Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2410.04331 · Physical Review A

Strongest quantum nonlocality in N -partite systems

Mengying Hu, Ting Gao, Fengli Yan

Published Aug 11, 2025• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/69z8-sk86

Related papers

Strongest nonlocal sets with small sizes

Jicun Li, Fei Shi, Xiande Zhang

A set of orthogonal states is the strongest nonlocal set if it is locally stable in every bipartition, which shows the strong quantum nonlocality proposed by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)]. The existence of the strongest nonlocal sets with the minimum size is an open question. In this work, we partially solve this question by constructing the strongest nonlocal sets with the minimum size in $2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}2$ and $2\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}{d}_{3}$, where $2\ensuremath{\le}{d}_{2}\ensuremath{\le}{d}_{3}$. Moreover, we also give the strongest nonlocal sets with size ${d}_{2}{d}_{3}+{d}_{1}\ensuremath{-}1$ in ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}{d}_{3}$ and the strongest nonlocal sets with size ${d}^{3}+d\ensuremath{-}1$ in $d\ensuremath{\bigotimes}d\ensuremath{\bigotimes}d\ensuremath{\bigotimes}d$. All the sizes of the strongest nonlocal sets are close to the minimum size and are smaller than all previously known constructions. As an application, our strongest nonlocal sets can be used to construct partially genuinely entangled subspaces in ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}{d}_{3}$ when $2\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}{d}_{3}$ and ${d}_{2}\ensuremath{\ge}3$.

Physical Review A•Dec 8, 2023•9 citations

Strongest nonlocal sets with minimum cardinality in multipartite systems

Li, Hong-Run, Hui-Juan Zuo, Fei Shi +1 more

Quantum nonlocality based on state discrimination describes the global property of the set of orthogonal states and has a wide range of applications in quantum cryptographic protocols. Strongest nonlocality is the strongest form of quantum nonlocality recently presented in multipartite quantum systems: a set of orthogonal multipartite quantum states is strongest nonlocal if the only orthogonality-preserving local measurements on the subsystems in every bipartition are trivial. In this work, we found a construction of strongest nonlocal sets in $\mathbb{C}^{d_{1}}\otimes \mathbb{C}^{d_{2}}\otimes \mathbb{C}^{d_{3}}$ $(2\leq d_{1}\leq d_{2}\leq d_{3})$ of size $d_2d_3+1$ without stopper states. Then we obtain the strongest nonlocal sets in four-partite systems with $d^3+1$ orthogonal states in $\mathbb{C}^d\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}$ $(d\geq2)$ and $d_{2}d_{3}d_{4}+1$ orthogonal states in $\mathbb{C}^{d_{1}}\otimes \mathbb{C}^{d_{2}}\otimes \mathbb{C}^{d_{3}}\otimes \mathbb{C}^{d_{4}}$ $(2\leq d_{1}\leq d_{2}\leq d_{3}\leq d_{4})$. Surprisingly, the number of the elements in all above constructions perfectly reaches the recent conjectured lower bound and reduces the size of the strongest nonlocal set in $\mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}$ of [\href{https://doi.org/10.1103/PhysRevA.108.062407}{Phys. Rev. A \textbf{108}, 062407 (2023)}] by $d-2$. In particular, the general optimal construction of the strongest nonlocal set in four-partite system is completely solved for the first time, which further highlights the theory of quantum nonlocality from the perspective of state discrimination.

arXiv:2408.02894•Aug 6, 2024•1 citation

Strong quantum nonlocality in N -partite systems

Fei Shi, Zuo Ye, Lin Chen +1 more

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.

arXiv:2202.07139•Physical Review A•Feb 15, 2022•35 citations

Strongest nonlocal sets with minimum cardinality in tripartite systems

Xiao-Fan Zhen, Mao-Sheng Li, Hui-Juan Zuo

Strong nonlocality, proposed by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)], is a stronger manifestation than quantum nonlocality. Subsequently, Shi et al. presented the concept of the strongest nonlocality [Quantum 6, 619 (2022)]. Recently, Li and Wang [Quantum 7, 1101 (2023)] posed the conjecture about a lower bound to the cardinality of the strongest nonlocal set $\mathcal{S}$ in ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$, i.e., $|\mathcal{S}|\ensuremath{\le}{max}_{i}{{\ensuremath{\prod}}_{j=1}^{n}{d}_{j}/{d}_{i}+1}$. In this work, we construct the strongest nonlocal set of size ${d}^{2}+1$ in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$. Furthermore, we obtain the strongest nonlocal set of size ${d}_{2}{d}_{3}+1$ in ${\mathbb{C}}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{2}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{3}}$. Our construction reaches the lower bound, which provides an affirmative solution to Li and Wang's conjecture. ln particular, the strongest nonlocal sets we present here contain the least number of orthogonal states among the available results.

