Fundamental quantum limit in precision phase measurement

We show through a series of arguments that, given a total average photon number 〈N〉, the fundamental limit in the precision phase measurement is set by quantum mechanics to be the so-called Heisenberg limit, i.e., 1/〈N〉. Some specific types of phase measurement are considered in the discussion. However, the proof based on the general principle of complementarity of quantum mechanics applies to any scheme of phase measurement. From the general argument by the complementarity principle, we are able to find a necessary condition for those states that can achieve the Heisenberg limit if they are employed for precision phase measurement. A general guideline is given for the search of the measurement schemes in which the Heisenberg limit is achieved. We demonstrate the procedure by applying it to a few specific examples.