  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. This suggests a canonical local tester for entanglement
spectra (i.e., Schmidt coefficients), and reveals that purified samples offer
no advantage in property testing of mixed states. As applications, we show new
sample lower bounds, e.g.:
  - The first general lower bound $\Omega(r/\epsilon^2)$ for testing whether
the Schmidt rank of a bipartite state is at most $r$ or $\epsilon$-far,
settling an open question raised in Montanaro and de Wolf (ToC 2016).
  - A lower bound $\Omega((\sqrt n+\sqrt r)\cdot\sqrt r/\epsilon^2)$ for
testing whether an $n$-partite state is a matrix product state of bond
dimension $r$ or $\epsilon$-far, improving the prior lower bound $\Omega(\sqrt
n/\epsilon^2)$ by Soleimanifar and Wright (SODA 2022) and $\Omega(\sqrt r)$ by
Aaronson et al. (ITCS 2024). Further, when perfect completeness is required, we
provide a matching lower bound $\Omega(r^2/\epsilon^2)$ with respect to $r$ and
$\epsilon$.
  - A matching lower bound $\Omega(d/\epsilon^2)$ for testing whether a
$d$-dimensional bipartite state is maximally entangled or $\epsilon$-far,
showing that the algorithm of O'Donnell and Wright (STOC 2015) is optimal for
this task.
  Beyond sample complexity, we also contribute new query lower bounds:
  - A query lower bound $\tilde\Omega(\sqrt{d/\Delta})$ for the $d$-dimensional
entanglement entropy problem with gap $\Delta$, improving the prior best
$\Omega(\sqrt[4]{d})$ by She and Yuen (ITCS 2023) and
$\tilde\Omega(1/\sqrt\Delta)$ by Wang and Zhang (2023) and Weggemans (2024).
  Further, our central result can be extended when the tested state is mixed:
one-way LOCC is sufficient to realize the optimal tester.
