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Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement

David P. DiVincenzo·Tal Mor·Peter W. Shor·John A. Smolin·Barbara M. Terhal
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Abstract

We report new results and generalizations of our work on unextendible product bases (UPB), uncompletable product bases and bound entanglement. We present a new construction for bound entangled states based on product bases which are only completable in a locally extended Hilbert space. We introduce a very useful representation of a product basis, an orthogonality graph. Using this representation we give a complete characterization of unextendible product bases for two qutrits. We present several generalizations of UPBs to arbitrary high dimensions and multipartite systems. We present a sufficient condition for sets of orthogonal product states to be distinguishable by separable superoperators. We prove that bound entangled states cannot help increase the distillable entanglement of a state beyond its regularized entanglement of formation assisted by bound entanglement.

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David P. DiVincenzo, Tal Mor, Peter W. Shor, John A. Smolin, Barbara M. Terhal. 2000-11-06. Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement. https://doi.org/10.1007/s00220-003-0877-6

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