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arXiv:2212.14523 · Physical Review A

Nonlocality without entanglement in general multipartite quantum systems

Xiao-Fan Zhen, Shao-Ming Fei, Hui-Juan Zuo

Published Dec 23, 2022• 28 citations
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Abstract

The construction of nonlocal sets of quantum states has attracted much attention in recent years. We first introduce two lemmas related to the triviality of orthogonality-preserving local measurements. Then, we propose a general construction of nonlocal set of $n(d\ensuremath{-}1)+1$ orthogonal product states in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}n}$. The sets of nonlocal orthogonal product states are also put forward for multipartite quantum systems with arbitrary dimensions. Our construction gives rise to nonlocal sets of orthogonal product states with much less members and thus reveals the phenomenon of nonlocality without entanglement more efficiently.

DOI
10.1103/physreva.106.062432

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The construction of nonlocal sets of quantum states has attracted much attention in recent years. We first introduce two lemmas related to the triviality of orthogonality-preserving local measurements. Then, we propose a general construction of nonlocal set of n(d\ensuremath−1)+1n(d\ensuremath{-}1)+1n(d\ensuremath−1)+1 orthogonal product states in (Cd)\ensuremath⨂n{({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}n}(Cd)\ensuremath⨂n. The sets of nonlocal orthogonal product states are also put forward for multipartite quantum systems with arbitrary dimensions. Our construction gives rise to nonlocal sets of orthogonal product states with much less members and thus reveals the phenomenon of nonlocality without entanglement more efficiently.
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.
We study the constructions of nonlocal orthogonal product states in multipartite systems that cannot be distinguished by local operations and classical communication. We first present two constructions of nonlocal orthogonal product states in tripartite systems Cd⊗Cd⊗Cd (d≥3)\mathcal{C}^{d}\otimes\mathcal{C}^{d}\otimes\mathcal{C}^{d}~(d\geq3)Cd⊗Cd⊗Cd (d≥3) and Cd⊗Cd+1⊗Cd+2 (d≥3)\mathcal{C}^d\otimes \mathcal{C}^{d+1}\otimes \mathcal{C}^{d+2}~(d\geq 3)Cd⊗Cd+1⊗Cd+2 (d≥3). Then for general tripartite quantum system Cn1⊗Cn2⊗Cn3\mathcal{C}^{n_{1}}\otimes\mathcal{C}^{n_{2}}\otimes\mathcal{C}^{n_{3}}Cn1​⊗Cn2​⊗Cn3​ (3≤n1≤n2≤n3)(3\leq n_{1}\leq n_{2}\leq n_{3})(3≤n1​≤n2​≤n3​), we obtain 2(n2+n3−1)−n12(n_{2}+n_{3}-1)-n_{1}2(n2​+n3​−1)−n1​ nonlocal orthogonal product states. Finally, we put forward a new construction approach in Cd1⊗Cd2⊗⋯⊗Cdn\mathcal{C}^{d_{1}}\otimes \mathcal{C}^{d_{2}}\otimes\cdots\otimes \mathcal{C}^{d_{n}}Cd1​⊗Cd2​⊗⋯⊗Cdn​ (d1,d2,⋯dn≥3, n>6)(d_1,d_2,\cdots d_n\geq3,\, n>6)(d1​,d2​,⋯dn​≥3,n>6) multipartite systems. Remarkably, our indistinguishable sets contain less nonlocal product states than the existing ones, which improves the recent results and highlights their 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.
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.
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.
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.