Existence of product vectors and their partial conjugates in a pair of spaces

Abstract

Let D and E be subspaces of the tensor product of the m- and n-dimensional complex spaces, with co-dimensions k and ℓ, respectively. In order to give upper bounds for ranks of entangled edge states with positive partial transposes, we show that if k + ℓ < m + n − 2, then there must exist a product vector in D whose partial conjugate lies in E. If k + ℓ = m + n − 2, then such a product vector may or may not exist depending on k and ℓ.