Novel method for one-way local distinguishability of generalized Bell states in arbitrary dimension

In this paper the local distinguishability of generalized Bell states in arbitrary dimension is investigated. We firstly study the decomposition of a basis which consists of $d^{2}$ number of generalized Pauli matrices. We discover that this basis is equal to the union of $D$ number of different sets, where $D=\frac{2}{\phi(d)}\sum_{t\in \mathbb{Z}_{d} \atop gcd(t,d)=1}\sum_{i=2}^{\lfloor\frac{d}{t}\rfloor}\phi(i)+1$ and $\phi$ is Euler $\phi$-function. Then we define the generator of the matrices in this decomposition, and exhibit an algorithm to calculate generators of a given set of matrices. This algorithm shows that generators of a given set can be calculated simply and efficiently. Secondly, we show that a set $\mathcal {L}$ of GBSs can be distinguished by one-way LOCC if the cardinality of $\mathcal {G}_{\mathcal {L}}$ is less than $D\phi(d)$, where $\mathcal {G}_{\mathcal {L}}$ is a set of generators of all the elements in difference set of a set $\mathcal {L}$ of GBSs. The previous results in [2004 Phys. Rev. Lett. \textbf{92} 177905; 2019 Phys. Rev. A \textbf{99} 022307; 2021 Quant. Info. Proc. \textbf{20} 52] can be covered by our result. Finally, for the uncovered cases in [2021 Quant. Info. Proc. \textbf{20} 52], we give a new result to partly solve that problem.