartite entanglement in a measurement device independent way
• On the structure of mirrored operators obtained from optimal entanglement witnesses
• Physical realization of realignment criteria using structural physical approximation
• Structural physical approximation of partial transposition makes possible to distinguish SLOCC inequivalent classes of three-qubit system