The role of shared randomness in quantum state certification with unentangled measurements

Given $n$ copies of an unknown quantum state $\rho\in\mathbb{C}^{d\times d}$, quantum state certification is the task of determining whether $\rho=\rho_0$ or $\|\rho-\rho_0\|_1>\varepsilon$, where $\rho_0$ is a known reference state. We study quantum state certification using unentangled quantum measurements, namely measurements which operate only on one copy of $\rho$ 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 $\Theta(d^{3/2}/\varepsilon^2)$ 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 ${\Theta}(d^2/\varepsilon^2)$ 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 L\"uders rule. More precisely, we obtain lower bounds for randomized and fixed schemes as a function of the eigenvalues of the L\"uders channel which characterizes one possible post-measurement state transformation.