The Role of Randomness in Quantum State Certification with Unentangled Measurements

The Role of Randomness in Quantum State Certification with Unentangled Measurements (via Zoom)
Abstract: Given n copies of an unknown quantum state ρ∈C^{d×d}, quantum state certification is the task of determining whether ρ=ρ0 or ∥ρ−ρ0∥_1>ε, where ρ0 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 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 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.
Bio: Yuhan Liu is a 6th-year PhD candidate in ECE at Cornell University, advised by Prof. Jayadev Acharya. His research interest lies in federated learning, differential privacy, and quantum information theory.