Necessary and sufficient conditions for local discrimination of generalized Bell states: finding out all locally indistinguishable sets of generalized Bell states

In general, for a bipartite quantum system $mathbb{C}^{d}otimesmathbb{C}^{d}$ and an integer $k$ such that $4leq kle d$,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.In this paper, we consider the local discrimination of GBS sets and the purpose is to completely solve the problem of local discrimination of GBS sets in some bipartite quantum systems,specifically, we show some necessary and sufficient conditions for local discrimination of GBS sets by which the local discrimination of GBS sets can be quickly determined.Firstly some sufficient conditions are given, these sufficient conditions are practical and effective.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 $mathbb{C}^{4}otimesmathbb{C}^{4}$, a necessary and sufficient condition for local discrimination of GBS sets is provided,all locally indistinguishable 4-GBS sets are found,and then we can quickly determine the local discriminability of an arbitrary GBS set.In $mathbb{C}^{5}otimesmathbb{C}^{5}$, 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. (Phys Rev A 99:022307, 2019).