An Optimal Analysis of the Product Test

Product testing, i.e., deciding whether a pure multipartite quantum state is fully unentangled across a specified tensor decomposition, serves as a bridge between quantum property testing, unentangled quantum proof systems, and tensor optimization. Despite being a fundamental property testing task and having many applications, the product test's exact (worst-case) acceptance probability curve has yet to be fully determined. In this work, we determine this curve exactly. Let $\omega$ be the maximum squared overlap of the input with a product state, and let $\mathrm{PT}_n(\omega)$ be the largest possible acceptance probability of the product test over all $n$-partite pure states with product overlap $\omega$, allowing arbitrary finite local dimensions. We prove that, for every $n\ge 2 $ and every $\omega\in(0,1] $, $$ \mathrm{PT}_n(\omega)=\frac12\left(1+m\omega^2+(1-m\omega)^2\right), $$ where $m=\lfloor1/\omega\rfloor $. The formula recovers the previously known tight section of the curve for $\omega\ge 1/2 $, resolves all low-overlap regimes $\omega<1/2 $, and implies $\mathrm{PT}_n(\omega)\to 1/2 $ as $\omega\to 0$ answering an open problem in [Soleimanifar and Wright, SODA 2022]. As a complexity-theoretic application, our results improve the one-shot soundness parameter in the Harrow-Montanaro reduction from $\mathsf{QMA}(k)$ to $\mathsf{QMA}(2)$. Our techniques, built upon those of Soleimanifar and Wright, allow us to resolve these open questions while remaining surprisingly elementary.