In recent research, Doosti et al. introduced the problem of distributed quantum state verification, where $m$ distributed nodes each receive a copy of an unknown state $\rho$ and can send limited one-way communication to a central node that possesses a complete description of a known state $\sigma$. They investigate how many distributed nodes $m$ are needed for the central node to successfully distinguish between $\rho = \sigma$ or $\|\rho - \sigma\|_1 \geq \varepsilon$ with high probability. In the setting where only quantum communication is allowed, Doosti et al. exhibited conditional lower bounds in both the public and private-coin settings, along with a matching upper bound in the public-coin setting. We extend these results by showing unconditional lower bounds when both classical and quantum communication are permitted. We prove the public-coin lower bound is tight by providing an algorithm with a matching upper bound. Additionally, we demonstrate an almost tight upper bound in the private-coin setting when only quantum communication is allowed.
Blogger's Review: This paper delves into the bounds of distributed quantum state verification, offering a comprehensive analysis of the interplay between classical and quantum communication. The introduction of a new algorithm not only enhances the understanding of state certification but also paves the way for future research in quantum communication.