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Zhongguo kexue. Wulixue Lixue Tianwenxue

Constructions of orthogonal product state sets with genuine hidden nonlocality in bipartite system

ZHANG Qing, ZUO Hui-Juan

Published Nov 17, 2024• 1 citation
Publisher

Abstract

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.

DOI
10.1360/sspma-2024-0496

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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.
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.
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 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 ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$ $[3\ensuremath{\le}({d}_{1}\ensuremath{-}1)\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n},\phantom{\rule{4pt}{0ex}}n\ensuremath{\ge}3]$. 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.
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.
The research on minimal nonlocality aims to determine the minimal cardinality of nonlocal sets of quantum states. However, the construction of a nonlocal set of states in bipartite or tripartite quantum systems with unequal local dimensions remains unsolved. In this paper, we first give a method to construct a set of orthogonal quantum states with minimal nonlocality in C4⊗C7\mathbb{C}^{4} \otimes \mathbb{C}^{7}C4⊗C7 quantum system. Then we give a general method to construct a set of orthogonal quantum states with minimal nonlocality in a bipartite quantum system with unequal local dimensions. Furthermore, we generalize the construction method to tripartite quantum system with unequal local dimensions, and construct a set of orthogonal quantum states with minimal nonlocality in Cd1⊗Cd2⊗Cd3\mathbb C^{d_1} \otimes \mathbb C^{d_2}\otimes \mathbb C^{d_3}Cd1​⊗Cd2​⊗Cd3​ quantum system for 5≤d1<d2<d35\le d_1 < d_2 < d_35≤d1​<d2​<d3​. Our work settles the construction problem of a set of orthogonal states with minimal nonlocality in both bipartite and tripartite systems with unequal local dimensions.
Quantum nonlocality is usually associated with entangled states by their violations of Bell-type inequalities. However, even unentangled systems, whose parts may have been prepared separately, can show nonlocal properties. In particular, a set of product states is said to exhibit "quantum nonlocality without entanglement" if the states are locally indistinguishable; i.e., it is not possible to optimally distinguish the states by any sequence of local operations and classical communication. Here, we present a stronger manifestation of this kind of nonlocality in multiparty systems through the notion of local irreducibility. A set of multiparty orthogonal quantum states is defined to be locally irreducible if it is not possible to locally eliminate one or more states from the set while preserving orthogonality of the postmeasurement states. Such a set, by definition, is locally indistinguishable, but we show that the converse does not always hold. We provide the first examples of orthogonal product bases on C^{d}⊗C^{d}⊗C^{d} for d=3, 4 that are locally irreducible in all bipartitions, where the construction for d=3 achieves the minimum dimension necessary for such product states to exist. The existence of such product bases implies that local implementation of a multiparty separable measurement may require entangled resources across all bipartitions.