Heisenberg-scaling measurement of the single-photon Kerr non-linearity using mixed states

Improving the precision of measurements is a significant scientific challenge. Previous works suggest that in a photon-coupling scenario the quantum fisher information shows a quantum-enhanced scaling of N², which in theory allows a better-than-classical scaling in practical measurements. In this work, utilizing mixed states with a large uncertainty and a post-selection of an additional pure system, we present a scheme to extract this amount of quantum fisher information and experimentally attain a practical Heisenberg scaling. We performed a measurement of a single-photon's Kerr non-linearity with a Heisenberg scaling, where an ultra-small Kerr phase of ≃6 × 10^-8 rad was observed with a precision of ≃3.6 × 10^-10 rad. From the use of mixed states, the upper bound of quantum fisher information is improved to 2N². Moreover, by using an imaginary weak-value the scheme is robust to noise originating from the self-phase modulation.