Heisenberg Scaling of Agnostic Phase Estimation

Entanglement-enhanced metrology outperforms strategies that do not harness entanglement. We recently demonstrated a phase-estimation strategy that is agnostic to the generator of rotations [1]. Even without knowing the generator, we precisely estimate the rotation angle α in an unknown rotation operator U_α. As limited by the quantum Cramér-Rao bound, the variance of our estimate of α, after multiple independent measurements, is governed by the standard quantum limit var(α) ∝ 1/ν, where ν represents the total number of trials. In this work, we exhibit Heisenberg scaling, which surpasses the standard quantum limit in agnostic phase estimation. By applying U_α multiple times, we achieve Heisenberg scaling, reducing the estimator's variance as var(α) ∝ 1/ν². We also explore extensions that harness ancilla and probe rotations to further boost the measurement’s sensitivity. [1] Phys. Rev. Lett. 132, 260801 (2024)