Local Test for Unitarily Invariant Properties of Bipartite Quantum States

We study the power of local test for bipartite quantum states. Our central result is that, for properties of bipartite pure states, unitary invariance on one part implies an optimal (over all global testers) local tester acting only on the other part. As an application, we show that - Purified samples offer no advantage in property testing of mixed states. - A matching lower bound $\Omega(r^2/\varepsilon^2)$ for testing the Schmidt rank of bipartite states with perfect completeness, settling an open question raised in the survey of Montanaro and de Wolf (ToC 2016). - A lower bound $\Omega((\sqrt{n}+\sqrt{r})\cdot\sqrt{r}/\varepsilon^2)$ for testing whether an $n$-partite state is a matrix product state of bond dimension $r$ or $\varepsilon$-far, improving the prior lower bounds $\Omega(\sqrt{n}/\varepsilon^2)$ by Soleimanifar and Wright (SODA 2022) and $\Omega(\sqrt{r})$ by Aaronson et al. (ITCS 2024). - A matching lower bound $\Omega(d/\varepsilon^2)$ for testing whether a $d$-dimensional bipartite state is maximally entangled or $\varepsilon$-far, showing that the algorithm of O'Donnell and Wright (STOC 2015) is optimal for this task. We also show other applications in sample complexity and query complexity. In addition, our central result can be extended when the tested state is mixed: one-way LOCC is sufficient to realize the optimal tester.