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Physical Review A

Local distinguishability of orthogonal quantum states with multiple copies of 2 ⊗ 2 maximally entangled states

Zhichao Zhang, Yan-Qi Song, Tingting Song, Fei Gao, Su‐Juan Qin, Qiaoyan Wen

Published Feb 23, 2018• 42 citations
Publisher

Abstract

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 $d\ensuremath{\bigotimes}d$ ($d$ 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.

DOI
10.1103/physreva.97.022334

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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⨂55\ensuremath{\bigotimes}55\ensuremath⨂5 by using two copies of 2\ensuremath⨂22\ensuremath{\bigotimes}22\ensuremath⨂2 maximally entangled states. Then we generalize the distinguishing method for a class of orthogonal product states in d\ensuremath⨂dd\ensuremath{\bigotimes}dd\ensuremath⨂d (ddd is odd). Furthermore, for a class of nonlocality of orthogonal product states in d\ensuremath⨂dd\ensuremath{\bigotimes}dd\ensuremath⨂d (d\ensuremath≥5d\ensuremath{\ge}5d\ensuremath≥5), we prove that these states can be distinguished by LOCC with two copies of 2\ensuremath⨂22\ensuremath{\bigotimes}22\ensuremath⨂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⨂22\ensuremath{\bigotimes}22\ensuremath⨂2 maximally entangled states. These results also reveal the phenomenon of less nonlocality with more 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.
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.
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 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.
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 $d\ensuremath{\bigotimes}d$ (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.
It is known that many sets of orthogonal quantum states, which cannot be distinguished perfectly by local operations and classical communication (LOCC), have been constructed. However, these discussions have left the following open question: How many entanglement resources are necessary and/or sufficient for this task to be possible with LOCC? In this paper, we present a local discrimination protocol which uses entanglement more efficiently than teleportation to distinguish one class of orthogonal product states in 3⊗3⊗3. Then, we generalize it for 3⊗3⊗...⊗3 quantum systems which has 3n parties. 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.