Local Distinguishability of Multipartite Orthogonal Quantum States: Generalized and Simplified

TL;DR

This work generalizes the Walgate et al. result by proving that any pair of orthogonal bipartite pure states can be distinguished with one-way LOCC in infinite dimensions and providing a constructive finite-dimension protocol with time complexity $O(d_A^2 d_B^2)$. It introduces a concrete method to find the LOCC protocol by reducing to a unitary flattening problem for the overlap matrix $M_{|\phi\rangle}^* M_{|\psi\rangle}$ using vec mappings, and it delivers an explicit algorithm that outputs the necessary measurements. Additionally, the paper establishes an equivalence between one-way LOCC discriminability and one-shot environment-assisted classical capacity, clarifying how LOCC discrimination underpins capacity results for quantum channels. Collectively, the results offer both a practical, efficient construction of LOCC discrimination protocols in finite dimensions and a unifying perspective linking state discrimination with environment-assisted communication theory.

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.

Figures (2)

• Figure 1: Visualization of Algorithm \ref{['alg:computing-unitary']} for $d$ such that $\lceil \log_2 (d) \rceil=3$. Black lines depict the pairs of computational basis states used to define the diagonal entries being flattened at iterate $p$. Blocks of colour denote diagonal elements of $M$ with the same value after iterate $p$. Because at each iterate pairs of constant value blocks are combined, the number of distinct diagonal values is cut in half each iteration. As such it takes $\lceil \log_2 (d) \rceil$ iterations (three in this case).
• Figure 2: Depiction of environment-assisted classical communication and one-way LOCC discrimination. Black arrows denote quantum communication and black double-bars denote classical communication. Once the encoder $\mathcal{E}$ is fixed, environment-assisted classical communication is a one-way LOCC state discrimination protocol with the output states generated by the isometry $\mathcal{V}^{\mathcal{N}}$. The one-way LOCC protocol is in the dashed blue box. For concreteness, here we depict the case where $\mathcal{E}$ prepares pure states $\{{|{\phi_{m}}\rangle}\}_{m}$ reducing the problem to finding the best one-way LOCC discrimination protocol for $\{{|{\tilde{\psi}_{m}}\rangle} \coloneq \mathcal{V}^{\mathcal{N}}{|{\phi_{m}}\rangle}\}_{m}$.

Theorems & Definitions (16)

• Lemma 1
• Theorem 2
• proof
• Corollary 3
• Proposition 4
• Proposition 5
• proof
• Proposition 6
• proof
• Proposition 7