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.
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.
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 n-partite systems for all even n, 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 n-partite systems for all even n. 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.
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⊗d quantum system, which contains 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⊗4 and 5⊗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⊗3 and 4⊗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.
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 n-dimensional hypercube for n=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 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.
The strongest quantum nonlocality is a stronger form of nonlocality than that manifested by locally indistinguishable quantum states. It has efficient applications in quantum information hiding and quantum secret sharing. Constructing a smaller-size set with the strongest nonlocality contributes to reducing resource consumption. In this paper we establish a sufficient and necessary condition for the strongest nonlocality in general N-partite systems. Then we derive the minimum size of the strongest nonlocal set with symmetry in the system ${({\mathbb{C}}^{3})}^{\ensuremath{\bigotimes}N}$. Notably, the genuinely entangled sets constructed in our previous work [Phys. Rev. A 109, 022220 (2024)] achieved this minimum size. In addition, we construct the strongest nonlocal genuinely entangled sets in the system ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$ for $d\ensuremath{\ge}4$, which have a smaller size than the existing sets as N increases. Our results provide a meaningful complement to the current understanding of the strongest nonlocality.