Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement

• DOI: 10.1007/s00220-003-0877-6
• Publication Date: 2004-10-15T01:25:16Z
• Authors: David P. DiVincenzo; Tal Mor; Peter W. Shor; John A. Smolin; Barbara M. Terhal
• Keywords: Quantum Physics; 0103 physical sciences; FOS: Physical sciences; Quantum Physics (quant-ph); 01 natural sciences

24 pages RevTex, 15 figures; appendix removed, several small corrections, to appear in Comm. Math. Phys<br/>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.<br/>