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.

本文研究了在双部分子量子系统中包含高度纠缠态的子空间可以有多大，重点考虑了 Schmidt rank 作为联系纠缠度量的情况，给出了对于局域尺寸 dA 和 dB 以及 r 对应的只由 Schmidt rank 为 r 或更高的纠缠态构成的子空间的最大维度。