On the dimension of subspaces with bounded Schmidt rank

Understand

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.