Improving the precision of measurements is a significant scientific challenge. The challenge is twofold: first, overcoming noise that limits the precision given a fixed amount of a resource, N, and second, improving the scaling of precision over the standard quantum limit (SQL), 1/sqrt{N}, and ultimately reaching a Heisenberg scaling (HS), 1/N. Here we present and experimentally implement a new scheme for precision measurements. Our scheme is based on a probe in a mixed state with a large uncertainty, combined with a post-selection of an additional pure system, such that the precision of the estimated coupling strength between the probe and the system is enhanced. We performed a measurement of a single photons Kerr non-linearity with an HS, where an ultra-small Kerr phase of around 6 *10^{-8} rad was observed with an unprecedented precision of around 3.6* 10^{-10} rad. Moreover, our scheme utilizes an imaginary weak-value, the Kerr non-linearity results in a shift of the mean photon number of the probe, and hence, the scheme is robust to noise originating from the self-phase modulation.
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