aiqbench analysis
This research establishes fundamental limits on local testing for bipartite quantum states, proving that unitary invariance guarantees optimality for testers acting on a single subsystem. The study resolves several open questions regarding property testing for mixed states, Schmidt rank, and matrix product states by deriving tight lower bounds that match or improve upon previous algorithms. These theoretical advances clarify the efficiency of quantum verification protocols without relying on purified samples or global access to the state.
Local Test for Unitarily Invariant Properties of Bipartite Quantum States
arXiv:2404.04599v4 Announce Type: replace 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 \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 $\Omega(r^2/\varepsilon^2)$ 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 $\Omega((\sqrt{n}+\sqrt{r})\cdot\sqrt{r}/\varepsilon^2)$ for testing whether an $n$-partite state is a matrix product state of bond dimension $r$ or $\varepsilon$-far, improving the prior lower bounds $\Omega(\sqrt{n}/\varepsilon^2)$ by Soleimanifar and Wright (SODA 2022) and $\Omega(\sqrt{r})$ by Aaronson et al. (ITCS 2024). - A matching lower bound $\Omega(d/\varepsilon^2)$ 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 $\widetilde\Omega(\sqrt{d/\Delta})$ for the $d$-dimensional entanglement entropy problem with gap $\Delta$, improving the prior lower bounds $\Omega(\sqrt[4]{d})$ by She and Yuen (ITCS 2023) and $\widetilde{\Omega}(1/\sqrt{\Delta})$ 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.
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 \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 $\Omega(r^2/\varepsilon^2)$ 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 $\Omega((\sqrt{n}+\sqrt{r})\cdot\sqrt{r}/\varepsilon^2)$ for testing whether an $n$-partite state is a matrix product state of bond dimension $r$ or $\varepsilon$-far, improving the prior lower bounds $\Omega(\sqrt{n}/\varepsilon^2)$ by Soleimanifar and Wright (SODA 2022) and $\Omega(\sqrt{r})$ by Aaronson et al. (ITCS 2024). - A matching lower bound $\Omega(d/\varepsilon^2)$ 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 $\widetilde\Omega(\sqrt{d/\Delta})$ for the $d$-dimensional entanglement entropy problem with gap $\Delta$, improving the prior lower bounds $\Omega(\sqrt[4]{d})$ by She and Yuen (ITCS 2023) and $\widetilde{\Omega}(1/\sqrt{\Delta})$ 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.
56 pages. [v2]: extended testers with parameterized completeness and soundness, added new lower bounds for testing the bond dimension of matrix product states (MPS), and improved the lower bounds for testing Schmidt rank; [v3] and [v4]: minor revision
Quantum Physics (quant-ph); Computational Complexity (cs.CC)
arXiv:2404.04599 [quant-ph]
(or arXiv:2404.04599v4 [quant-ph] for this version)
https://doi.org/10.1109/TIT.2026.3697790 DOI(s) linking to related resources
Full text under the publisher's open licence · 300 words
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