 
Summary: Unextendible Product Bases
N. Alon
L. Lov´asz
February 22, 2002
Abstract
Let C denote the complex field. A vector v in the tensor product m
i=1Cki
is called a pure product
vector if it is a vector of the form v1 v2 ˇ ˇ ˇ 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 m
i=1Cki
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(ki  1) for every
sequence of integers k1, k2, . . . , km 2 unless either (i) m = 2 and 2 {k1, k2} or (ii)1+
m
