Local Distinguishability of Multipartite Orthogonal Quantum States: Generalized and Simplified

TL;DR

This paper extends the fundamental result on local distinguishability of orthogonal quantum states to infinite dimensions, providing a simplified proof, an efficient algorithm, and clarifying the relationship with quantum channel capacities.

Contribution

It generalizes the distinguishability result to infinite dimensions, offers a polynomial-time algorithm for constructing LOCC protocols, and links state distinguishability to quantum channel capacities.

Findings

Extended distinguishability proof to infinite dimensions

Developed an $O(d_A^2 d_B^2)$-time algorithm for LOCC protocols

Established equivalence between state distinguishability and quantum channel capacity

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 $\mathbb{C}^{d_A \times d_A} \otimes \mathbb{C}^{d_B \times d_B}$, we strengthen this existence result by constructing an $O(d_A^2 d_B^2)$-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.