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arXiv:2408.11566 · Physical Review A

Genuinely nonlocal sets without entanglement in multipartite systems

Ying-Ying Lu, Hai-Qing Cao, Hui-Juan Zuo, Shao-Ming Fei

Published Aug 16, 2024• 7 citations
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Abstract

A set of multipartite orthogonal states is genuinely nonlocal if it is locally indistinguishable in every bipartition of the subsystems. If the set is locally reducible, we say it has genuine nonlocality of type I. Otherwise, we say it has genuine nonlocality of type II. Due to the complexity of the problem, the construction of genuinely nonlocal sets in general multipartite systems has not been completely solved so far. In this paper, we first provide a nonlocal set of product states in bipartite systems. We obtain a genuinely nonlocal set of type I without entanglement in general $n$-partite systems ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$ $[3\ensuremath{\le}({d}_{1}\ensuremath{-}1)\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n},\phantom{\rule{4pt}{0ex}}n\ensuremath{\ge}3]$. Then we present two constructions with genuine nonlocality of type II in ${\mathbb{C}}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{2}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{3}}$ $(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}{d}_{3})$ and ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$ $(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n},\phantom{\rule{4pt}{0ex}}n\ensuremath{\ge}4)$. Our results further positively answer the open problem that there does exist a genuinely nonlocal set of type II in multipartite systems [M. S. Li, Y. L. Wang, F. Shi, and M. H. Yung, J. Phys. A: Math. Theor. 54, 445301 (2021)] and highlight its related applications in quantum information processing.

DOI
10.1103/physreva.110.022427

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A set of multipartite orthogonal states is genuinely nonlocal if it is locally indistinguishable in every bipartition of the subsystems. If the set is locally reducible, we say it has genuine nonlocality of type I. Otherwise, we say it has genuine nonlocality of type II. Due to the complexity of the problem, the construction of genuinely nonlocal sets in general multipartite systems has not been completely solved so far. In this paper, we first provide a nonlocal set of product states in bipartite systems. We obtain a genuinely nonlocal set of type I without entanglement in general nnn-partite systems \ensuremath⨂i=1nCdi{\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}\ensuremath⨂i=1n​Cdi​ [3\ensuremath≤(d1\ensuremath−1)\ensuremath≤d2\ensuremath≤\ensuremath⋯\ensuremath≤dn,n\ensuremath≥3][3\ensuremath{\le}({d}_{1}\ensuremath{-}1)\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n},\phantom{\rule{4pt}{0ex}}n\ensuremath{\ge}3][3\ensuremath≤(d1​\ensuremath−1)\ensuremath≤d2​\ensuremath≤\ensuremath⋯\ensuremath≤dn​,n\ensuremath≥3]. Then we present two constructions with genuine nonlocality of type II in Cd1\ensuremath⨂Cd2\ensuremath⨂Cd3{\mathbb{C}}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{2}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{3}}Cd1​\ensuremath⨂Cd2​\ensuremath⨂Cd3​ (3\ensuremath≤d1\ensuremath≤d2\ensuremath≤d3)(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}{d}_{3})(3\ensuremath≤d1​\ensuremath≤d2​\ensuremath≤d3​) and \ensuremath⨂i=1nCdi{\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}\ensuremath⨂i=1n​Cdi​ (3\ensuremath≤d1\ensuremath≤d2\ensuremath≤\ensuremath⋯\ensuremath≤dn,n\ensuremath≥4)(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n},\phantom{\rule{4pt}{0ex}}n\ensuremath{\ge}4)(3\ensuremath≤d1​\ensuremath≤d2​\ensuremath≤\ensuremath⋯\ensuremath≤dn​,n\ensuremath≥4). Our results further positively answer the open problem that there does exist a genuinely nonlocal set of type II in multipartite systems [M. S. Li, Y. L. Wang, F. Shi, and M. H. Yung, J. Phys. A: Math. Theor. 54, 445301 (2021)] and highlight its related applications in quantum information processing.
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 .
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.
A set of orthogonal quantum states in multipartite systems is of genuine nonlocality if it is locally indistinguishable in every bipartition. If it is locally reducible when the parties are separated, we say that it has genuine nonlocality of type~\uppercase\expandafter{\romannumeral 1}; otherwise, it has genuine nonlocality of type~\uppercase\expandafter{\romannumeral 2}. For a locally distinguishable set without local redundancy, if there exist some orthogonality preserving local measurements such that each outcome leads to a locally indistinguishable set, then we say that it exhibits the activation of nonlocality. We activate type-\uppercase\expandafter{\romannumeral 1} and type-\uppercase\expandafter{\romannumeral 2} genuine nonlocality of orthogonal product state sets in tripartite systems. In particular, we tackle the local irredundancy problem with partial trace operation and ppp-ary numeral systems to significantly simplify the proofs. Our results also address the open question raised by S. Bandyopadhyay \textit{et al.}[\href{https://link.aps.org/doi/10.1103/PhysRevA.104.L050201}{Phys. Rev. A \textbf{104}, L050201 (2021)}]. Furthermore, we observe the activation of hidden genuine nonlocality in multipartite systems, which highlights the applications of nonlocality based on state discrimination in different practical scenarios.
A set of orthogonal quantum states has genuine hidden nonlocality, if there is an orthogonality preserving local measurement such that the locally distinguishable and locally redundant set become locally indistinguishable under it.If the cardinality of the postmeasurement sets is equal to the cardinality of the original set, then we say the original set has genuine hidden nonlocality of type uppercaseexpandafterromannumeral 1; otherwise, we say it has genuine hidden nonlocality of type uppercaseexpandafterromannumeral 2 In this study, we mainly study the constructions of orthogonal product state sets with genuine hidden nonlocality, and the main results are as follows. We first construct a product set with genuine hidden nonlocality of type uppercaseexpandafterromannumeral 1in mathbbC^d_1⊗mathbbC^d_2d_1geq8, d_2geq10, d_iis even). In addition, we present a type-uppercaseexpandafterromannumeral 2genuine hidden nonlocal set without entanglement in mathbbC^d⊗mathbbC^2ddgeq3). In particular, we found the product state set with genuine hidden nonlocality of type uppercaseexpandafterromannumeral2in mathbbC³⊗ mathbbC^6, which has the smallest local dimension.
Recently, much attention have been paid to the constructions of nonlocal multipartite orthogonal product states. Among the existing results, some are relatively complex in structure while others have many constraint conditions. In this paper, we first give a simple method to construct a nonlocal set of orthogonal product states in ${\ensuremath{\bigotimes}}_{j=1}^{n}{\mathbb{C}}^{d}$, where d,nd,\phantom{\rule{0.16em}{0ex}}nd,n are two positive integers, $d\ensuremath{\ge}2$ and $n\ensuremath{\ge}3$. Then we give a proof for local indistinguishability of the set constructed by our method. According to the characteristics of this construction method, we get a new construction of nonlocal set with fewer states in the same quantum system. Furthermore, we generalize these two results to a more general ${\ensuremath{\bigotimes}}_{j=1}^{n}{\mathbb{C}}^{{d}_{j}}$ quantum system, where n,djn,\phantom{\rule{0.16em}{0ex}}{d}_{j}n,dj​ are two positive integers, $n\ensuremath{\ge}3$ and ${d}_{j}\ensuremath{\ge}2$. Compared with the existing results, the nonlocal set of multipartite orthogonal product states constructed by our method has fewer elements and is simpler. Most of all, our multipartite constructions are more general than the ones reported earlier.
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.