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

Mao-Sheng Li; Yan-Ling Wang

doi:10.22331/q-2023-09-07-1101

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

Published:	2023-09-07, volume 7, page 1101
Eprint:	arXiv:2202.09034v4
Doi:	https://doi.org/10.22331/q-2023-09-07-1101
Citation:	Quantum 7, 1101 (2023).

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Abstract

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.

► BibTeX data

@article{Li2023boundssmallestsets,
  doi = {10.22331/q-2023-09-07-1101},
  url = {https://doi.org/10.22331/q-2023-09-07-1101},
  title = {Bounds on the smallest sets of quantum states with special quantum nonlocality},
  author = {Li, Mao-Sheng and Wang, Yan-Ling},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1101},
  month = sep,
  year = {2023}
}

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