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Chinese Physics B

Construction methods of nonlocal sets of orthogonal product states on multipartite quantum systems

Guang‐Bao Xu, Zhao-Xia 昭霞 Zhong 仲, Yu‐Guang Yang, Dong‐Huan Jiang

Published May 26, 2025• 0 citations
Publisher

Abstract

Abstract Nonlocal set of orthogonal product states (OPSs) can improve the confidentiality of information when it is used to design quantum cryptographic protocols. It is a difficult question how to construct a nonlocal set of OPSs on general multipartite and high dimensional quantum systems. Different from the previous works, we first present a novel method for constructing a nonlocal product set with 3 d – 2 members on C d ⊗ C d ⊗ C d quantum system for d ≥ 3. Then, we extend this construction method to C d 1 ⊗ C d 2 ⊗ C d 3 quantum system and ⊗ i = 1 n C d i quantum system respectively, where 3 ≤ d 1 ≤ d 2 ≤ d 3 ≤ … ≤ d n and n ≥ 3. The nonlocal set of OPSs constructed by our method contains fewer elements than those constructed by the existing methods, except for one special case. More importantly, the set of states constructed by our method has a completely different structure from those constructed by the existing methods since our nonlocal set does not contain a “stopper” state. Our result is helpful to further understand the different structures of nonlocal sets on multipartite systems.

DOI
10.1088/1674-1056/addcd5

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Abstract Nonlocal set of orthogonal product states (OPSs) can improve the confidentiality of information when it is used to design quantum cryptographic protocols. It is a difficult question how to construct a nonlocal set of OPSs on general multipartite and high dimensional quantum systems. Different from the previous works, we first present a novel method for constructing a nonlocal product set with 3 d – 2 members on C d ⊗ C d ⊗ C d quantum system for d ≥ 3. Then, we extend this construction method to C d 1 ⊗ C d 2 ⊗ C d 3 quantum system and ⊗ i = 1 n C d i quantum system respectively, where 3 ≤ d 1 ≤ d 2 ≤ d 3 ≤ … ≤ d n and n ≥ 3. The nonlocal set of OPSs constructed by our method contains fewer elements than those constructed by the existing methods, except for one special case. More importantly, the set of states constructed by our method has a completely different structure from those constructed by the existing methods since our nonlocal set does not contain a “stopper” state. Our result is helpful to further understand the different structures of nonlocal sets on multipartite systems.
Nonlocal sets of orthogonal product states (OPSs) are widely used in quantum protocols owing to their good property. Thus a lot of attention are paid to how to construct a nonlocal set of orthogonal product states though it is a difficult problem. In this paper, we propose a novel general method to construct a nonlocal set of orthogonal product states in Cd⊗Cd\mathbb{C}^{d} \otimes \mathbb{C}^{d}Cd⊗Cd for d≥3d\geq3d≥3. We give an ingenious proof for the local indistinguishability of those product states. The set of product states, which are constructed by our method, has a very good structure. Subsequently, we give a construction of nonlocal set of OPSs with smaller members in Cd⊗Cd\mathbb{C}^{d} \otimes \mathbb{C}^{d}Cd⊗Cd for d≥3d\geq3d≥3. On the other hand, we present two construction methods of nonlocal sets of OPSs in Cm⊗Cn\mathbb{C}^{m} \otimes \mathbb{C}^{n}Cm⊗Cn, where m≥3m\geq3m≥3 and n≥3.n\geq3.n≥3. Furthermore, we propose the concept of isomorphism for two nonlocal sets of OPSs. Our work is of great help to understand the structure and classification of locally indistinguishable OPSs.
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.
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 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 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.