arXiv:2307.08330 · Physical Review A

Locally stable sets with minimum cardinality

Hai-Qing Cao, Mao-Sheng Li, Hui-Juan Zuo

Published Jul 13, 2023• 18 citations

Abstract

The nonlocal set has received wide attention over recent years. Shortly before, Li and Wang arXiv:2202.09034 proposed the concept of a locally stable set: The only possible orthogonality preserving measurement on each subsystem is trivial. Locally stable sets present stronger nonlocality than those sets that are just locally indistinguishable. In this work, we focus on the constructions of locally stable sets in multipartite quantum systems. First, two lemmas are put forward to prove that an orthogonality-preserving local measurement must be trivial. Then we present the constructions of locally stable sets with minimum cardinality in bipartite quantum systems ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\phantom{\rule{4pt}{0ex}}(d\ensuremath{\ge}3)$ and ${\mathbb{C}}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{2}}\phantom{\rule{4pt}{0ex}}(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2})$. Moreover, for the multipartite quantum systems ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}n}\phantom{\rule{4pt}{0ex}}(d\ensuremath{\ge}2)$ and ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}\phantom{\rule{4pt}{0ex}}(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n})$, we also obtain $d+1$ and ${d}_{n}+1$ locally stable orthogonal states, respectively. Fortunately, our constructions reach the lower bound of the cardinality on the locally stable sets, which provides a positive and complete answer to an open problem raised in arXiv:2202.09034.

DOI: 10.1103/physreva.108.012418

Related papers

Bounds on the smallest sets of quantum states with special quantum nonlocality

Mao-Sheng Li, Yanling Wang

arXiv:2202.09034QuantumSep 7, 202321 citations

Locally stable sets without entanglement with minimum cardinality

Xiao-Fan Zhen, Su‐Juan Qin, Zhengping Jin +2 more

Physical Review AMay 19, 20253 citations

Strongest nonlocal sets with minimum cardinality in multipartite systems

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

arXiv:2408.02894Aug 6, 20241 citation

Strongest nonlocal sets with minimum cardinality in tripartite systems

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

arXiv:2405.15298Physical Review AMay 13, 202410 citations

Nonlocality without entanglement in general multipartite quantum systems

Xiao-Fan Zhen, Shao-Ming Fei, Hui-Juan Zuo

arXiv:2212.14523Physical Review ADec 23, 202228 citations

Genuinely nonlocal sets without entanglement in multipartite systems

Ying-Ying Lu, Hai-Qing Cao, Hui-Juan Zuo +1 more

arXiv:2408.11566Physical Review AAug 16, 20247 citations

arXiv:2307.08330 · Physical Review A

Locally stable sets with minimum cardinality

Hai-Qing Cao, Mao-Sheng Li, Hui-Juan Zuo

Published Jul 13, 2023• 18 citations

arXiv HTML PDF

Abstract

{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\phantom{\rule{4pt}{0ex}}(d\ensuremath{\ge}3)

and

{\mathbb{C}}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{2}}\phantom{\rule{4pt}{0ex}}(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2})

. Moreover, for the multipartite quantum systems

{({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}n}\phantom{\rule{4pt}{0ex}}(d\ensuremath{\ge}2)

and

{\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}\phantom{\rule{4pt}{0ex}}(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n})

, we also obtain

d+1

and

{d}_{n}+1

locally stable orthogonal states, respectively. Fortunately, our constructions reach the lower bound of the cardinality on the locally stable sets, which provides a positive and complete answer to an open problem raised in arXiv:2202.09034.

DOI: 10.1103/physreva.108.012418

Related papers

Bounds on the smallest sets of quantum states with special quantum nonlocality

Mao-Sheng Li, Yanling Wang

An orthogonal set of states in multipartite systems is called to be strong quantum nonlocality if it is locally irreducible under every bipartition of the subsystems \cite{Halder19}. In this work, we study a subclass of locally irreducible sets: the only possible orthogonality preserving measurement on each subsystems are trivial measurements. We call the set with this property is locally stable. We find that in the case of two qubits systems locally stable sets are coincide with locally indistinguishable sets. Then we present a characterization of locally stable sets via the dimensions of some states depended spaces. Moreover, we construct two orthogonal sets in general multipartite quantum systems which are locally stable under every bipartition of the subsystems. As a consequence, we obtain a lower bound and an upper bound on the size of the smallest set which is locally stable for each bipartition of the subsystems. Our results provide a complete answer to an open question (that is, can we show strong quantum nonlocality in C d 1 ⊗ C d 1 ⊗ ⋯ ⊗ C d N for any d i ≥ 2 and 1 ≤ i ≤ N ?) raised in a recent paper \cite{Shi22N}. Compared with all previous relevant proofs, our proof here is quite concise.An orthogonal set of states in multipartite systems is called to be strong quantum nonlocality if it is locally irreducible under every bipartition of the subsystems \cite{Halder19}. In this work, we study a subclass of locally irreducible sets: the only possible orthogonality preserving measurement on each subsystems are trivial measurements. We call the set with this property is locally stable. We find that in the case of two qubits systems locally stable sets are coincide with locally indistinguishable sets. Then we present a characterization of locally stable sets via the dimensions of some states depended spaces. Moreover, we construct two orthogonal sets in general multipartite quantum systems which are locally stable under every bipartition of the subsystems. As a consequence, we obtain a lower bound and an upper bound on the size of the smallest set which is locally stable for each bipartition of the subsystems. Our results provide a complete answer to an open question (that is, can we show strong quantum nonlocality in C d 1 ⊗ C d 1 ⊗ ⋯ ⊗ C d N for any d i ≥ 2 and 1 ≤ i ≤ N ?) raised in a recent paper \cite{Shi22N}. Compared with all previous relevant proofs, our proof here is quite concise.

