Simple criterion for local distinguishability of generalized Bell states in prime dimension

Local distinguishability of sets generalized Bell states (GBSs) is investigated. We first clarify the conditions such that a set GBSs can be locally transformed to certain type GBS easily distinguishable within local operations and one-way classical communication. then show that, if space dimension $d$ prime, these are necessary sufficient for in $\mathbb{C}^d \otimes \mathbb{C}^d$ distinguishable. Thus we obtain simple computable criterion prime $d$.