A Subspace of Maximal Dimension with Bounded Schmidt Rank

TL;DR

This paper investigates the properties of subspaces in bipartite quantum systems that exclude vectors of low Schmidt rank, establishing bounds on entanglement measures and constructing maximal subspaces with specific entanglement properties.

Contribution

It introduces a construction of maximal dimension subspaces in bipartite systems that contain no vectors of Schmidt rank less than 3, advancing understanding of entanglement structure.

Findings

Subspaces with no vectors of Schmidt rank ≤k imply high Schmidt number for supported states.

Constructed maximal subspace in rac{m}{n}inity with no vectors of Schmidt rank <3.

Provides bounds and explicit examples for entanglement measures in bipartite systems.

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 $\leqslant k$, 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.