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

TL;DR

This paper investigates conditions under which product vectors with certain conjugate properties exist in subspaces of tensor product spaces, revealing a threshold related to their codimensions.

Contribution

It establishes a precise codimension condition determining the existence of product vectors with partial conjugates in tensor subspaces.

Findings

Existence of product vectors depends on the sum of codimensions relative to space dimensions.

If $k+fell < m+n-2$, such vectors always exist.

If $k+fell > m+n-2$, such vectors may not exist.

Abstract

Let $D$ and $E$ be subspaces of the tensor product of the $m$ and $n$ dimensional complex spaces, with codimensions $k$ and \ell$, respectively. We show that if $k+\ell<m+n-2$ then there must exist a product vector in $D$ whose partial conjugate lies in $E$. If $k+\ell >m+n-2$ then there may not exist such a product vector. If $k+\ell=m+n-2$ then both cases may occur depending on $k$ and $\ell$.