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Physical Review A

Local distinguishability of orthogonal quantum states with no more than one ebit of entanglement

Lvjun Li, Fei Gao, Zhichao Zhang, Qiaoyan Wen

Published Jan 28, 2019• 27 citations
Publisher

Abstract

In this paper, we study the local distinguishability of mutually orthogonal product quantum states with entanglement as a valuable resource. We present protocols in which entanglement is used more efficiently than in the previous protocols presented by Zhang et al. [Phys. Rev. A 97, 022334 (2018)]. Specifically, the main technical contributions are as follows: First, we present a method to locally distinguish a set of orthogonal product quantum states in $d\ensuremath{\bigotimes}d$ ($d$ is odd) by using only one ancillary two-qubit maximally entangled state. Then we present a similar method to locally distinguish multipartite orthogonal product states by using entanglement efficiently. Moreover, we propose a method to study the local distinguishability of this kind of orthogonal product state with less than one ebit of entanglement, and analyze the relationship between the probability of error and the quantity of entanglement, which reveals the phenomenon of less nonlocality with more entanglement.

DOI
10.1103/physreva.99.012343

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In this paper, we study the local distinguishability of mutually orthogonal product quantum states with entanglement as a valuable resource. We present protocols in which entanglement is used more efficiently than in the previous protocols presented by Zhang et al. [Phys. Rev. A 97, 022334 (2018)]. Specifically, the main technical contributions are as follows: First, we present a method to locally distinguish a set of orthogonal product quantum states in d\ensuremath⨂dd\ensuremath{\bigotimes}dd\ensuremath⨂d (ddd is odd) by using only one ancillary two-qubit maximally entangled state. Then we present a similar method to locally distinguish multipartite orthogonal product states by using entanglement efficiently. Moreover, we propose a method to study the local distinguishability of this kind of orthogonal product state with less than one ebit of entanglement, and analyze the relationship between the probability of error and the quantity of entanglement, which reveals the phenomenon of less nonlocality with more entanglement.
For general quantum systems, many sets of locally indistinguishable orthogonal quantum states have been constructed so far. However, it is interesting how much entanglement resources are sufficient and/or necessary to distinguish these states by local operations and classical communication (LOCC). Here we first present a method to locally distinguish a set of orthogonal product states in $5\ensuremath{\bigotimes}5$ by using two copies of $2\ensuremath{\bigotimes}2$ maximally entangled states. Then we generalize the distinguishing method for a class of orthogonal product states in $d\ensuremath{\bigotimes}d$ (ddd is odd). Furthermore, for a class of nonlocality of orthogonal product states in $d\ensuremath{\bigotimes}d$ ($d\ensuremath{\ge}5$), we prove that these states can be distinguished by LOCC with two copies of $2\ensuremath{\bigotimes}2$ maximally entangled states. Finally, for some multipartite orthogonal product states, we also present a similar method to locally distinguish these states with multiple copies of $2\ensuremath{\bigotimes}2$ maximally entangled states. These results also reveal the phenomenon of less nonlocality with more entanglement.
Any set of states which cannot be perfectly distinguished by local operations and classical communication (LOCC) alone can always be locally distinguished using quantum teleportation with a maximally entangled state as resource. However, in quantum information theory, entanglement is a very valuable resource, so it leaves the following problem: How to accomplish this task more efficiently than teleportation, that is, design the local discrimination protocol using less entanglement. In this paper, we first present two protocols to locally distinguish a particular unextendible product bases (UPB) in $5\ensuremath{\bigotimes}5$ by using different entanglement resource. Then, we generalize the distinguishing methods for a class of UPB in $d\ensuremath{\bigotimes}d$, where ddd is odd and $d\ensuremath{\geqslant}3$. Furthermore, for a class of UPB in $d\ensuremath{\bigotimes}d$, where ddd is even and $d\ensuremath{\geqslant}4$, we prove that these states can also be distinguished by LOCC with multiple copies of low-dimensional entanglement resource. These results offer rather general insight into how to use entanglement resource more efficiently and also reveal the phenomenon of less nonlocality with more entanglement.
