Quantum-classical boundary for precision optical phase estimation

Understanding the fundamental limits on the precision to which an optical phase can be estimated is of key interest for many investigative techniques utilized across science and technology. We study the estimation of a fixed optical phase shift due to a sample which has an associated optical loss, and compare phase estimation strategies using classical and non-classical probe states. These comparisons are based on the attainable (quantum) Fisher information calculated per number of photons absorbed or scattered by the sample throughout the sensing process. We find that, for a given number of incident photons upon the unknown phase, non-classical techniques in principle provide less than a 20% reduction in root-mean-square-error (RMSE) in comparison with ideal classical techniques in multi-pass optical setups. Using classical techniques in a new optical setup we devise, which incorporates additional stages of interference during the sensing process, the achievable reduction in RMSE afforded by non-classical techniques falls to only ' 4%. We explain how these conclusions change when non-classical techniques are compared to classical probe states in non-ideal multi-pass optical setups, with additional photon losses due to the measurement apparatus.