A set of orthogonal multipartite quantum states are called (distinguishability-based) genuinely nonlocal if they are locally indistinguishable across any bipartition of the subsystems. In this work, we consider the problem of constructing small genuinely nonlocal sets consisting of generalized Greenberger-Horne-Zeilinger (GHZ) states in multipartite systems. For system ${({\mathbb{C}}^{2})}^{\ensuremath{\bigotimes}N}$ where $N$ is large, using the language of group theory, we show that a tiny proportion $\mathrm{\ensuremath{\Theta}}(1/\sqrt{{2}^{N}})$ of the states among the $N$ -qubit GHZ basis suffice to exhibit genuine nonlocality. Similar arguments also hold for the canonical generalized GHZ bases in systems ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$, wherever $d$ is even and $N$ is large. What is more, moving to the condition that any fixed $N$ is given, we show that $d+1$ genuinely nonlocal generalized GHZ states exist in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$, provided the local dimension $d$ is sufficiently large. As an additional merit, within and beyond an asymptotic sense, the latter result also indicates some evident limitations of the ``trivial othogonality-preserving local measurements'' (TOPLM) technique that has been utilized frequently for detecting genuine nonlocality.A set of orthogonal multipartite quantum states are called (distinguishability-based) genuinely nonlocal if they are locally indistinguishable across any bipartition of the subsystems. In this work, we consider the problem of constructing small genuinely nonlocal sets consisting of generalized Greenberger-Horne-Zeilinger (GHZ) states in multipartite systems. For system ${({\mathbb{C}}^{2})}^{\ensuremath{\bigotimes}N}$ where $N$ is large, using the language of group theory, we show that a tiny proportion $\mathrm{\ensuremath{\Theta}}(1/\sqrt{{2}^{N}})$ of the states among the $N$-qubit GHZ basis suffice to exhibit genuine nonlocality. Similar arguments also hold for the canonical generalized GHZ bases in systems ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$, wherever $d$ is even and $N$ is large. What is more, moving to the condition that any fixed $N$ is given, we show that $d+1$ genuinely nonlocal generalized GHZ states exist in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$, provided the local dimension $d$ is sufficiently large. As an additional merit, within and beyond an asymptotic sense, the latter result also indicates some evident limitations of the ``trivial othogonality-preserving local measurements'' (TOPLM) technique that has been utilized frequently for detecting genuine nonlocality.

Physical Review AFeb 21, 20249 citations

Distinguishability-based genuine nonlocality with genuine multipartite entanglement

Zong-Xing Xiong, Mao-Sheng Li, Zhu‐Jun Zheng +1 more

A set of orthogonal multipartite quantum states is said to be distinguishability-based genuinely nonlocal (also genuinely nonlocal, for abbreviation) if the states are locally indistinguishable across any bipartition of the subsystems. This form of multipartite nonlocality, although more naturally arising than the recently popular ``strong nonlocality'' in the context of local distinguishability, receives much less attention. In this work, we study the distinguishability-based genuine nonlocality of a typical type of genuine multipartite entangled states---the $d$ -dimensional Greenberger-Horne-Zeilinger (GHZ) states, featuring systems with local dimension not limited to two. In the three-partite case, we find the existence of small genuinely nonlocal sets consisting of these states: we show that the cardinality can at least scale down to linear in the local dimension $d$ , with the linear factor $l=1$ . Specifically, the method we use is semidefinite programming and the GHZ states to construct these sets are special ones which we call ``GHZ lattices''. This result might arguably suggest a significant gap between the strength of strong nonlocality and the distinguishability-based genuine nonlocality. Moreover, we put forward the notion of $(s,n)$ -threshold distinguishability and, utilizing a similar method, we successfully construct (2,3)-threshold sets consisting of GHZ states in three-partite systems.A set of orthogonal multipartite quantum states is said to be distinguishability-based genuinely nonlocal (also genuinely nonlocal, for abbreviation) if the states are locally indistinguishable across any bipartition of the subsystems. This form of multipartite nonlocality, although more naturally arising than the recently popular ``strong nonlocality'' in the context of local distinguishability, receives much less attention. In this work, we study the distinguishability-based genuine nonlocality of a typical type of genuine multipartite entangled states---the $d$-dimensional Greenberger-Horne-Zeilinger (GHZ) states, featuring systems with local dimension not limited to two. In the three-partite case, we find the existence of small genuinely nonlocal sets consisting of these states: we show that the cardinality can at least scale down to linear in the local dimension $d$, with the linear factor $l=1$. Specifically, the method we use is semidefinite programming and the GHZ states to construct these sets are special ones which we call ``GHZ lattices''. This result might arguably suggest a significant gap between the strength of strong nonlocality and the distinguishability-based genuine nonlocality. Moreover, we put forward the notion of $(s,n)$-threshold distinguishability and, utilizing a similar method, we successfully construct (2,3)-threshold sets consisting of GHZ states in three-partite systems.

