Locally indistinguishable subspaces spanned by three-qubit unextendible product bases

Abstract

We study the local distinguishability of general multi-qubit 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 . When n = 2 such a condition is also sufficient. 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. In particular, 3 is the minimal possible dimension of locally indistinguishable subspaces. Combining with the previous results, we conclude that any positive integer between 3 and 7 is the possible dimension of some three-qubit locally indistinguishable subspace.

Any2⊗nsubspace is locally distinguishable

“…We show that any 2 ⊗ n subspace is locally distinguishable. Combining with the previous results [4,12,13], we conclude that there is no locally indistinguishable m ⊗ n subspace if and only if m 2 or n 2. Our key techniques can be used to study the distinguishability of three-dimensional bipartite subspace which contains a product state.…”

“…Watrous also proved that there is no 2 ⊗ 2 locally indistinguishable subspace, by directly employing the results from [11]. Winter's result [5] implies that the existence of bipartite subspace Q such that Q ⊗k is locally indistinguishable for any k. Duan et al generalized Watrous's result to the most general m ⊗ n systems for m = n and the multipartite setting, and found locally indistinguishable subspaces with smaller dimensions [12,13]. Most notably, it was shown that any subspace spanned by three-qubit UPB is locally indistinguishable, and there exists a three-dimensional * nengkunyu@gmail.com † runyao.duan@uts.edu.au ‡ mying@it.uts.edu.au three-qubit locally indistinguishable subspace [13].…”

Evaluation of two different entanglement measures on a bound entangled state

“…This is a typical example of a multipartite mixed state whose entanglement can be detected by entanglement witnesses built from local observables [18]. Recently, a scheme for studying local distinguishability of three-qubit UPB states has also been proposed [19]. Here we use a unified strategy to compute two measures, namely, the geometric measure of entanglement and the generalized concurrence.…”

The structure of qubit unextendible product bases

“…While they were originally introduced as a tool for constructing bound entangled states [BDF + 99, DMS + 03], they can also be used to construct indecomposible positive maps [Ter01] and to demonstrate the existence of nonlocality without entanglement-that is, they can not be perfectly distinguished by local quantum operations and classical communication, even though they contain no entanglement. Furthermore, in the qubit case (i.e., the case where each local space has dimension 2), unextendible product bases can be used to construct tight Bell inequalities with no quantum violation [AFK + 12, ASH + 11] and subspaces of small dimension that are locally indistinguishable [DXY10].…”