The role of shared randomness in quantum state certification with unentangled measurements
Yuhan Liu, Jayadev Acharya
TL;DR
The paper analyzes quantum state certification with unentangled measurements, revealing a fundamental separation between fixed deterministic schemes and randomized schemes that use shared randomness. It proves a tight lower bound of $n=\Theta(d^2/\varepsilon^2)$ for fixed measurements and $n=\Theta(d^{3/2}/\varepsilon^2)$ for randomized schemes, showing that randomness substantially reduces copy complexity. A unified lower-bound framework based on the average Lüders channel connects the hardness of testing to the eigenstructure of the measurement ensemble, enabling tight bounds for both fixed and randomized non-adaptive schemes. The authors also provide an $O(d^2/\varepsilon^2)$ upper bound for fixed measurements by employing quantum 2-designs and reducing to classical distribution closeness testing, highlighting the practical viability of fixed schemes with structured measurements. Overall, the work clarifies the resource role of shared randomness in quantum state certification and offers a cohesive analytic approach that links measurement post-processing (Lüders channel) to information-theoretic limits.
Abstract
Given $n$ copies of an unknown quantum state $ρ\in\mathbb{C}^{d\times d}$, quantum state certification is the task of determining whether $ρ=ρ_0$ or $\|ρ-ρ_0\|_1>\varepsilon$, where $ρ_0$ is a known reference state. We study quantum state certification using unentangled quantum measurements, namely measurements which operate only on one copy of $ρ$ at a time. When there is a common source of shared randomness available and the unentangled measurements are chosen based on this randomness, prior work has shown that $Θ(d^{3/2}/\varepsilon^2)$ copies are necessary and sufficient. This holds even when the measurements are allowed to be chosen adaptively. We consider deterministic measurement schemes (as opposed to randomized) and demonstrate that $Θ(d^2/\varepsilon^2)$ copies are necessary and sufficient for state certification. This shows a separation between algorithms with and without shared randomness. We develop a unified lower bound framework for both fixed and randomized measurements, under the same theoretical framework that relates the hardness of testing to the well-established Lüders rule. More precisely, we obtain lower bounds for randomized and fixed schemes as a function of the eigenvalues of the Lüders channel which characterizes one possible post-measurement state transformation.