A Subspace of Maximal Dimension with Bounded Schmidt Rank

Abstract

We study Schmidt rank for a vector (i.e., a pure state) and Schmidt number for a mixed state which are entanglement measures. We show that if a subspace of a certain bipartite system contains no vector of Schmidt rank $⩽k$, then any state supported on that space has Schmidt number at least $k+1$. A construction of subspace of ${\mathbb{C}}^m\otimes {\mathbb{C}}^n$ of maximal dimension, which does not contain any vector of Schmidt rank less than $3$, is given here.

6 pages