Unextendible Product Bases

Let C denote the complex field. A vector v in the tensor product ⊗^m_i=1C^ki is called a pure product vector if it is a vector of the form v₁⊗v₂...⊗v_m, with v_i∈C^ki. 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 ⊗^m_i=1C^ki 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+∑^m_i=1(k_i-1) for every sequence of integers k₁, k₂, ..., k_m≥2 unless either (i) m=2 and 2∈{k₁, k₂} or (ii) 1+∑^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+∑^m_i=1(k_i-1).