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Physical Review A

Nonlocality of orthogonal product basis quantum states

Zhichao Zhang, Fei Gao, Guojing Tian, Tian-Qing Cao, Qiaoyan Wen

Published Aug 13, 2014• 99 citations
Publisher

Abstract

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 $d$ is odd, we construct ${d}^{2}$ 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.''

DOI
10.1103/physreva.90.022313

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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⨂33\ensuremath{\bigotimes}33\ensuremath⨂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⨂dd\ensuremath{\bigotimes}dd\ensuremath⨂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.''
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.
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.
Recently, based on some quantum states which are locally irreducible in all bipartitions, strong quantum nonlocality has been presented by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)]. However, the remaining questions are whether a set of quantum states can reveal strong quantum nonlocality when these states are locally irreducible in all tripartitions or more multipartitions and how to construct strong nonlocality of orthogonal product states in more than tripartite quantum systems. Here we present a general definition of strong quantum nonlocality based on the local irreducibility. Then, using a $3\ensuremath{\bigotimes}3\ensuremath{\bigotimes}3$ quantum system as an example, we show three sets of locally indistinguishable orthogonal product states to explain the general definition of strong quantum nonlocality. Furthermore, in $3\ensuremath{\bigotimes}3\ensuremath{\bigotimes}3\ensuremath{\bigotimes}3$, we construct a set of strong nonlocality of orthogonal product states; these states do not form a complete basis. Our results demonstrate the phenomenon of strong nonlocality without entanglement for the multipartite quantum systems.
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.
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.
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.