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arXiv:2405.15298 · Physical Review A

Strongest nonlocal sets with minimum cardinality in tripartite systems

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

Published May 13, 2024• 10 citations
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Abstract

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.

DOI
10.1103/physreva.109.052422

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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⨂i=1nCdi{\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}\ensuremath⨂i=1n​Cdi​, i.e., ∣S∣\ensuremath≤maxi\ensuremath∏j=1ndj/di+1|\mathcal{S}|\ensuremath{\le}{max}_{i}{{\ensuremath{\prod}}_{j=1}^{n}{d}_{j}/{d}_{i}+1}∣S∣\ensuremath≤maxi​\ensuremath∏j=1n​dj​/di​+1. In this work, we construct the strongest nonlocal set of size d2+1{d}^{2}+1d2+1 in Cd\ensuremath⨂Cd\ensuremath⨂Cd{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}Cd\ensuremath⨂Cd\ensuremath⨂Cd. Furthermore, we obtain the strongest nonlocal set of size d2d3+1{d}_{2}{d}_{3}+1d2​d3​+1 in Cd1\ensuremath⨂Cd2\ensuremath⨂Cd3{\mathbb{C}}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{2}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{3}}Cd1​\ensuremath⨂Cd2​\ensuremath⨂Cd3​. 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.
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 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 ${d}_{2}\ensuremath{\ge}3$.
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+1d+1d+1 and dn+1{d}_{n}+1dn​+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.
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 ${({\mathbb{C}}^{3})}^{\ensuremath{\bigotimes}N}$. 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 NNN increases. Our results provide a meaningful complement to the current understanding of the strongest nonlocality.
In this paper, we generalize the concept of strong quantum nonlocality from two aspects. First, in a tripartite quantum system, we present a construction of strongly nonlocal quantum states containing $6{(d\ensuremath{-}1)}^{2}$ orthogonal product states in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ and build $6{d}^{2}\ensuremath{-}8d+4$ product states in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d+1}$, which have been proved to be strongly nonlocal. Obviously, both results turn out to be one order of magnitude smaller than the number of basis states d3{d}^{3}d3 for $d\ensuremath{\ge}3$. Second, we give explicit forms of strongly nonlocal orthogonal product basis in ${\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$ and ${\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}$ quantum systems, where four is the largest known number of subsystems in which there exists strong quantum nonlocality without entanglement up to now. All the results positively answer the open problems raised by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)]; that is, there do exist a small number of quantum states that can demonstrate strong quantum nonlocality without entanglement.
Recently, based on some quantum states which are locally irreducible in all bipartitions, strong quantum nonlocality has been presented by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)]. However, the remaining questions are whether a set of quantum states can reveal strong quantum nonlocality when these states are locally irreducible in all tripartitions or more multipartitions and how to construct strong nonlocality of orthogonal product states in more than tripartite quantum systems. Here we present a general definition of strong quantum nonlocality based on the local irreducibility. Then, using a $3\ensuremath{\bigotimes}3\ensuremath{\bigotimes}3$ quantum system as an example, we show three sets of locally indistinguishable orthogonal product states to explain the general definition of strong quantum nonlocality. Furthermore, in $3\ensuremath{\bigotimes}3\ensuremath{\bigotimes}3\ensuremath{\bigotimes}3$, we construct a set of strong nonlocality of orthogonal product states; these states do not form a complete basis. Our results demonstrate the phenomenon of strong nonlocality without entanglement for the multipartite quantum systems.