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Quantum Information Processing

Novel methods to construct nonlocal sets of orthogonal product states in an arbitrary bipartite high-dimensional system

Guang‐Bao Xu, Dong‐Huan Jiang

Published Apr 1, 2021• 56 citations
Publisher
DOI
10.1007/s11128-021-03062-8

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Novel methods to construct nonlocal sets of orthogonal product states in arbitrary bipartite high-dimensional system

Guang‐Bao Xu, Dong‐Huan Jiang

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 $\mathbb{C}^{d} \otimes \mathbb{C}^{d}$ for $d\geq3$. 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 $\mathbb{C}^{d} \otimes \mathbb{C}^{d}$ for $d\geq3$. On the other hand, we present two construction methods of nonlocal sets of OPSs in $\mathbb{C}^{m} \otimes \mathbb{C}^{n}$, where $m\geq3$ and $n\geq3.$ 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.

arXiv:2003.08291•Mar 16, 2020•0 citations

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 +1 more

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.

Chinese Physics B•May 26, 2025•0 citations
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.
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.