Locally indistinguishable subspaces spanned by three-qubit unextendible product bases

Abstract

We study the local distinguishability of general multiqubit states and show that local projective measurements and classical communication are as powerful as the most general local measurements and classical communication. Remarkably, this indicates that the local distinguishability of multiqubit states can be decided efficiently. Another useful consequence is that a set of orthogonal $n$-qubit states is locally distinguishable only if the summation of their orthogonal Schmidt numbers is less than the total dimension ${2}^{n}$. Employing these results, we show that any orthonormal basis of a subspace spanned by arbitrary three-qubit orthogonal unextendible product bases (UPB) cannot be exactly distinguishable by local operations and classical communication. This not only reveals another intrinsic property of three-qubit orthogonal UPB but also provides a class of locally indistinguishable subspaces with dimension $4$. We also explicitly construct locally indistinguishable subspaces with dimensions $3$ and $5$, respectively. Similar to the bipartite case, these results on multipartite locally indistinguishable subspaces can be used to estimate the one-shot environment-assisted classical capacity of a class of quantum broadcast channels.