Local Test for Unitarily Invariant Properties of Bipartite Quantum States

Kean Chen, Qisheng Wang, Zhicheng Zhang

Published 2024-04-06Version 1

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 settle two open questions raised in the survey of Montanaro and de Wolf (2016) by providing: 1. A matching lower bound $\Omega(1/\varepsilon^2)$ for testing whether a multipartite pure state is product or $\varepsilon$-far, showing that the algorithm of Harrow and Montanaro (2010) is optimal, even for bipartite states; 2. The first non-trivial lower bound $\Omega(r/\varepsilon)$ for testing whether the Schmidt rank of a bipartite pure state is at most $r$ or $\varepsilon$-far. We also show other new sample lower bounds, for example: - A matching lower bound $\Omega(d/\varepsilon^2)$ for testing whether a $d$-dimensional bipartite pure state is maximally entangled or $\varepsilon$-far, showing that the algorithm of O'Donnell and Wright (2015) is optimal for this task. Beyond sample complexity, we also contribute new quantum query lower bounds: - A query lower bound $\widetilde \Omega(\sqrt{d/\Delta})$ for the $d$-dimensional entanglement entropy problem with gap $\Delta$, improving the prior best $\Omega(\sqrt[4]{d})$ by She and Yuen (2023) and $\widetilde \Omega(1/\sqrt{\Delta})$ by Wang and Zhang (2023) and Weggemans (2024). Furthermore, our central result can be extended when the tested state is mixed: one-way LOCC is sufficient to realize the optimal tester.