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 anoptimal(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(r2/ε2) 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(( √n+ √r)· √r/ε2) for testing whether ann-partite state is a matrix product state of bond dimensionror ε-far, improving the prior lower bounds Ω( √n/ε2) by Soleimanifar and Wright (SODA 2022) and Ω( √r) by Aaronson et al. (ITCS 2024). • A matching lower bound Ω(d/ε2) for testing whether ad-dimensional bipartite state is maximally entangled or ε-far, showing that the algorithm of O’Donnell and Wright (STOC 2015) is optimal for this task. • A query lower bound Ω(d/ √ Δ) for thed-dimensional entanglement entropy problem with gap Δ, improving the prior lower bounds Ω(4√d) by She and Yuen (ITCS 2023) and Ω(1/ √ Δ) 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.