Finding out all locally indistinguishable sets of generalized Bell states

Jiang-Tao Yuan1,2, Ying-Hui Yang2, and Cai-Hong Wang1,2

1College of Science, Wuxi University, Wuxi, 214105, China
2School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454000, China

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In general, for a bipartite quantum system $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ and an integer $k$ such that $4\leq k\le d$,there are few necessary and sufficient conditions for local discrimination of sets of $k$ generalized Bell states (GBSs) and it is difficult to locally distinguish $k$-GBS sets.The purpose of this paper is to completely solve the problem of local discrimination of GBS sets in some bipartite quantum systems.Firstly three practical and effective sufficient conditions are given,Fan$^{,}$s and Wang et al.$^{,}$s results [Phys Rev Lett 92, 177905 (2004); Phys Rev A 99, 022307 (2019)] can be deduced as special cases of these conditions.Secondly in $\mathbb{C}^{4}\otimes\mathbb{C}^{4}$, a necessary and sufficient condition for local discrimination of GBS sets is provided, and a list of all locally indistinguishable 4-GBS sets is provided,and then the problem of local discrimination of GBS sets is completely
$\mathbb{C}^{5}\otimes\mathbb{C}^{5}$, a concise necessary and sufficient condition for one-way local discrimination of GBS sets is obtained,which gives an affirmative answer to the case $d=5$ of the problem proposed by Wang et al.

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Cited by

[1] Xing-Chen Guo and Mao-Sheng Li, "Application of Ramsey theory to localization of set of product states via multicopies", The European Physical Journal Plus 138 1, 50 (2023).

[2] Mao-Sheng Li, Fei Shi, and Yan-Ling Wang, "Local discrimination of generalized Bell states via commutativity", Physical Review A 105 3, 032455 (2022).

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