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.

中文翻译：

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找出所有局部不可区分的广义贝尔状态集

一般来说，对于一个二分量子系统 $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ 和一个整数 $k$ 使得 $4\leq k\le d$，有几个$k$广义贝尔状态（GBS）集合的局部判别的充要条件，$k$-GBS集合的局部判别困难。本文的目的是彻底解决GBS集合的局部判别问题在一些二分量子系统中。首先给出三个实际有效的充分条件，Fan$^{,}$s 和 Wang et al.$^{,}$s 结果 [Phys Rev Lett 92, 177905 (2004); Phys Rev A 99, 022307 (2019)]可以推导出为这些条件的特例。其次在$\mathbb{C}^{4}\otimes\mathbb{C}^{4}$中，一个充要条件提供用于 GBS 集的本地区分，
$\mathbb{C}^{5}\otimes\mathbb{C}^{5}$，得到了GBS集合单向局部判别的简明充要条件，对$\otimes\mathbb{C}^{5}$的情况给出了肯定的回答Wang 等人提出的问题的 d=5$。