Locally indistinguishable orthogonal product bases in arbitrary bipartite quantum system.

Guang-Bao Xu,Fei Gao,Su-Juan Qin,Ying-Hui Yang,Qiao-Yan Wen

doi:10.1038/srep31048

Abstract

As we know, unextendible product basis (UPB) is an incomplete basis whose members cannot be perfectly distinguished by local operations and classical communication. However, very little is known about those incomplete and locally indistinguishable product bases that are not UPBs. In this paper, we first construct a series of orthogonal product bases that are completable but not locally distinguishable in a general m ⊗ n (m ≥ 3 and n ≥ 3) quantum system. In particular, we give so far the smallest number of locally indistinguishable states of a completable orthogonal product basis in arbitrary quantum systems. Furthermore, we construct a series of small and locally indistinguishable orthogonal product bases in m ⊗ n (m ≥ 3 and n ≥ 3). All the results lead to a better understanding of the structures of locally indistinguishable product bases in arbitrary bipartite quantum system.

Highlights

The local indistinguishability of orthogonal quantum states provides an effective tool to explore the relationship between quantum nonlocality and quantum entanglement
Bennett et al [1] firstly constructed a set of nine orthogonal product states that cannot be perfectly distinguished by local operations and classical communication (LOCC) in 3 ⊗ 3
An uncompletable orthogonal product basis (UCPB) is a PB whose complementary subspace HS⊥, i.e., the subspace in H spanned by vectors that are orthogonal to all the vectors in HS, contains fewer mutually orthogonal product states than its dimension

Summary

INTRODUCTION

The local indistinguishability of orthogonal quantum states provides an effective tool to explore the relationship between quantum nonlocality and quantum entanglement. Numerous results [3,4,5,6,7,8,9,10,11,12,13,14,15,16] have been presented up to now In spite of these huge advances, some basic problems are still incompletely solved, such as the smallest number of the states of an orthogonal product basis that can be completable and cannot be locally distinguished in a high-dimensional bipartite quantum system. All the results show it is a meaningful work to research the structure of the locally indistinguishable product basis and the smallest number of locally indistinguishable orthogonal product states in a high-dimensional quantum system. We construct a small orthogonal product basis that contains 2p − 1 orthogonal product states and is maybe uncompletable in m ⊗ n , and prove its local indistinguishability, where 3 ≤ m ≤ n and p is an arbitrary integer from 3 to m

PRELIMINARIES

LOCAL INDISTINGUISHABILITY OF COMPLETABLE ORTHOGONAL PRODUCT

LOCAL INDISTINGUISHABILITY OF SMALL ORTHOGONAL PRODUCT BASIS

CONCLUSION