Hilbert-Schmidt distance and entanglement witnessing

Abstract

Gilbert proposed an algorithm for bounding the distance between a given point and a convex set. In this article we apply the Gilbert's algorithm to get an upper bound on the Hilbert-Schmidt distance between a given state and the set of separable states. While Hilbert Schmidt Distance does not form a proper entanglement measure, it can nevertheless be useful for witnessing entanglement. We provide here a few methods based on the Gilbert's algorithm that can reliably qualify a given state as strongly entangled or practically separable, while being computationally efficient. The method also outputs successively improved approximations to the Closest Separable State for the given state. We demonstrate the efficacy of the method with examples.

arXiv comment6 pages, 7 figures (on the occasion of opening ICTQT, Gdansk). Updated figures and data

References in corpus (21)

• Mixed State Entanglement and Quantum Error Correction
• Separability Criterion for Density Matrices
• Separability of Mixed States: Necessary and Sufficient Conditions
• Concentrating Partial Entanglement by Local Operations
• Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels
• Mixed-state entanglement and distillation: is there a ``bound'' entanglement in nature?
• Bell inequalities for arbitrarily high dimensional systems
• Unextendible Product Bases and Bound Entanglement
• Classification of mixed three-qubit states
• A complete family of separability criteria
• Induced measures in the space of mixed quantum states
• The asymptotic entanglement cost of preparing a quantum state
• Limits for entanglement measures
• Separability of n-particle mixed states: necessary and sufficient conditions in terms of linear maps
• Entanglement measures and the Hilbert-Schmidt distance
• Detecting multipartite entanglement
• A new entanglement measure induced by the Hilbert-Schmidt norm
• Convex optimization over classes of multiparticle entanglement
• Non-full rank bound entangled states satisfying the range criterion
• Convex separation from convex optimization for large-scale problems
• Distance between Bound Entangled States from Unextendible Product Bases and Separable States

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• Long-distance genuine multipartite Entanglement between Magnetic Defects in Spin Chains
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