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Physical Review A

Minimal nonlocality in multipartite quantum systems

Guang‐Bao Xu, Yong-Qi Zhang, Dong‐Huan Jiang

Published Feb 10, 2025• 11 citations
Publisher

Abstract

Quantum nonlocality, rooted in the quantum state discrimination task, contributes to the development of the theory of local distinguishability of quantum states. Recently, Zhu et al. [Phys. A (Amsterdam, Neth.) 624, 128956 (2023)] found that a nonlocal set of orthogonal product states (OPSs) can be perfectly identified by local operations and classical communication after removing a specific quantum state in a bipartite quantum system. They called this phenomenon minimal nonlocality. It is obvious that minimal nonlocality effectively reflects the lower boundary of the elements contained in a nonlocal set of OPSs. In this paper, we generalize the methods of constructing nonlocal sets with minimal nonlocality from bipartite systems to multipartite systems and from orthogonal product states to entangled states. First, we give a method to construct a set of OPSs with minimal nonlocality in the ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ quantum system, and then we extend it to the ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{d}$ quantum system for $d\ensuremath{\ge}4$ and $n\ensuremath{\ge}3$. In addition, we give a method to construct a nonlocal set of orthogonal states with entanglement relations in the ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ quantum system, and we prove that this set has minimal nonlocality, which can also be extended to the ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{d}$ quantum system.

DOI
10.1103/physreva.111.022416

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Quantum nonlocality, rooted in the quantum state discrimination task, contributes to the development of the theory of local distinguishability of quantum states. Recently, Zhu et al. [Phys. A (Amsterdam, Neth.) 624, 128956 (2023)] found that a nonlocal set of orthogonal product states (OPSs) can be perfectly identified by local operations and classical communication after removing a specific quantum state in a bipartite quantum system. They called this phenomenon minimal nonlocality. It is obvious that minimal nonlocality effectively reflects the lower boundary of the elements contained in a nonlocal set of OPSs. In this paper, we generalize the methods of constructing nonlocal sets with minimal nonlocality from bipartite systems to multipartite systems and from orthogonal product states to entangled states. First, we give a method to construct a set of OPSs with minimal nonlocality in the Cd\ensuremath⨂Cd\ensuremath⨂Cd{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}Cd\ensuremath⨂Cd\ensuremath⨂Cd quantum system, and then we extend it to the \ensuremath⨂i=1nCd{\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{d}\ensuremath⨂i=1n​Cd quantum system for d\ensuremath≥4d\ensuremath{\ge}4d\ensuremath≥4 and n\ensuremath≥3n\ensuremath{\ge}3n\ensuremath≥3. In addition, we give a method to construct a nonlocal set of orthogonal states with entanglement relations in the Cd\ensuremath⨂Cd{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}Cd\ensuremath⨂Cd quantum system, and we prove that this set has minimal nonlocality, which can also be extended to the \ensuremath⨂i=1nCd{\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{d}\ensuremath⨂i=1n​Cd quantum system.
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.
A set of multipartite orthogonal product states is locally irreducible, if it is not possible to eliminate one or more states from the set by orthogonality-preserving local measurements. An effective way to prove that a set is locally irreducible is to show that only trivial orthogonality-preserving measurement can be performed to this set. In general, it is difficult to show such a orthogonality-preserving measurement must be trivial. In this work, we develop two basic techniques to deal with this problem. Using these techniques, we successfully construct orthogonal product sets (OPSs) in any possible three, four and five-partite systems that are locally irreducible in every bipartition. Moreover, we also show the existence of unextendible product bases (UPBs) with such properties in d⊗d⊗dd\otimes d\otimes dd⊗d⊗d for any d≥3d\geq 3d≥3. Therefore, these OPSs and UPBs exhibit the phenomenon of strong quantum nonlocality without entanglement. Both results solve the open questions given by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)] and Yuan et al. [Phys. Rev. A 102, 042228 (2020)]. Our results shed a new light on the strong quantum nonlocality without entanglement.
Quantum nonlocality based on state discrimination describes the global property of the set of orthogonal states and has a wide range of applications in quantum cryptographic protocols. Strongest nonlocality is the strongest form of quantum nonlocality recently presented in multipartite quantum systems: a set of orthogonal multipartite quantum states is strongest nonlocal if the only orthogonality-preserving local measurements on the subsystems in every bipartition are trivial. In this work, we found a construction of strongest nonlocal sets in Cd1⊗Cd2⊗Cd3\mathbb{C}^{d_{1}}\otimes \mathbb{C}^{d_{2}}\otimes \mathbb{C}^{d_{3}}Cd1​⊗Cd2​⊗Cd3​ (2≤d1≤d2≤d3)(2\leq d_{1}\leq d_{2}\leq d_{3})(2≤d1​≤d2​≤d3​) of size d2d3+1d_2d_3+1d2​d3​+1 without stopper states. Then we obtain the strongest nonlocal sets in four-partite systems with d3+1d^3+1d3+1 orthogonal states in Cd⊗Cd⊗Cd⊗Cd\mathbb{C}^d\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}Cd⊗Cd⊗Cd⊗Cd (d≥2)(d\geq2)(d≥2) and d2d3d4+1d_{2}d_{3}d_{4}+1d2​d3​d4​+1 orthogonal states in Cd1⊗Cd2⊗Cd3⊗Cd4\mathbb{C}^{d_{1}}\otimes \mathbb{C}^{d_{2}}\otimes \mathbb{C}^{d_{3}}\otimes \mathbb{C}^{d_{4}}Cd1​⊗Cd2​⊗Cd3​⊗Cd4​ (2≤d1≤d2≤d3≤d4)(2\leq d_{1}\leq d_{2}\leq d_{3}\leq d_{4})(2≤d1​≤d2​≤d3​≤d4​). Surprisingly, the number of the elements in all above constructions perfectly reaches the recent conjectured lower bound and reduces the size of the strongest nonlocal set in Cd⊗Cd⊗Cd⊗Cd\mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}Cd⊗Cd⊗Cd⊗Cd of [\href{https://doi.org/10.1103/PhysRevA.108.062407}{Phys. Rev. A \textbf{108}, 062407 (2023)}] by d−2d-2d−2. In particular, the general optimal construction of the strongest nonlocal set in four-partite system is completely solved for the first time, which further highlights the theory of quantum nonlocality from the perspective of state discrimination.
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 .
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.