arXiv:2211.02388Physical Review AAug 8, 202317 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

Local distinguishability based genuinely quantum nonlocality without entanglement

Mao-Sheng Li, Yan-Ling Wang, Fei Shi +1 more

Abstract Recently, Halder et al (2019 Phys. Rev. Lett. 122 040403) proposed the concept of strong nonlocality without entanglement: an orthogonal set of fully product states in multipartite quantum systems that is locally irreducible for every bipartition of its subsystems. Due to the complexity of the problem, most results are limited to tripartite systems. Here we consider a weaker form of nonlocality which is called local distinguishability based genuine nonlocality. A set of orthogonal multipartite quantum states is said to be genuinely nonlocal if it is locally indistinguishable for every bipartition of the subsystems. In this work, we study how to construct sets of orthogonal product states which are genuinely nonlocal. Firstly, we present a set of product states with simple structure in bipartite systems that is locally indistinguishable. After that, based on a simple observation, we present a general method to construct genuinely nonlocal sets of multipartite product states by using those sets that are genuinely nonlocal but with less parties. As a consequence, we obtain that genuinely nonlocal sets of fully product states exist in any L parties systems ⨂ i = 1 L C d i provided L ⩾ 3 and d i ⩾ 3 for all i .Abstract Recently, Halder et al (2019 Phys. Rev. Lett. 122 040403) proposed the concept of strong nonlocality without entanglement: an orthogonal set of fully product states in multipartite quantum systems that is locally irreducible for every bipartition of its subsystems. Due to the complexity of the problem, most results are limited to tripartite systems. Here we consider a weaker form of nonlocality which is called local distinguishability based genuine nonlocality. A set of orthogonal multipartite quantum states is said to be genuinely nonlocal if it is locally indistinguishable for every bipartition of the subsystems. In this work, we study how to construct sets of orthogonal product states which are genuinely nonlocal. Firstly, we present a set of product states with simple structure in bipartite systems that is locally indistinguishable. After that, based on a simple observation, we present a general method to construct genuinely nonlocal sets of multipartite product states by using those sets that are genuinely nonlocal but with less parties. As a consequence, we obtain that genuinely nonlocal sets of fully product states exist in any L parties systems ⨂ i = 1 L C d i provided L ⩾ 3 and d i ⩾ 3 for all i .

arXiv:2011.10286Journal of Physics A Mathematical and TheoreticalSep 21, 202126 citations

