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

Given n copies of an unknown quantum state ρ∈ℂ^d× d, quantum state certification is the task of determining whether ρ=ρ_0 or ρ-ρ_0_1>ε, 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/ε^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/ε^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.