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arXiv:2011.00924

The construction of sets with strong quantum nonlocality using fewer states

Bichen Che, Zhao Dou, Min Lei, Yixian Yang

Published Nov 2, 2020• 2 citations
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Abstract

In this paper, we investigate the construction of orthogonal product states with strong nonlocality in multiparty quantum systems. Firstly, we focus on the tripartite system and propose a general set of orthogonal product states exhibiting strong nonlocality in $d\otimes d\otimes d$ quantum system, which contains $6{{\left( d-1 \right)}^{2}}$ states. Secondly, we find that the number of the sets constructed in this way could be further reduced. Then using $4\otimes 4\otimes 4$ and $5\otimes 5\otimes 5$ quantum systems as examples, it can be seen that when d increases, the reduced quantum state is considerable. Thirdly, by imitating the construction method of the tripartite system, two 3-divisible four-party quantum systems are proposed, $3\otimes 3\otimes 3\otimes 3$ and $4\otimes 4\otimes 4\otimes 4$, both of which contains fewer states than the existing ones. Our research gives a positive answer to an open question raised in [Halder, et al., PRL, 122, 040403 (2019)], indicating that there do exist fewer quantum states that can exhibit strong quantum nonlocality without entanglement.

DOI
10.48550/arxiv.2011.00924

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In this paper, we investigate the construction of orthogonal product states with strong nonlocality in multiparty quantum systems. Firstly, we focus on the tripartite system and propose a general set of orthogonal product states exhibiting strong nonlocality in d⊗d⊗dd\otimes d\otimes dd⊗d⊗d quantum system, which contains 6(d−1)26{{\left( d-1 \right)}^{2}}6(d−1)2 states. Secondly, we find that the number of the sets constructed in this way could be further reduced. Then using 4⊗4⊗44\otimes 4\otimes 44⊗4⊗4 and 5⊗5⊗55\otimes 5\otimes 55⊗5⊗5 quantum systems as examples, it can be seen that when d increases, the reduced quantum state is considerable. Thirdly, by imitating the construction method of the tripartite system, two 3-divisible four-party quantum systems are proposed, 3⊗3⊗3⊗33\otimes 3\otimes 3\otimes 33⊗3⊗3⊗3 and 4⊗4⊗4⊗44\otimes 4\otimes 4\otimes 44⊗4⊗4⊗4, both of which contains fewer states than the existing ones. Our research gives a positive answer to an open question raised in [Halder, et al., PRL, 122, 040403 (2019)], indicating that there do exist fewer quantum states that can exhibit strong quantum nonlocality without entanglement.
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.
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.
A set of multipartite orthogonal product states is strongly nonlocal if it is locally irreducible in every bipartition. Most known constructions of strongly nonlocal orthogonal product set (OPS) are limited to tripartite systems, and they are lack of intuitive structures. In this work, based on the decomposition for the outermost layer of an nnn-dimensional hypercube for n=3,4,5n= 3,4,5n=3,4,5, we successfully construct strongly nonlocal OPSs in any possible three, four and five-partite systems, which answers an open question given by Halder et al. [Phys. Rev. Lett.122, 040403 (2019)] and Yuan et al. [Phys. Rev. A102, 042228 (2020)] for any possible three, four and five-partite systems. Our results build the connection between hypercubes and strongly nonlocal OPSs, and exhibit the phenomenon of strong quantum nonlocality without entanglement in multipartite systems.
In multipartite systems, great progress has been made recently on the study of strong quantum nonlocality without entanglement. However, the existence of orthogonal product sets with strong quantum nonlocality in even party systems remains unknown. Here the even number is greater than 4. In this paper, we successfully construct strongly nonlocal orthogonal product sets in nnn-partite systems for all even nnn, which answers the open questions given by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)] and Yuan et al. [Phys. Rev. A 102, 042228 (2020)] for any possible even party systems. Thus, we find general construction of strongly nonlocal orthogonal product sets in space ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathcal{C}}^{{d}_{i}}$ $(n,{d}_{i}\ensuremath{\ge}3)$ and show that there do exist incomplete orthogonal product bases that can be strongly nonlocal in any possible nnn-partite systems for all even nnn. Our newly constructed orthogonal product sets are asymmetric. We analyze the differences and connections between these sets and the known orthogonal product sets in odd party systems. In addition, we present a local state discrimination protocol for our sets by using additional entangled resources. When at least two subsystems have dimensions greater than three, the protocol consumes less entanglement than a teleportation-based protocol. A strongly nonlocal set implies that the information cannot be completely accessed as long as it does not happen that all parties are together. As an application, we connect our sets with local information hiding in multipartite 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.
The orthogonal product set with quantum nonlocality can enhance the confidentiality of information without consuming entanglement resources. The confidentiality increases with the reinforcement of its nonlocality. However, the orthogonal product sets with the strongest nonlocality need an enormous number of quantum states. In this paper, we propose a sufficient condition for orthogonal product set being strongly nonlocal and construct some strongly nonlocal sets in every ( n − 1 ) -partition on n -qudit systems, where n is greater than 3. These newly constructed sets possess stronger properties than nonlocality and reduce the exponential number of required quantum states compared with the strongest nonlocal sets. Our results not only partially answer the open question "how to construct different strength nonlocality of orthogonal product states for general multipartite and high-dimensional quantum systems" but also provide significant theoretical fundamental for quantum secure communication.