Relaxations of separability in multipartite systems: Semidefinite programs, witnesses and volumes

Abstract

While entanglement is believed to be an important ingredient in understanding quantum many-body physics, the complexity of its characterization scales very unfavorably with the size of the system. Finding super-sets of the set of separable states that admit a simpler description has proven to be a fruitful approach in the bipartite setting. In this paper we discuss a systematic way of characterizing multiparticle entanglement via various relaxations. We furthermore describe an operational witness construction arising from such relaxations that is capable of detecting every entangled state. Finally, we also derive an analytic upper-bound on the volume of biseparable states and show that the volume of the states with a positive partial transpose for any split rapidly outgrows this volume. This proves that simple semi-definite relaxations in the multiparticle case cannot be an equally good approximation for any scenario.

26 pages. In v2: proposed SDP implemented, analytical example included, typos corrected, references added (published version)

References in corpus (7)

• Entanglement detection
• A complete family of separability criteria
• Taming multiparticle entanglement
• The structure of multidimensional entanglement in multipartite systems
• Witnessing genuine multipartite entanglement with positive maps
• Separable Multipartite Mixed States - Operational Asymptotically Necessary and Sufficient Conditions
• A Robust Semidefinite Programming Approach to the Separability Problem

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