2006/02/28 by Lawrence M. Ioannou, B. C. Travaglione, Benjamin C. Travaglione
Computer Science · Mathematics · Physics and Astronomy · #Algorithm #Computer science #Discrete mathematics #Entanglement witness #Mathematics #Multipartite entanglement #Observable #Physics #Polytope #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum discord #Quantum entanglement #Quantum mechanics #Quantum state #Separable state #Squashed entanglement #State (computer science) #quant-ph
paper · pdf · doi:10.1103/physreva.73.052314
published as Phys. Rev. A 73, 052314 (2006)
openalex publication_date 2006/05/23 · arxiv created 2006/06/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We focus on determining the separability of an unknown bipartite quantum state \ensuremathρ by invoking a sufficiently large subset of all possible entanglement witnesses given the expected value of each element of a set of mutually orthogonal observables. We review the concept of an entanglement witness from the geometrical point of view and use this geometry to show that the set of separable states is not a polytope and to characterize the class of entanglement witnesses (observables) that detect entangled states on opposite sides of the set of separable states. All this serves to motivate a classical algorithm which, given the expected values of a subset of an orthogonal basis of observables of an otherwise unknown quantum state, searches for an entanglement witness in the span of the subset of observables. The idea of such an algorithm, which is an efficient reduction of the quantum separability problem to a global optimization problem, was introduced by [Ioannou et al., Phys. Rev. A 70, 060303(R)], where it was shown to be an improvement on the naive approach for the quantum separability problem (exhaustive search for a decomposition of the given state into a convex combination of separable states). The last section of the paper discusses in more generality such algorithms, which, in our case, assume a subroutine that computes the global maximum of a real function of several variables. Despite this, we anticipate that such algorithms will perform sufficiently well on small instances that they will render a feasible test for separability in some cases of interest (e.g., in 3\ifmmode×\else\texttimes\fi3 dimensional systems).