Investigating a Class of $2\otimes 2\otimes d$ Chessboard Density Matrices via Linear and Non-linear Entanglement Witnesses Constructed by Exact Convex Optimization

Abstract

Here we consider a class of $2\otimes 2\otimes d$ chessboard density matrices starting with three-qubit ones which have positive partial transposes with respect to all subsystems. To investigate the entanglement of these density matrices, we use the entanglement witness approach. For constructing entanglement witnesses (EWs) detecting these density matrices, we attempt to convert the problem to an exact convex optimization problem. To this aim, we map the convex set of separable states into a convex region, named feasible region, and consider cases that the exact geometrical shape of feasible region can be obtained. In this way, various linear and non-linear EWs are constructed. The optimality and decomposability of some of introduced EWs are also considered. Furthermore, the detection of the density matrices by introduced EWs are discussed analytically and numerically. {\bf Keywords: chessboard density matrices, optimal non-linear entanglement witnesses, convex optimization} {\bf PACs Index: 03.65.Ud}

arXiv comment35 pages, 1 figures

References in corpus (25)

• Separability of Mixed States: Necessary and Sufficient Conditions
• A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound
• Bell Inequalities and the Separability Criterion
• Classification of mixed three-qubit states
• A complete family of separability criteria
• Bound entanglement can be activated
• Distinguishing separable and entangled states
• Secure key from bound entanglement
• Detecting multipartite entanglement
• Optimum Quantum Error Recovery using Semidefinite Programming
• Complete hierarchies of efficient approximations to problems in entanglement theory
• The asymptotic relative entropy of entanglement
• Optimizing Completely Positive Maps using Semidefinite Programming
• Bounds on Quantum Correlations in Bell Inequality Experiments
• Activating NPPT distillation with an infinitesimal amount of bound entanglement
• A Robust Semidefinite Programming Approach to the Separability Problem
• The geometry of entanglement witnesses and local detection of entanglement
• Generation and detection of bound entanglement
• Bell states diagonal entanglement witnesses
• Manipulating Multi-qudit Entanglement Witnesses by Using Linear Programming
• Multipartite unlockable bound entanglement in the stabilizer formalism
• Bell-states diagonal entanglement witnesses for relativistic and non-relativistic multispinor systems in arbitrary dimensions
• Generalized qudit Choi maps
• Multi-qubit stabilizer and cluster entanglement witnesses
• Finite quantum tomography via semidefinite programming

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