arXiv:2202.09034QuantumSep 7, 202321 citations

Locally stable sets without entanglement with minimum cardinality

Xiao-Fan Zhen, Su‐Juan Qin, Zhengping Jin +2 more

Over the past two decades, much progress has been made in the construction of nonlocal sets without entanglement. Recently, Li and Wang [Quantum 7, 1101 (2023)] proposed an open question that an orthogonal product set (OPS) $\mathcal{S}$ is locally stable with minimum cardinality in ${⨂}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$ , and the numerical lower bound for such a set was given by Wang et al. [Quantum Inf. Process. 22, 15 (2023)], $(|\mathcal{S}|\ensuremath{-}1)|\mathcal{S}|\ensuremath{\ge}{\ensuremath{\sum}}_{i=1}^{n}({d}_{i}^{2}\ensuremath{-}1)$, but the existence of the construction remains unresolved. In this work we construct locally stable sets with minimum cardinalities ${d}^{2}$ and ${d}^{2}+1$ via 1-orthogonal cyclic product sets in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}{d}^{2}}$ for odd $d$ and ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}{d}^{2}+1}$ for even $d$ , respectively, thereby partly addressing Li and Wang's open question and confirming the existence of the locally stable OPS with the lower bound. Furthermore, we present a locally stable orthogonal cyclic product set of size $n$ in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}n}$ and a locally stable OPS with smaller size in general multipartite systems. Our results show a perfectly cyclic structure of locally stable OPSs and provide new insight into the role of local stability without entanglement in quantum information theory.Over the past two decades, much progress has been made in the construction of nonlocal sets without entanglement. Recently, Li and Wang [Quantum 7, 1101 (2023)] proposed an open question that an orthogonal product set (OPS) $\mathcal{S}$ is locally stable with minimum cardinality in ${⨂}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$, and the numerical lower bound for such a set was given by Wang et al. [Quantum Inf. Process. 22, 15 (2023)], $(|\mathcal{S}|\ensuremath{-}1)|\mathcal{S}|\ensuremath{\ge}{\ensuremath{\sum}}_{i=1}^{n}({d}_{i}^{2}\ensuremath{-}1)$, but the existence of the construction remains unresolved. In this work we construct locally stable sets with minimum cardinalities ${d}^{2}$ and ${d}^{2}+1$ via 1-orthogonal cyclic product sets in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}{d}^{2}}$ for odd $d$ and ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}{d}^{2}+1}$ for even $d$, respectively, thereby partly addressing Li and Wang's open question and confirming the existence of the locally stable OPS with the lower bound. Furthermore, we present a locally stable orthogonal cyclic product set of size $n$ in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}n}$ and a locally stable OPS with smaller size in general multipartite systems. Our results show a perfectly cyclic structure of locally stable OPSs and provide new insight into the role of local stability without entanglement in quantum information theory.

Physical Review AMay 19, 20253 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.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:2408.02894Aug 6, 20241 citation

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.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:2405.15298Physical Review AMay 13, 202410 citations

Nonlocality without entanglement in general multipartite quantum systems

Xiao-Fan Zhen, Shao-Ming Fei, Hui-Juan Zuo

The construction of nonlocal sets of quantum states has attracted much attention in recent years. We first introduce two lemmas related to the triviality of orthogonality-preserving local measurements. Then, we propose a general construction of nonlocal set of $n(d\ensuremath{-}1)+1$ orthogonal product states in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}n}$. The sets of nonlocal orthogonal product states are also put forward for multipartite quantum systems with arbitrary dimensions. Our construction gives rise to nonlocal sets of orthogonal product states with much less members and thus reveals the phenomenon of nonlocality without entanglement more efficiently.The construction of nonlocal sets of quantum states has attracted much attention in recent years. We first introduce two lemmas related to the triviality of orthogonality-preserving local measurements. Then, we propose a general construction of nonlocal set of $n(d\ensuremath{-}1)+1$ orthogonal product states in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}n}$. The sets of nonlocal orthogonal product states are also put forward for multipartite quantum systems with arbitrary dimensions. Our construction gives rise to nonlocal sets of orthogonal product states with much less members and thus reveals the phenomenon of nonlocality without entanglement more efficiently.

