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 \textit{optimal} (over all global testers) local tester acting only on the other part. As an application, we demonstrate - Purified samples offer no advantage in property testing of mixed states. - A matching lower bound $Ω(r^2/\varepsilon^2)$ 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 $Ω((\sqrt{n}+\sqrt{r})\cdot\sqrt{r}/\varepsilon^2)$ for testing whether an $n$-partite state is a matrix product state of bond dimension $r$ or $\varepsilon$-far, improving the prior lower bounds $Ω(\sqrt{n}/\varepsilon^2)$ by Soleimanifar and Wright (SODA 2022) and $Ω(\sqrt{r})$ by Aaronson et al. (ITCS 2024). - A matching lower bound $Ω(d/\varepsilon^2)$ 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. - A query lower bound $\widetildeΩ(\sqrt{d/Δ})$ for the $d$-dimensional entanglement entropy problem with gap $Δ$, improving the prior lower bounds $Ω(\sqrt[4]{d})$ by She and Yuen (ITCS 2023) and $\widetildeΩ(1/\sqrtΔ)$ by Wang and Zhang (SICOMP 2025) and Weggemans (Quantum 2025). Moreover, we extend our central result to a robust version where the tested states are subject to noise and are not guaranteed to be pure: in this case, one-way LOCC is sufficient to realize the optimal tester.

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