On the dimension of subspaces with bounded Schmidt rank

TL;DR

This paper determines the maximum dimension of bipartite subspaces containing only states with a specified minimum or maximum Schmidt rank, advancing understanding of entanglement structure in quantum systems.

Contribution

It provides exact bounds on the largest subspace dimensions with states of Schmidt rank at least or at most a given value, improving upon previous probabilistic bounds.

Findings

Exact maximum dimension for subspaces with Schmidt rank ≥ r

Exact maximum dimension for subspaces with Schmidt rank ≤ r

Discussion on subspaces with states of fixed Schmidt rank r

Abstract

We consider the question of how large a subspace of a given bipartite quantum system can be when the subspace contains only highly entangled states. This is motivated in part by results of Hayden et al., which show that in large d x d--dimensional systems there exist random subspaces of dimension almost d^2, all of whose states have entropy of entanglement at least log d - O(1). It is also related to results due to Parthasarathy on the dimension of completely entangled subspaces, which have connections with the construction of unextendible product bases. Here we take as entanglement measure the Schmidt rank, and determine, for every pair of local dimensions dA and dB, and every r, the largest dimension of a subspace consisting only of entangled states of Schmidt rank r or larger. This exact answer is a significant improvement on the best bounds that can be obtained using random subspace techniques. We also determine the converse: the largest dimension of a subspace with an upper bound on the Schmidt rank. Finally, we discuss the question of subspaces containing only states with Schmidt equal to r.