The Minimum Size of Unextendible Product Bases in the Bipartite Case (and Some Multipartite Cases)
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A long-standing open question asks for the minimum number of vectors needed to form an unextendible product basis in a given bipartite or multipartite Hilbert space. A partial solution was found by Alon and Lovász (J. Comb. Theory Ser. A, 95:169–179, 2001), but since then only a few other cases have been solved. We solve all remaining bipartite cases, as well as a large family of multipartite cases.
KeywordsMinimum Size Orthogonality Condition Permutation Matrix Positive Partial Transpose Nonzero Product
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