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Physical Review A

Locally stable sets without entanglement with minimum cardinality

Xiao-Fan Zhen, Su‐Juan Qin, Zhengping Jin, Hui-Juan Zuo, Fei Gao

Published May 19, 2025• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.111.052434

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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) S\mathcal{S}S is locally stable with minimum cardinality in ⨂i=1nCdi{⨂}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}⨂i=1n​Cdi​, and the numerical lower bound for such a set was given by Wang et al. [Quantum Inf. Process. 22, 15 (2023)], (∣S∣\ensuremath−1)∣S∣\ensuremath≥\ensuremath∑i=1n(di2\ensuremath−1)(|\mathcal{S}|\ensuremath{-}1)|\mathcal{S}|\ensuremath{\ge}{\ensuremath{\sum}}_{i=1}^{n}({d}_{i}^{2}\ensuremath{-}1)(∣S∣\ensuremath−1)∣S∣\ensuremath≥\ensuremath∑i=1n​(di2​\ensuremath−1), but the existence of the construction remains unresolved. In this work we construct locally stable sets with minimum cardinalities d2{d}^{2}d2 and d2+1{d}^{2}+1d2+1 via 1-orthogonal cyclic product sets in (Cd)\ensuremath⨂d2{({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}{d}^{2}}(Cd)\ensuremath⨂d2 for odd ddd and (Cd)\ensuremath⨂d2+1{({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}{d}^{2}+1}(Cd)\ensuremath⨂d2+1 for even ddd, 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 nnn in (Cd)\ensuremath⨂n{({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}n}(Cd)\ensuremath⨂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.
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.
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.
In quantum information theory, the state space of a multipartite quantum system is modeled by a tensor product. In the tensor-product space $\C^{d_1}\otimes\cdots\otimes\C^{d_p}$, a nonzero vector is a \emph{product state} if it can be written as ∣φ1⟩⊗⋯⊗∣φp⟩\ket{\varphi_1}\otimes\cdots\otimes\ket{\varphi_p}∣φ1​⟩⊗⋯⊗∣φp​⟩ with $\ket{\varphi_j}\in\C^{d_j}\setminus\{0\}$. An \emph{unextendible product basis} (UPB) is a finite family of pairwise orthogonal product states such that no nonzero product state is orthogonal to all of them. UPBs play a key role in investigating quantum entanglement and nonlocal phenomena. Finding a smallest UPB is a natural extremal problem: it asks how few pairwise orthogonal product states suffice to prevent any further product state from being added. The general minimum-size problem for UPBs has been studied for over two decades since the seminal work of Alon and Lovász. Let fm(d1,…,dp)f_m(d_1,\ldots,d_p)fm​(d1​,…,dp​) be the minimum cardinality of a UPB in the space, and let fLB(d1,…,dp)=1+∑j=1p(dj−1)f_{LB}(d_1,\ldots,d_p)=1+\sum_{j=1}^{p}(d_j-1)fLB​(d1​,…,dp​)=1+∑j=1p​(dj​−1) be the natural lower bound. Alon and Lovász determined exactly when fmf_mfm​ attains the lower bound fLBf_{LB}fLB​, but the obstructed multipartite cases remained open in general. We prove a stabilization theorem: for every non-all-qubit system with p≥3p\ge3p≥3, whenever the natural lower bound fLBf_{LB}fLB​ is unattainable, the true minimum is exactly fLB+1f_{LB}+1fLB​+1. Precisely, if the number of even local dimensions did_idi​ is positive and even, and at least one local dimension is greater than two, then fm(d1,…,dp)=fLB(d1,…,dp)+1f_m(d_1,\ldots,d_p)=f_{LB}(d_1,\ldots,d_p)+1fm​(d1​,…,dp​)=fLB​(d1​,…,dp​)+1. The proof is built on a unified graph-theoretic framework. Our result, together with earlier work, settles the minimum size problem for UPBs in all finite quantum systems.
Quantum nonlocality, rooted in the quantum state discrimination task, contributes to the development of the theory of local distinguishability of quantum states. Recently, Zhu et al. [Phys. A (Amsterdam, Neth.) 624, 128956 (2023)] found that a nonlocal set of orthogonal product states (OPSs) can be perfectly identified by local operations and classical communication after removing a specific quantum state in a bipartite quantum system. They called this phenomenon minimal nonlocality. It is obvious that minimal nonlocality effectively reflects the lower boundary of the elements contained in a nonlocal set of OPSs. In this paper, we generalize the methods of constructing nonlocal sets with minimal nonlocality from bipartite systems to multipartite systems and from orthogonal product states to entangled states. First, we give a method to construct a set of OPSs with minimal nonlocality in the ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ quantum system, and then we extend it to the ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{d}$ quantum system for $d\ensuremath{\ge}4$ and $n\ensuremath{\ge}3$. In addition, we give a method to construct a nonlocal set of orthogonal states with entanglement relations in the ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ quantum system, and we prove that this set has minimal nonlocality, which can also be extended to the ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{d}$ quantum system.
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 nnn-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 product states is locally irreducible, if it is not possible to eliminate one or more states from the set by orthogonality-preserving local measurements. An effective way to prove that a set is locally irreducible is to show that only trivial orthogonality-preserving measurement can be performed to this set. In general, it is difficult to show such a orthogonality-preserving measurement must be trivial. In this work, we develop two basic techniques to deal with this problem. Using these techniques, we successfully construct orthogonal product sets (OPSs) in any possible three, four and five-partite systems that are locally irreducible in every bipartition. Moreover, we also show the existence of unextendible product bases (UPBs) with such properties in d⊗d⊗dd\otimes d\otimes dd⊗d⊗d for any d≥3d\geq 3d≥3. Therefore, these OPSs and UPBs exhibit the phenomenon of strong quantum nonlocality without entanglement. Both results solve the open questions given by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)] and Yuan et al. [Phys. Rev. A 102, 042228 (2020)]. Our results shed a new light on the strong quantum nonlocality without entanglement.