Subspaces without locally distinguishable orthonormal bases
Wei Jiang, Xi-Jun Ren, Xingxiang Zhou, Zheng-Wei Zhou, Guang‐Can Guo · Physical Review A · 2009
In this paper, we find subspaces that have no locally distinguishable orthonormal bases. For a ${C}^{m}\ensuremath{\bigotimes}{C}^{n}$ $(m,n\ensuremath{\ge}3)$ bipartite tensor system, we show that the orthogonal complement of the space spanned by a Schmidt rank-3 maximally entangled state is such a subspace. We further demonstrate that, when both parties' dimensions are greater than three, the orthogonal complement of the space spanned by any entangled state whose Schmidt rank is greater than three has no locally distinguishable orthonormal bases. Although derived using a scheme simpler than previous methods, our results are much more general than known before.