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arXiv:2202.07139 · Physical Review A

Strong quantum nonlocality in N -partite systems

Fei Shi, Zuo Ye, Lin Chen, Xiande Zhang

Published Feb 15, 2022• 35 citations
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Abstract

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 $N$-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 $N$-partite systems. For $N=3$ 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.

DOI
10.1103/physreva.105.022209

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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≥6N\ensuremath{\ge}6N\ensuremath≥6. In this paper, we successfully show that a strongly nonlocal set of orthogonal entangled states exists in (Cd)\ensuremath⨂N{({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}(Cd)\ensuremath⨂N for all N\ensuremath≥3N\ensuremath{\ge}3N\ensuremath≥3 and d\ensuremath≥2d\ensuremath{\ge}2d\ensuremath≥2, which reveals the strong quantum nonlocality in general NNN-partite systems. For N=3N=3N=3 or 4 and d\ensuremath≥3d\ensuremath{\ge}3d\ensuremath≥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.
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 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.
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 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.
Strong quantum nonlocality was introduced recently as a stronger manifestation of nonlocality in multipartite systems through the notion of local irreducibility in all bipartitions. Known existing results for sets of strongly nonlocal orthogonal states are limited to product states. In this paper, based on the Rubik's cube, we give a construction of such sets consisting of entangled states in $d\ensuremath{\bigotimes}d\ensuremath{\bigotimes}d$ for all $d\ensuremath{\ge}3$. Consequently, we answer an open problem given by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)], that is, orthogonal entangled bases that are strongly nonlocal do exist. Furthermore, we propose two entanglement-assisted protocols for local discrimination of our results. Each protocol consumes less entanglement resources than the teleportation-based protocol on average. Our results exhibit the phenomenon of strong quantum nonlocality with 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.