arXiv:2405.15298•Physical Review A•May 13, 2024•10 citations

Strong quantum nonlocality without entanglement in every ( n − 1 ) -partition

Huaqi Zhou, Ting Gao, Fengli Yan

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.

arXiv:2407.03114•iScience•Dec 5, 2024•5 citations

Strong nonlocality with genuine entanglement based on GHZ-like states in multipartite quantum systems

Ying‐Hui Yang, Guang-Wei Mi, Shi-Jiao Geng +2 more

Abstract Strong nonlocality with genuine entanglement was first shown by Wang et al using sets of GHZ-like states in tripartite quantum systems (2021 Phys. Rev. A 104, 012 424). However, it is an open problem whether there exists strong nonlocality with genuine entanglement in four or more partite systems. In this paper, we unify two different concepts of strong nonlocality introduced by Halder et al (2019 Phys. Rev. Lett. 122, 040 403) and by Zhang et al (2019 Phys. Rev. A 99, 062108). That is, we use a concept of k-strong nonlocality instead of these two different types of strong nonlocality. A set of orthogonal quantum states is k-strong nonlocal if it is locally irreducible in every k-partition. In fact, the strong nonlocality that is usually said is 2-strong nonlocality. The smaller the k is, the stronger the nonlocality will be. A set of states is k +-strong nonlocal if the strong nonlocality of this set is stronger than k-strong nonlocality but weaker than (k − 1)-strong nonlocality. Based on these concepts, firstly, we show 2-strong nonlocality with genuine entanglement by some sets of GHZ-like states with weight d in tripartite systems. These sets are not necessarily complete bases. Secondly, we present 2-strong nonlocality with genuine entanglement for systems with four or more parties. These results solve the open problem raised by Wang et al Finally, we construct a set of GHZ-like states with n +-strong nonlocality in n-partite quantum systems.

