Unextendible Product Bases

Let C denote the complex field. A vector v in the tensor product ⊗mi=1 Cki is called a pure product vector if it is a vector of the form v1⊗v2…⊗vm, with vi∈Cki. A set F of pure product vectors is called an unextendible product basis if F consists of orthogonal nonzero vectors, and there is no nonzero pure product vector in ⊗mi=1 Cki which is orthogonal to all members of F. The construction of such sets of small cardinality is motivated by a problem in quantum information theory. Here it is shown that the minimum possible cardinality of such a set F is precisely 1+∑mi=1 (ki−1) for every sequence of integers k1, k2, …, km⩾2 unless either (i) m=2 and 2∈{k1, k2} or (ii) 1+∑mi=1(ki−1) is odd and at least one ki is even. In each of these two cases, the minimum cardinality of the corresponding F is strictly bigger than 1+∑mi=1(ki−1).