Communications in Mathematical Physics

, Volume 238, Issue 3, pp 379–410 | Cite as

Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement

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


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.


Hilbert Space High Dimension Entangle State Product State Complete Characterization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • David P. DiVincenzo
    • 1
  • Tal Mor
    • 2
  • Peter W. Shor
    • 3
  • John A. Smolin
    • 1
  • Barbara M. Terhal
    • 1
  1. 1.IBM T.J. Watson Research CenterUSA
  2. 2.Dept. of Electrical EngineeringUCLALos AngelesUSA
  3. 3.AT&T ResearchFlorham ParkUSA

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