Product testing with single-copy measurements
Jacob Beckey, Luke Coffman, Ariel Shlosberg, Louis Schatzki, Felix Leditzky
TL;DR
The paper addresses the sample complexity of product testing on $n$-qudit states when restricted to single-copy measurements, focusing on BP and MP variants. It develops an information-theoretic lower-bounds framework by relating overlaps to Gram-permanents and leveraging Le Cam-style arguments to show $T=\Omega(d^{n/4})$ and $T=\Omega(\sqrt{d/n})$ lower bounds for BP and MP testing, respectively, highlighting an exponential separation from known multi-copy algorithms. It also provides a nontrivial single-copy, local MP-testing algorithm with upper bound $O(n \log n \sqrt{d}/\varepsilon^2)$, illustrating a practical pathway under stringent measurement constraints. Overall, the results demonstrate a substantial gap between single-copy and multi-copy strategies for entanglement learning tasks and raise open questions about optimality and extensions to broader testing tasks.
Abstract
In this work, we study the sample complexity of two variants of product testing when restricted to single-copy measurements. In particular, we consider both bipartite product testing (i.e., does there exist at least one non-trivial cut across which the state is product) and multipartite product testing (i.e., is the state fully product across every cut). For the first variant, we prove an exponential lower bound on the sample complexity of any algorithm for this task which utilizes only single-copy measurements. When comparing this with known efficient algorithms that utilize multi-copy measurements, this establishes an exponential separation for this and several related entanglement learning tasks. For the second variant, we prove another sample lower bound that establishes a separation between single- and multi-copy strategies. To obtain our results, we prove a crucial technical lemma that gives a lower bound on the overlap between tensor products of permutation operators acting on subsystems of states that themselves carry a tensor structure. Finally, we provide an algorithm for multipartite product testing using only single-copy, local measurements, and we highlight several interesting open questions arising from this work.