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arXiv:2205.10813 · Physical Review A

Orthogonal product sets with strong quantum nonlocality on a plane structure

Huaqi Zhou, Ting Gao, Fengli Yan

Published Nov 15, 2022• 14 citations
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Abstract

In this paper, we consider the orthogonal product set (OPS) with strong quantum nonlocality in multipartite quantum systems. Based on the decomposition of plane geometry, we present a sufficient condition for the triviality of orthogonality-preserving positive operator-valued measures on fixed subsystem and partially answer an open question given by Yuan et al. [Phys. Rev. A 102, 042228 (2020)]. The connection between the nonlocality and the plane structure of OPSs is established. We successfully construct a strongly nonlocal OPS in ${\mathcal{C}}^{{d}_{A}}\ensuremath{\bigotimes}{\mathcal{C}}^{{d}_{B}}\ensuremath{\bigotimes}{\mathcal{C}}^{{d}_{C}}\phantom{\rule{4pt}{0ex}}({d}_{A},{d}_{B},{d}_{C}\ensuremath{\ge}4)$, which contains fewer quantum states, and generalize the structures of known OPSs to any possible three and four-partite systems. In addition, we propose several entanglement-assisted protocols for perfectly local discrimination of the sets. It is shown that the protocols without teleportation use less entanglement resources that on average and these sets can always be discriminated locally with multiple copies of two-qubit maximally entangled states. These results also exhibit nontrivial signification of maximally entangled states in the local discrimination of quantum states.

DOI
10.1103/physreva.106.052209

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In this paper, we consider the orthogonal product set (OPS) with strong quantum nonlocality in multipartite quantum systems. Based on the decomposition of plane geometry, we present a sufficient condition for the triviality of orthogonality-preserving positive operator-valued measures on fixed subsystem and partially answer an open question given by Yuan et al. [Phys. Rev. A 102, 042228 (2020)]. The connection between the nonlocality and the plane structure of OPSs is established. We successfully construct a strongly nonlocal OPS in CdA\ensuremath⨂CdB\ensuremath⨂CdC(dA,dB,dC\ensuremath≥4){\mathcal{C}}^{{d}_{A}}\ensuremath{\bigotimes}{\mathcal{C}}^{{d}_{B}}\ensuremath{\bigotimes}{\mathcal{C}}^{{d}_{C}}\phantom{\rule{4pt}{0ex}}({d}_{A},{d}_{B},{d}_{C}\ensuremath{\ge}4)CdA​\ensuremath⨂CdB​\ensuremath⨂CdC​(dA​,dB​,dC​\ensuremath≥4), which contains fewer quantum states, and generalize the structures of known OPSs to any possible three and four-partite systems. In addition, we propose several entanglement-assisted protocols for perfectly local discrimination of the sets. It is shown that the protocols without teleportation use less entanglement resources that on average and these sets can always be discriminated locally with multiple copies of two-qubit maximally entangled states. These results also exhibit nontrivial signification of maximally entangled states in the local discrimination of quantum states.
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.
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.
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.
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.
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 $N\ensuremath{\ge}6$. 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 NNN-partite systems. For N=3N=3N=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.