Optimal Quantum State Testing Even with Limited Entanglement

2026-04-08

AI Summary

This work investigates the sample complexity of quantum state certification, mixedness testing, and purity estimation under a restricted entangled measurement model where only a limited number of copies of the quantum state—specifically, $t \ll d/\varepsilon^2$—can be jointly measured. By reducing these testing tasks to learning problems and leveraging non-black-box quantum state tomography together with a refined smooth complexity analysis in the high-accuracy regime ($\varepsilon < 1/\sqrt{d}$), the authors achieve near-optimal sample complexity under an entanglement constraint of $t = d^2$, thereby overcoming the prior reliance on fully joint measurements across all copies. Moreover, they establish a matching lower bound, demonstrating that $t \geq d^{\Omega(1)}$ is necessary to attain optimal rates.

Abstract

In this work, we consider the fundamental task of quantum state certification: given copies of an unknown quantum state $ρ$, test whether it matches some target state $σ$ or is $ε$-far from it. For certifying $d$-dimensional states, $Θ(d/ε^2)$ copies of $ρ$ are known to be necessary and sufficient. However, the algorithm achieving this complexity makes fully entangled measurements over all $O(d/ε^2)$ copies of $ρ$. Often, one is interested in certifying states to a high precision; this makes such joint measurements intractable even for low-dimensional states. Thus, we study whether one can obtain optimal rates for quantum state certification and related testing problems while only performing measurements on $t$ copies at once, for some $1 < t \ll d/ε^2$. While it is well-understood how to use intermediate entanglement to achieve optimal quantum state learning, the only protocol known to achieve optimal testing is the one using fully entangled measurements. Our main result is a smooth copy complexity upper bound for state certification as a function of $t$, which achieves a near-optimal rate at $t = d^2$. In the high-precision regime, i.e., for $ε< \frac{1}{\sqrt{d}}$, this is a strict improvement over the entanglement used by the aforementioned optimal protocol. We also extend our techniques to develop new algorithms for the related tasks of mixedness testing and purity estimation, and show tradeoffs achieving the optimal rates for these problems at $t = d^2$ as well. Our algorithms are based on novel reductions from testing to learning and leverage recent advances in quantum state tomography in a non-black-box fashion. We complement our upper bounds with smooth lower bounds that imply joint measurements on $t \geq d^{Ω(1)}$ copies are necessary to achieve optimal rates for certification in the high-precision regime.

Problem

• quantum state certification
• entanglement-limited measurements
• sample complexity
• high-precision regime
• quantum testing

Innovation

• quantum state certification
• limited entanglement
• sample complexity
• quantum testing
• quantum tomography