Activate genuine nonlocality from distinguishable sets in tripartite systems

Hui-Juan Zuo, Ying-Ying Lu, Shao-Ming Fei

A set of orthogonal quantum states in multipartite systems is of genuine nonlocality if it is locally indistinguishable in every bipartition. If it is locally reducible when the parties are separated, we say that it has genuine nonlocality of type~\uppercase\expandafter{\romannumeral 1}; otherwise, it has genuine nonlocality of type~\uppercase\expandafter{\romannumeral 2}. For a locally distinguishable set without local redundancy, if there exist some orthogonality preserving local measurements such that each outcome leads to a locally indistinguishable set, then we say that it exhibits the activation of nonlocality. We activate type-\uppercase\expandafter{\romannumeral 1} and type-\uppercase\expandafter{\romannumeral 2} genuine nonlocality of orthogonal product state sets in tripartite systems. In particular, we tackle the local irredundancy problem with partial trace operation and $p$ -ary numeral systems to significantly simplify the proofs. Our results also address the open question raised by S. Bandyopadhyay \textit{et al.}[\href{https://link.aps.org/doi/10.1103/PhysRevA.104.L050201}{Phys. Rev. A \textbf{104}, L050201 (2021)}]. Furthermore, we observe the activation of hidden genuine nonlocality in multipartite systems, which highlights the applications of nonlocality based on state discrimination in different practical scenarios.A set of orthogonal quantum states in multipartite systems is of genuine nonlocality if it is locally indistinguishable in every bipartition. If it is locally reducible when the parties are separated, we say that it has genuine nonlocality of type~\uppercase\expandafter{\romannumeral 1}; otherwise, it has genuine nonlocality of type~\uppercase\expandafter{\romannumeral 2}. For a locally distinguishable set without local redundancy, if there exist some orthogonality preserving local measurements such that each outcome leads to a locally indistinguishable set, then we say that it exhibits the activation of nonlocality. We activate type-\uppercase\expandafter{\romannumeral 1} and type-\uppercase\expandafter{\romannumeral 2} genuine nonlocality of orthogonal product state sets in tripartite systems. In particular, we tackle the local irredundancy problem with partial trace operation and $p$-ary numeral systems to significantly simplify the proofs. Our results also address the open question raised by S. Bandyopadhyay \textit{et al.}[\href{https://link.aps.org/doi/10.1103/PhysRevA.104.L050201}{Phys. Rev. A \textbf{104}, L050201 (2021)}]. Furthermore, we observe the activation of hidden genuine nonlocality in multipartite systems, which highlights the applications of nonlocality based on state discrimination in different practical scenarios.

quant-pharXiv:2608.20741Advanced Quantum TechnologiesAug 24, 20260 citations

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

N

is large, using the language of group theory, we show that a tiny proportion $\mathrm{\ensuremath{\Theta}}(1/\sqrt{{2}^{N}})$ of the states among the

N

-qubit GHZ basis suffice to exhibit genuine nonlocality. Similar arguments also hold for the canonical generalized GHZ bases in systems ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$, wherever

d

is even and

N

is large. What is more, moving to the condition that any fixed

N

is given, we show that

d+1

genuinely nonlocal generalized GHZ states exist in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$, provided the local dimension

d

is sufficiently large. As an additional merit, within and beyond an asymptotic sense, the latter result also indicates some evident limitations of the ``trivial othogonality-preserving local measurements'' (TOPLM) technique that has been utilized frequently for detecting genuine nonlocality.

d

-dimensional Greenberger-Horne-Zeilinger (GHZ) states, featuring systems with local dimension not limited to two. In the three-partite case, we find the existence of small genuinely nonlocal sets consisting of these states: we show that the cardinality can at least scale down to linear in the local dimension

d

, with the linear factor

l=1

. Specifically, the method we use is semidefinite programming and the GHZ states to construct these sets are special ones which we call ``GHZ lattices''. This result might arguably suggest a significant gap between the strength of strong nonlocality and the distinguishability-based genuine nonlocality. Moreover, we put forward the notion of

(s,n)

-threshold distinguishability and, utilizing a similar method, we successfully construct (2,3)-threshold sets consisting of GHZ states in three-partite systems.

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.

p

-ary numeral systems to significantly simplify the proofs. Our results also address the open question raised by S. Bandyopadhyay \textit{et al.}[\href{https://link.aps.org/doi/10.1103/PhysRevA.104.L050201}{Phys. Rev. A \textbf{104}, L050201 (2021)}]. Furthermore, we observe the activation of hidden genuine nonlocality in multipartite systems, which highlights the applications of nonlocality based on state discrimination in different practical scenarios.