In this paper, we mainly study the local indistinguishability of mutually orthogonal product basis quantum states in $d\ensuremath{\bigotimes}d$. In $3\ensuremath{\bigotimes}3$, Bennett et al. [ Phys. Rev. A 59, 1070 (1999)] presented nine orthogonal product basis quantum states which cannot be distinguished by local operations and classical communication (LOCC). In the work by Zhang et al. [Z.-C. Zhang et al., Phys. Rev. A 90, 022313 (2014)], this result was generalized in $d\ensuremath{\bigotimes}d$, where ddd is odd. In this paper, we aim to construct locally indistinguishable orthogonal product basis quantum states in $d\ensuremath{\bigotimes}d$. For the general $d\ensuremath{\bigotimes}d\phantom{\rule{0.28em}{0ex}}(d>2)$ quantum system, we first construct $4d\ensuremath{-}4$ orthogonal product states, and prove these states are locally indistinguishable using a very simple but quite effective method. Then, based on these states, we construct some classes of locally indistinguishable orthogonal product basis quantum states (OPBS) in $d\ensuremath{\bigotimes}d\phantom{\rule{0.28em}{0ex}}(d>2)$. Finally, we construct some LOCC indistinguishable OPBS in multipartite quantum systems. All of the above results demonstrate the phenomenon of nonlocality without entanglement.
Abstract It is an interesting topic how much entanglement resources are necessary to distinguish locally indistinguishable orthogonal quantum states by local operations and classical communication (LOCC). Using only one ebit of entanglement some locally indistinguishable orthogonal product bases can be distinguished by LOCC [2019 Phys. Rev. A 99 012343]. However, it is still an open question whether one ebit of entanglement is sufficient for distinguishing a locally indistinguishable set of orthogonal maximally entangled states. We focus on the generalized Bell states (GBSs), and firstly find that any four GBSs in a 4 ⊗ 4 quantum system are always distinguished by one-way LOCC using one ebit of entanglement. Then we consider a general case, i.e., d ⊗ d quantum system ( d is even), and show that the set L of any GBSs can be distinguished by one-way LOCC using one ebit of entanglement if S A \ S L is nonempty, where S A is the admissible solutions set of L and S L is the genuinely nonadmissible solutions set of L . This criterion is very simple and convenient since only some congruence equations are calculated. Finally, we find a curious and counterintuitive case, i.e., in a 3 ⊗ 3 quantum system there exist four GBSs which cannot be distinguished by one-way LOCC using one ebit of entanglement, although the dimension of the subsystem is less than four. These results can help us better understand the role of entanglement as a resource in local state discrimination problems.
For general bipartite quantum systems, many sets of locally indistinguishable orthogonal product states have been constructed so far. Here, we first present a general method to construct multipartite orthogonal product states in ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{d}_{n}({d}_{1,2,\ensuremath{\cdots},n}\ensuremath{\ge}3,n\ensuremath{\ge}4)$ by using some locally indistinguishable bipartite orthogonal product states. And we prove that these multipartite orthogonal quantum states cannot be distinguished by local operations and classical communication. Furthermore, in ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{d}_{n}({d}_{1,2,\ensuremath{\cdots},n}\ensuremath{\ge}3,n\ensuremath{\ge}5)$, we give a general method to construct a much smaller number of locally indistinguishable multipartite orthogonal product states for even and odd nnn separately. In addition, we also present a general method to construct complete orthogonal product bases for the multipartite quantum systems. Our results demonstrate the phenomenon of nonlocality without entanglement for the multipartite quantum systems.
It is known that there are many sets of orthogonal product states which cannot be distinguished perfectly by local operations and classical communication (LOCC). However, these discussions have left the following open question: What entanglement resources are necessary and/or sufficient for this task to be possible with LOCC? In m ⊗ n, certain classes of unextendible product bases (UPB) which can be distinguished perfectly using entanglement as a resource, had been presented in 2008. In this paper, we present protocols which use entanglement more efficiently than teleportation to distinguish some classes of orthogonal product states in m ⊗ n, which are not UPB. For the open question, our results offer rather general insight into why entanglement is useful for such tasks, and present a better understanding of the relationship between entanglement and nonlocality.