unextendible product basis
In quantum mechanics, an unextendible product basis is a set of orthogonal, non-entangled state vectors for a multipartite system, with the property that local operations and classical communication are insufficient to distinguish one member of the set from the others. Because these states are product states and yet local measurements cannot tell them apart, they are sometimes said to exhibit "nonlocality without entanglement".{{Cite journal |last=Bennett |first=Charles H. |author-link=Charles H. Bennett (physicist) |last2=DiVincenzo |first2=David P. |author-link2=David DiVincenzo |last3=Fuchs |first3=Christopher A. |last4=Mor |first4=Tal |last5=Rains |first5=Eric |author-link5=Eric Rains |last6=Shor |first6=Peter W. |author-link6=Peter Shor |last7=Smolin |first7=John A. |author-link7=John A. Smolin |last8=Wootters |first8=William K. |author-link8=William Wootters |date=1999-02-01 |title=Quantum nonlocality without entanglement |url=https://journals.aps.org/pra/abstract/10.1103/PhysRevA.59.1070 |journal=Physical Review A |language=en |volume=59 |issue=2 |pages=1070–1091 |arxiv=quant-ph/9804053 |doi=10.1103/PhysRevA.59.1070}}{{Cite journal |last=Bennett |first=Charles H. |author-link=Charles H. Bennett (physicist) |last2=DiVincenzo |first2=David P. |author-link2=David DiVincenzo |last3=Mor |first3=Tal |last4=Shor |first4=Peter W. |author-link4=Peter Shor |last5=Smolin |first5=John A. |author-link5=John Smolin |last6=Terhal |first6=Barbara M. |author-link6=Barbara Terhal |date=1999-06-28 |title=Unextendible Product Bases and Bound Entanglement |url=https://link.aps.org/doi/10.1103/PhysRevLett.82.5385 |journal=Physical Review Letters |language=en |volume=82 |issue=26 |pages=5385–5388 |arxiv=quant-ph/9808030 |doi=10.1103/PhysRevLett.82.5385}} They provide examples of non-entangled states that pass the Peres–Horodecki criterion for entanglement.{{Cite book |last=Bengtsson |first=Ingemar |title=Geometry of Quantum States: An Introduction to Quantum Entanglement |title-link=Geometry of Quantum States |last2=Życzkowski |first2=Karol |author-link2=Karol Życzkowski |date= |publisher=Cambridge University Press |year=2017 |isbn=978-1-107-02625-4 |edition=2nd |location=Cambridge, UK |pages=458-460}}