Fourier analytic approach to phase estimation in quantum systems
Hiroshi Imai, Masahito Hayashi · New Journal of Physics · 2009
For a unified analysis of phase estimation, we focus on the limiting distribution. It is shown that the limiting distribution can be expressed as the absolute square of the Fourier transform of an L 2 function whose support belongs to [−1, 1]. Using this result, we study the relation between the variance of the limiting distribution and its tail probability. We prove that the protocol which minimizes the asymptotic variance does not minimize the tail probability. We derive the estimation protocol minimizing the tail probability outside a given interval, depending on the width of the interval. Such an optimal protocol is given by a prolate spheroidal wave function, which often appears in wavelet or time-limited Fourier analysis. Also, within the framework of interval estimation, we derive the minimum confidence interval that guarantees a given confidence coefficient.