Distributed Property Testing with (Quantum) Carrier Pigeons: Tight Bounds on State Certification

Recently, Doosti et al. introduced the problem of distributed quantum state verification, where $m$ distributed nodes are given a copy of an unknown state $\rho$, and can send limited one way communication to a central node, who has a complete description of a known state $\sigma$. They ask how many distributed nodes $m$ are required, before the central node can succeed at distinguishing whether $\rho=\sigma$ or $\|\rho-\sigma\|_1\geq\varepsilon$ with high probability. In the setting where only quantum communication is allowed, Doosti et al. exhibit conditional lower bounds in both the public and private-coin settings, and a matching upper bound in the public-coin setting. We extend these results, and show unconditional lower bounds for when both classical and quantum communication are permitted. We show the public-coin lower bound is tight by giving an algorithm with a matching upper bound. We also show an almost tight upper bound in the private-coin setting when only quantum communication is permitted.