Unextendible product bases

Let C denote the complex field. A vector upsilon in the tensor product circle times (m)(i=1) C-k,C- is called a pure produc vector if it is a vector of the form nu (1) circle times upsilon (2) ... circle times nu (m), with nu is an element of C-k,C-. 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 circle times (m)(i=1) C-k,C- 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 + Sigma (m)(i=1)(k(i-1)) for every sequence of integers k(1), k(2).... k(m) greater than or equal to 2 unless either (i) m = 2 and 2 is an element of {k(1), k(2)} or (ii) 1 +Sigma (m)(i=1) (k(i-1)) is odd and at least one k(i) is even. In each of these two cases. the minimum cardinality of the corresponding F is strictly bigger than 1 + Sigma (m)(i=1)(k(i) - 1). (C) 2001 Academic Press.