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

Mao-Sheng Li1 and Yan-Ling Wang2

1School of Mathematics, South China University of Technology, Guangzhou 510641, China
2School of Computer Science and Technology, Dongguan University of Technology, Dongguan, 523808, China

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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 [46]. 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 $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_1}\otimes \cdots \otimes \mathbb{C}^{d_N} $ for any $d_i \geq 2$ and $1\leq i\leq N$?) raised in a recent paper [54]. Compared with all previous relevant proofs, our proof here is quite concise.

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Cited by

[1] Mengying Hu, Ting Gao, and Fengli Yan, "Strong quantum nonlocality with genuine entanglement in an N -qutrit system", Physical Review A 109 2, 022220 (2024).

[2] Jicun Li, Fei Shi, and Xiande Zhang, "Strongest nonlocal sets with small sizes", Physical Review A 108 6, 062407 (2023).

[3] Zong-Xing Xiong, Yongli Zhang, Mao-Sheng Li, and Lvzhou Li, "Small sets of genuinely nonlocal Greenberger-Horne-Zeilinger states in multipartite systems", Physical Review A 109 2, 022428 (2024).

[4] Yiyun He, Fei Shi, and Xiande Zhang, "Strong quantum nonlocality and unextendibility without entanglement in N-partite systems with odd N", Quantum 8, 1349 (2024).

[5] Xiao-Fan Zhen, Mao-Sheng Li, and Hui-Juan Zuo, "Strongest nonlocal sets with minimum cardinality in tripartite systems", Physical Review A 109 5, 052422 (2024).

[6] Huaqi Zhou, Ting Gao, and Fengli Yan, "Orthogonal product sets with strong quantum nonlocality on a plane structure", Physical Review A 106 5, 052209 (2022).

[7] Zong-Xing Xiong, Mao-Sheng Li, Zhu-Jun Zheng, and Lvzhou Li, "Distinguishability-based genuine nonlocality with genuine multipartite entanglement", Physical Review A 108 2, 022405 (2023).

[8] Hai-Qing Cao and Hui-Juan Zuo, "Locally distinguishing nonlocal sets with entanglement resource", Physica A Statistical Mechanics and its Applications 623, 128852 (2023).

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