Quantum phase measurements and a general method for the simulation of random processes

Abstract

Quantum interferences offer the potential for improving the effective resolution wavelength of many measurements by a factor of N . Coincidence detection methods to date have been limited to N = 4 due to an increase in complexity of the apparatus with N . We offer an alternative method of extracting this higher-order phase information from a standard (first-order) interferometer. This “phase function fitting” algorithm also eliminates the need to null the interferometer. We compare the interferometer’s statistics to those of the quantum phase measurement, which we also utilize to derive Heisenberg limits for the quantum noon states. States which surpass this noon state limit are then demonstrated; and we discuss how state optimization for various signal environments might proceed. The phase representation is also shown to yield computationally efficient and conceptually revealing forms for calculating the statistics of the interferometer itself. We define a static limit in which the signal varies slowly enough that we can make the integration time long enough to approach perfect measurement of the interferometer statistics. We discuss the sense in which the phase function fitting algorithm can approach zero error in this limit, while still at finite N —satisfying the power constraint (but approaching infinite energy as we collect over longer times). To incorporate the signal dynamics we present a general method in which we can approximately prescribe the autocorrelation, as well as the probability distribution, of a random process; and illustrate how probabilistic effects can mitigate the spectral distortions of a nonlinear mapping. Mathematical challenges in applying these techniques to the simulation of the quantum phase measurement are circumvented—enabling the simulation of a variety of quantum algorithms for the estimation and tracking of various signal models in this exciting application.