Local Test for Unitarily Invariant Properties of Bipartite Quantum States
Kean Chen, Qisheng Wang, Zhicheng Zhang
TL;DR
The paper resolves how local testers can optimally identify properties of bipartite quantum states by exploiting unitary invariance on one side. It shows that for bipartite pure-state properties, a local tester on the other party achieves the same sample complexity as any global tester, with a construction based on extended Schur-Weyl duality and randomization. The results extend to mixed inputs via one-way LOCC, yielding a canonical tester for entanglement-spectrum properties and enabling strong new lower bounds for Schmidt rank, MPS bond dimension, maximal entanglement, uniform Schmidt coefficients, and productness, as well as a quantum query lower bound for entanglement entropy. The work clarifies the power and limits of purified samples and provides a suite of technical tools (block-encoding, LCU, and LOCC constructions) that advance quantum property testing and complexity theory.
Abstract
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 $Ω(r^2/\varepsilon^2)$ 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. 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.