Physica Scripta•Nov 24, 2022•0 citations
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 NNN-partite systems. Then we derive the minimum size of the strongest nonlocal set with symmetry in the system (C3)\ensuremath⨂N{({\mathbb{C}}^{3})}^{\ensuremath{\bigotimes}N}(C3)\ensuremath⨂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 (Cd)\ensuremath⨂N{({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}(Cd)\ensuremath⨂N for d\ensuremath≥4d\ensuremath{\ge}4d\ensuremath≥4, which have a smaller size than the existing sets as NNN increases. Our results provide a meaningful complement to the current understanding of the strongest nonlocality.
A set of orthogonal states is the strongest nonlocal set if it is locally stable in every bipartition, which shows the strong quantum nonlocality proposed by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)]. The existence of the strongest nonlocal sets with the minimum size is an open question. In this work, we partially solve this question by constructing the strongest nonlocal sets with the minimum size in $2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}2$ and $2\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}{d}_{3}$, where $2\ensuremath{\le}{d}_{2}\ensuremath{\le}{d}_{3}$. Moreover, we also give the strongest nonlocal sets with size ${d}_{2}{d}_{3}+{d}_{1}\ensuremath{-}1$ in ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}{d}_{3}$ and the strongest nonlocal sets with size ${d}^{3}+d\ensuremath{-}1$ in $d\ensuremath{\bigotimes}d\ensuremath{\bigotimes}d\ensuremath{\bigotimes}d$. All the sizes of the strongest nonlocal sets are close to the minimum size and are smaller than all previously known constructions. As an application, our strongest nonlocal sets can be used to construct partially genuinely entangled subspaces in ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}{d}_{3}$ when $2\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}{d}_{3}$ and ${d}_{2}\ensuremath{\ge}3$.
Quantum nonlocality based on state discrimination describes the global property of the set of orthogonal states and has a wide range of applications in quantum cryptographic protocols. Strongest nonlocality is the strongest form of quantum nonlocality recently presented in multipartite quantum systems: a set of orthogonal multipartite quantum states is strongest nonlocal if the only orthogonality-preserving local measurements on the subsystems in every bipartition are trivial. In this work, we found a construction of strongest nonlocal sets in Cd1⊗Cd2⊗Cd3\mathbb{C}^{d_{1}}\otimes \mathbb{C}^{d_{2}}\otimes \mathbb{C}^{d_{3}}Cd1​⊗Cd2​⊗Cd3​ (2≤d1≤d2≤d3)(2\leq d_{1}\leq d_{2}\leq d_{3})(2≤d1​≤d2​≤d3​) of size d2d3+1d_2d_3+1d2​d3​+1 without stopper states. Then we obtain the strongest nonlocal sets in four-partite systems with d3+1d^3+1d3+1 orthogonal states in Cd⊗Cd⊗Cd⊗Cd\mathbb{C}^d\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}Cd⊗Cd⊗Cd⊗Cd (d≥2)(d\geq2)(d≥2) and d2d3d4+1d_{2}d_{3}d_{4}+1d2​d3​d4​+1 orthogonal states in Cd1⊗Cd2⊗Cd3⊗Cd4\mathbb{C}^{d_{1}}\otimes \mathbb{C}^{d_{2}}\otimes \mathbb{C}^{d_{3}}\otimes \mathbb{C}^{d_{4}}Cd1​⊗Cd2​⊗Cd3​⊗Cd4​ (2≤d1≤d2≤d3≤d4)(2\leq d_{1}\leq d_{2}\leq d_{3}\leq d_{4})(2≤d1​≤d2​≤d3​≤d4​). Surprisingly, the number of the elements in all above constructions perfectly reaches the recent conjectured lower bound and reduces the size of the strongest nonlocal set in Cd⊗Cd⊗Cd⊗Cd\mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}Cd⊗Cd⊗Cd⊗Cd of [\href{https://doi.org/10.1103/PhysRevA.108.062407}{Phys. Rev. A \textbf{108}, 062407 (2023)}] by d−2d-2d−2. In particular, the general optimal construction of the strongest nonlocal set in four-partite system is completely solved for the first time, which further highlights the theory of quantum nonlocality from the perspective of state discrimination.
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.
Strong nonlocality, proposed by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)], is a stronger manifestation than quantum nonlocality. Subsequently, Shi et al. presented the concept of the strongest nonlocality [Quantum 6, 619 (2022)]. Recently, Li and Wang [Quantum 7, 1101 (2023)] posed the conjecture about a lower bound to the cardinality of the strongest nonlocal set S\mathcal{S}S in ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$, i.e., $|\mathcal{S}|\ensuremath{\le}{max}_{i}{{\ensuremath{\prod}}_{j=1}^{n}{d}_{j}/{d}_{i}+1}$. In this work, we construct the strongest nonlocal set of size d2+1{d}^{2}+1d2+1 in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$. Furthermore, we obtain the strongest nonlocal set of size d2d3+1{d}_{2}{d}_{3}+1d2​d3​+1 in ${\mathbb{C}}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{2}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{3}}$. Our construction reaches the lower bound, which provides an affirmative solution to Li and Wang's conjecture. ln particular, the strongest nonlocal sets we present here contain the least number of orthogonal states among the available results.
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.
Abstract Strong nonlocality with genuine entanglement was first shown by Wang et al using sets of GHZ-like states in tripartite quantum systems (2021 Phys. Rev. A 104, 012 424). However, it is an open problem whether there exists strong nonlocality with genuine entanglement in four or more partite systems. In this paper, we unify two different concepts of strong nonlocality introduced by Halder et al (2019 Phys. Rev. Lett. 122, 040 403) and by Zhang et al (2019 Phys. Rev. A 99, 062108). That is, we use a concept of k-strong nonlocality instead of these two different types of strong nonlocality. A set of orthogonal quantum states is k-strong nonlocal if it is locally irreducible in every k-partition. In fact, the strong nonlocality that is usually said is 2-strong nonlocality. The smaller the k is, the stronger the nonlocality will be. A set of states is k +-strong nonlocal if the strong nonlocality of this set is stronger than k-strong nonlocality but weaker than (k − 1)-strong nonlocality. Based on these concepts, firstly, we show 2-strong nonlocality with genuine entanglement by some sets of GHZ-like states with weight d in tripartite systems. These sets are not necessarily complete bases. Secondly, we present 2-strong nonlocality with genuine entanglement for systems with four or more parties. These results solve the open problem raised by Wang et al Finally, we construct a set of GHZ-like states with n +-strong nonlocality in n-partite quantum systems.