Complementarity and Fundamental Limit in Precision Phase Measurement

By analyzing a single-photon interference experiment together with a quantum nondemolition measurement scheme for the which-path information, we prove that given the total mean number $〈n〉$ of available photons, the fundamental limit in precision measurement of a phase shift is the Heisenberg limit, i.e., $1/〈n〉$. The analysis is based on the complementarity principle and is independent of the scheme for the measurement of the phase shift. We also show that to achieve the Heisenberg limit states with photon number fluctuations of the order of or larger than the mean photon number have to be exploited.