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Physical Review A

Nonlocality of orthogonal product states

Zhichao Zhang, Fei Gao, Su‐Juan Qin, Ying‐Hui Yang, Qiaoyan Wen

Published Jul 29, 2015• 81 citations
Publisher

Abstract

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 $d$ 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.

DOI
10.1103/physreva.92.012332

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In this paper, we mainly study the local indistinguishability of mutually orthogonal product basis quantum states in d\ensuremath⨂dd\ensuremath{\bigotimes}dd\ensuremath⨂d. In 3\ensuremath⨂33\ensuremath{\bigotimes}33\ensuremath⨂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⨂dd\ensuremath{\bigotimes}dd\ensuremath⨂d, where ddd is odd. In this paper, we aim to construct locally indistinguishable orthogonal product basis quantum states in d\ensuremath⨂dd\ensuremath{\bigotimes}dd\ensuremath⨂d. For the general d\ensuremath⨂d(d>2)d\ensuremath{\bigotimes}d\phantom{\rule{0.28em}{0ex}}(d>2)d\ensuremath⨂d(d>2) quantum system, we first construct 4d\ensuremath−44d\ensuremath{-}44d\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⨂d(d>2)d\ensuremath{\bigotimes}d\phantom{\rule{0.28em}{0ex}}(d>2)d\ensuremath⨂d(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.
In the general bipartite quantum system $m\ensuremath{\bigotimes}n\phantom{\rule{0.16em}{0ex}}(m,n\ensuremath{\ge}3)$, Y.-L. Wang et al. [Phys. Rev. A 92, 032313 (2015)] presented $3(m+n)\ensuremath{-}9$ orthogonal product states which cannot be distinguished by local operations and classical communication (LOCC). In this paper, we aim to construct less locally indistinguishable orthogonal product states in $m\ensuremath{\bigotimes}n$. First, in the $3\ensuremath{\bigotimes}n\phantom{\rule{0.28em}{0ex}}(3<n)$ quantum system, we construct $3n\ensuremath{-}2$ locally indistinguishable orthogonal product states which are not unextendible product bases. Then, for $m\ensuremath{\bigotimes}n\phantom{\rule{0.28em}{0ex}}(4\ensuremath{\le}m\ensuremath{\le}n)$, we present $3n+m\ensuremath{-}4$ orthogonal product states which cannot be perfectly distinguished by LOCC. Finally, in the general bipartite quantum system $m\ensuremath{\bigotimes}n\phantom{\rule{0.28em}{0ex}}(3\ensuremath{\le}m\ensuremath{\le}n)$, we show a smaller set with $2n\ensuremath{-}1$ orthogonal product states and prove that these states are LOCC indistinguishable using a very simple but quite effective method. All of the above results demonstrate the phenomenon of nonlocality without entanglement.
We study the local indistinguishability of mutually orthogonal product basis quantum states in the high-dimensional quantum system. In the quantum system of ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$, where ddd is odd, Zhang et al. [Z.-C. Zhang et al., Phys. Rev. A 90, 022313 (2014)] have constructed d2{d}^{2}d2 orthogonal product basis quantum states that are locally indistinguishable. We find a subset that contains $6d\ensuremath{-}9$ orthogonal product states that are still locally indistinguishable. We generalize our method to an arbitrary bipartite quantum system ${\mathbb{C}}^{m}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$. We present a small set with only $3(m+n)\ensuremath{-}9$ orthogonal product states and prove that these states are local operations and classical communication (LOCC) indistinguishable. Even though these $3(m+n)\ensuremath{-}9$ product states are LOCC indistinguishable, they can be distinguished by separable measurements. This shows that separable operations are strictly stronger than the local operations and classical communication.
In this paper, we mainly study the local indistinguishability of mutually orthogonal product basis quantum states in the high-dimensional quantum systems. In the Hilbert space of $3\ensuremath{\bigotimes}3$, Walgate and Hardy [Phys. Rev. Lett. 89, 147901 (2002)] presented a very simple proof for nonlocality of nine orthogonal product basis quantum states which are given by Bennett et al. [Phys. Rev. A 59, 1070 (1999)]. In the quantum system of $d\ensuremath{\bigotimes}d$, where ddd is odd, we construct d2{d}^{2}d2 orthogonal product basis quantum states and prove these states are locally indistinguishable. Then we are able to construct some locally indistinguishable product basis quantum states in the multipartite systems. All these results reveal the phenomenon of ``nonlocality without entanglement.''
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.
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.
Local distinguishability of orthogonal quantum states is an area of active research in quantum information theory. However, most of the relevant results are about local distinguishability in bipartite Hilbert space and very little is known about the multipartite case. In this paper we present a generic method to construct a completable nnn-partite $(n\ensuremath{\ge}3)$ product basis with only 2n2n2n members, which exhibits nonlocality without entanglement with nnn parties, each holding a system of any finite dimension. We give an effective proof of the nonlocality of the completable multipartite product basis. In addition, we construct another incomplete multipartite product basis with a smaller number of members that cannot be distinguished by local operations and classical communication in a ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{d}_{n}$ quantum system, where $n\ensuremath{\ge}3$ and ${d}_{i}\ensuremath{\ge}2$ for i=1,2,...,ni=1,2,...,ni=1,2,...,n. The results can lead to a better understanding of the phenomenon of nonlocality without entanglement in any multipartite quantum system.