Abstract
In a seminal work [PRL85.4972], Walgate, Short, Hardy, and Vedral prove in finite dimensions that for every pair of pure multipartite orthogonal quantum states, there exists a one-way local operations and classical communication (LOCC) protocol that perfectly distinguishes the pair. We extend this result to infinite dimensions with a simpler proof. For states on C^d_A × d_A ⊗ C^d_B × d_B, we strengthen this existence result by constructing an O(d_A² d_B²)-time algorithm that specifies such a perfect one-way LOCC protocol. Finally, we establish the equivalence between Walgate et al.'s result and the fact that the one-shot environment-assisted classical capacity of every quantum channel is at least 1 bit per channel use, thereby clarifying the literature on these notions. At the core of all of these results is the fact that every operator with vanishing trace admits a basis where its diagonal entries are all zero.