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Physica Scripta

Strong nonlocality with genuine entanglement based on GHZ-like states in multipartite quantum systems

Ying‐Hui Yang, Guang-Wei Mi, Shi-Jiao Geng, Qianqian Liu, Hui-Juan Zuo

Published Nov 24, 2022• 0 citations
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Abstract

Abstract Strong nonlocality with genuine entanglement was first shown by Wang et al using sets of GHZ-like states in tripartite quantum systems (2021 Phys. Rev. A 104, 012 424). However, it is an open problem whether there exists strong nonlocality with genuine entanglement in four or more partite systems. In this paper, we unify two different concepts of strong nonlocality introduced by Halder et al (2019 Phys. Rev. Lett. 122, 040 403) and by Zhang et al (2019 Phys. Rev. A 99, 062108). That is, we use a concept of k-strong nonlocality instead of these two different types of strong nonlocality. A set of orthogonal quantum states is k-strong nonlocal if it is locally irreducible in every k-partition. In fact, the strong nonlocality that is usually said is 2-strong nonlocality. The smaller the k is, the stronger the nonlocality will be. A set of states is k +-strong nonlocal if the strong nonlocality of this set is stronger than k-strong nonlocality but weaker than (k − 1)-strong nonlocality. Based on these concepts, firstly, we show 2-strong nonlocality with genuine entanglement by some sets of GHZ-like states with weight d in tripartite systems. These sets are not necessarily complete bases. Secondly, we present 2-strong nonlocality with genuine entanglement for systems with four or more parties. These results solve the open problem raised by Wang et al Finally, we construct a set of GHZ-like states with n +-strong nonlocality in n-partite quantum systems.

DOI
10.1088/1402-4896/aca5c0

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Abstract Strong nonlocality with genuine entanglement was first shown by Wang et al using sets of GHZ-like states in tripartite quantum systems (2021 Phys. Rev. A 104, 012 424). However, it is an open problem whether there exists strong nonlocality with genuine entanglement in four or more partite systems. In this paper, we unify two different concepts of strong nonlocality introduced by Halder et al (2019 Phys. Rev. Lett. 122, 040 403) and by Zhang et al (2019 Phys. Rev. A 99, 062108). That is, we use a concept of k-strong nonlocality instead of these two different types of strong nonlocality. A set of orthogonal quantum states is k-strong nonlocal if it is locally irreducible in every k-partition. In fact, the strong nonlocality that is usually said is 2-strong nonlocality. The smaller the k is, the stronger the nonlocality will be. A set of states is k +-strong nonlocal if the strong nonlocality of this set is stronger than k-strong nonlocality but weaker than (k − 1)-strong nonlocality. Based on these concepts, firstly, we show 2-strong nonlocality with genuine entanglement by some sets of GHZ-like states with weight d in tripartite systems. These sets are not necessarily complete bases. Secondly, we present 2-strong nonlocality with genuine entanglement for systems with four or more parties. These results solve the open problem raised by Wang et al Finally, we construct a set of GHZ-like states with n +-strong nonlocality in n-partite quantum systems.
Strong nonlocality based on local distinguishability is a stronger form of quantum nonlocality recently introduced in multipartite quantum systems: an orthogonal set of multipartite quantum states is said to have the property of strong nonlocality if it is locally irreducible for every bipartition of the subsystems. Most of the known results are limited to sets with product states. Shi et al. presented the first result of strongly nonlocal sets with entanglement [Phys. Rev. A 102, 042202 (2020)], and they questioned the existence of a strongly nonlocal set with genuine entanglement. In this work, we relate the strong nonlocality of some special set of genuine entanglement to the connectivities of some graphs. Using this relation, we successfully construct sets of genuinely entangled states with strong nonlocality. As a consequence, our constructions give a negative answer to Shi et al.'s question and also provide an answer to the open problem raised by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)]. This work associates a physical quantity named strong nonlocality with a mathematical quantity called graph connectivity.
Abstract Recently, Halder et al (2019 Phys. Rev. Lett. 122 040403) proposed the concept of strong nonlocality without entanglement: an orthogonal set of fully product states in multipartite quantum systems that is locally irreducible for every bipartition of its subsystems. Due to the complexity of the problem, most results are limited to tripartite systems. Here we consider a weaker form of nonlocality which is called local distinguishability based genuine nonlocality. A set of orthogonal multipartite quantum states is said to be genuinely nonlocal if it is locally indistinguishable for every bipartition of the subsystems. In this work, we study how to construct sets of orthogonal product states which are genuinely nonlocal. Firstly, we present a set of product states with simple structure in bipartite systems that is locally indistinguishable. After that, based on a simple observation, we present a general method to construct genuinely nonlocal sets of multipartite product states by using those sets that are genuinely nonlocal but with less parties. As a consequence, we obtain that genuinely nonlocal sets of fully product states exist in any L parties systems ⨂ i = 1 L C d i provided L ⩾ 3 and d i ⩾ 3 for all i .
A set of multipartite orthogonal quantum states is strongly nonlocal if it is locally irreducible for every bipartition of the subsystems [S. Halder, M. Banik, S. Agrawal, and S. Bandyopadhyay, Phys. Rev. Lett. 122, 040403 (2019)]. Although this property has been shown in three-, four-, and five-partite systems, the existence of strongly nonlocal sets in NNN-partite systems remains unknown when $N\ensuremath{\ge}6$. In this paper, we successfully show that a strongly nonlocal set of orthogonal entangled states exists in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$ for all $N\ensuremath{\ge}3$ and $d\ensuremath{\ge}2$, which reveals the strong quantum nonlocality in general NNN-partite systems. For N=3N=3N=3 or 4 and $d\ensuremath{\ge}3$, we present a strongly nonlocal set consisting of genuinely entangled states, which has a smaller size than any known strongly nonlocal orthogonal product set. Finally, we connect strong quantum nonlocality with local hiding of information as an application.
Strong quantum nonlocality was introduced recently as a stronger manifestation of nonlocality in multipartite systems through the notion of local irreducibility in all bipartitions. Known existing results for sets of strongly nonlocal orthogonal states are limited to product states. In this paper, based on the Rubik's cube, we give a construction of such sets consisting of entangled states in $d\ensuremath{\bigotimes}d\ensuremath{\bigotimes}d$ for all $d\ensuremath{\ge}3$. Consequently, we answer an open problem given by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)], that is, orthogonal entangled bases that are strongly nonlocal do exist. Furthermore, we propose two entanglement-assisted protocols for local discrimination of our results. Each protocol consumes less entanglement resources than the teleportation-based protocol on average. Our results exhibit the phenomenon of strong quantum nonlocality with entanglement.
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.
In this paper, we generalize the concept of strong quantum nonlocality from two aspects. First, in a tripartite quantum system, we present a construction of strongly nonlocal quantum states containing $6{(d\ensuremath{-}1)}^{2}$ orthogonal product states in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ and build $6{d}^{2}\ensuremath{-}8d+4$ product states in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d+1}$, which have been proved to be strongly nonlocal. Obviously, both results turn out to be one order of magnitude smaller than the number of basis states d3{d}^{3}d3 for $d\ensuremath{\ge}3$. Second, we give explicit forms of strongly nonlocal orthogonal product basis in ${\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$ and ${\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}$ quantum systems, where four is the largest known number of subsystems in which there exists strong quantum nonlocality without entanglement up to now. All the results positively answer the open problems raised by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)]; that is, there do exist a small number of quantum states that can demonstrate strong quantum nonlocality without entanglement.