arXiv:2212.14523Physical Review ADec 23, 202228 citations

Genuinely nonlocal sets without entanglement in multipartite systems

Ying-Ying Lu, Hai-Qing Cao, Hui-Juan Zuo +1 more

A set of multipartite orthogonal states is genuinely nonlocal if it is locally indistinguishable in every bipartition of the subsystems. If the set is locally reducible, we say it has genuine nonlocality of type I. Otherwise, we say it has genuine nonlocality of type II. Due to the complexity of the problem, the construction of genuinely nonlocal sets in general multipartite systems has not been completely solved so far. In this paper, we first provide a nonlocal set of product states in bipartite systems. We obtain a genuinely nonlocal set of type I without entanglement in general $n$ -partite systems ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$ $[3\ensuremath{\le}({d}_{1}\ensuremath{-}1)\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n},\phantom{\rule{4pt}{0ex}}n\ensuremath{\ge}3]$. Then we present two constructions with genuine nonlocality of type II in ${\mathbb{C}}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{2}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{3}}$ $(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}{d}_{3})$ and ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$ $(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n},\phantom{\rule{4pt}{0ex}}n\ensuremath{\ge}4)$. Our results further positively answer the open problem that there does exist a genuinely nonlocal set of type II in multipartite systems [M. S. Li, Y. L. Wang, F. Shi, and M. H. Yung, J. Phys. A: Math. Theor. 54, 445301 (2021)] and highlight its related applications in quantum information processing.A set of multipartite orthogonal states is genuinely nonlocal if it is locally indistinguishable in every bipartition of the subsystems. If the set is locally reducible, we say it has genuine nonlocality of type I. Otherwise, we say it has genuine nonlocality of type II. Due to the complexity of the problem, the construction of genuinely nonlocal sets in general multipartite systems has not been completely solved so far. In this paper, we first provide a nonlocal set of product states in bipartite systems. We obtain a genuinely nonlocal set of type I without entanglement in general $n$-partite systems ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$ $[3\ensuremath{\le}({d}_{1}\ensuremath{-}1)\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n},\phantom{\rule{4pt}{0ex}}n\ensuremath{\ge}3]$. Then we present two constructions with genuine nonlocality of type II in ${\mathbb{C}}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{2}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{3}}$ $(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}{d}_{3})$ and ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$ $(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n},\phantom{\rule{4pt}{0ex}}n\ensuremath{\ge}4)$. Our results further positively answer the open problem that there does exist a genuinely nonlocal set of type II in multipartite systems [M. S. Li, Y. L. Wang, F. Shi, and M. H. Yung, J. Phys. A: Math. Theor. 54, 445301 (2021)] and highlight its related applications in quantum information processing.

arXiv:2408.11566Physical Review AAug 16, 20247 citations

{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\phantom{\rule{4pt}{0ex}}(d\ensuremath{\ge}3)

and

{\mathbb{C}}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{2}}\phantom{\rule{4pt}{0ex}}(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2})

. Moreover, for the multipartite quantum systems

{({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}n}\phantom{\rule{4pt}{0ex}}(d\ensuremath{\ge}2)

and

{\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}\phantom{\rule{4pt}{0ex}}(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n})

, we also obtain

d+1

and

{d}_{n}+1

\mathcal{S}

is locally stable with minimum cardinality in

{⨂}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}

, and the numerical lower bound for such a set was given by Wang et al. [Quantum Inf. Process. 22, 15 (2023)], $(|\mathcal{S}|\ensuremath{-}1)|\mathcal{S}|\ensuremath{\ge}{\ensuremath{\sum}}_{i=1}^{n}({d}_{i}^{2}\ensuremath{-}1)$, but the existence of the construction remains unresolved. In this work we construct locally stable sets with minimum cardinalities

{d}^{2}

and

{d}^{2}+1

via 1-orthogonal cyclic product sets in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}{d}^{2}}$ for odd

d

and ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}{d}^{2}+1}$ for even

d

, respectively, thereby partly addressing Li and Wang's open question and confirming the existence of the locally stable OPS with the lower bound. Furthermore, we present a locally stable orthogonal cyclic product set of size

n

in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}n}$ and a locally stable OPS with smaller size in general multipartite systems. Our results show a perfectly cyclic structure of locally stable OPSs and provide new insight into the role of local stability without entanglement in quantum information theory.

\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.

\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.

n

-partite systems ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$ $[3\ensuremath{\le}({d}_{1}\ensuremath{-}1)\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n},\phantom{\rule{4pt}{0ex}}n\ensuremath{\ge}3]$. Then we present two constructions with genuine nonlocality of type II in ${\mathbb{C}}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{2}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{3}}$ $(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}{d}_{3})$ and ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}$ $(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n},\phantom{\rule{4pt}{0ex}}n\ensuremath{\ge}4)$. Our results further positively answer the open problem that there does exist a genuinely nonlocal set of type II in multipartite systems [M. S. Li, Y. L. Wang, F. Shi, and M. H. Yung, J. Phys. A: Math. Theor. 54, 445301 (2021)] and highlight its related applications in quantum information processing.