Unextendible product bases, uncompletable product bases and bound entanglement

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

Research output: Contribution to journalArticleScientificpeer-review

200 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)379-410
Number of pages32
JournalCommunications in Mathematical Physics
Issue number3
Publication statusPublished - 2003
Externally publishedYes


  • Hilbert Space
  • High Dimension
  • Entangle State
  • Product State
  • Complete Characterization

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