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

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