Abstract

The Cramer-Rao (CR) inequality e²I ≥ 1 represents complimentarity between (i) the size of mean-squares error e² in estimation of a parameter θ from an observation y where y = θ + x, and (ii) the Fisher information I due to pdf p(x). In application to quantum mechanics, θ is the classical position of a particle, and I becomes the mean-squares momentum spread for the particle. Thus, the CR inequality becomes the Heisenberg uncertainty principle. The latter is, then, but one example of a general principle of error complimentarity. As applied to optical diffraction, θ is now the unknown centroid of a diffraction pattern. Here the CR inequality becomes e²z² ≥ (λf/4π)², where z² is the mean-squares photon position in the lens pupil, λ is the light wavelength, and f is the focal length. Interestingly, here the uncertainty product can be made arbitrarily small. Another use is in the case of a nonideal gas kept at a constant temperature T. The gas is inside a container of unknown drift velocity θ. If p(x) defines the probability law on velocity x for particles of the gas, the CR inequality attains a minimum for the uncertainty product e²I when p(x) is Gaussian, i.e., when p(x) is a Boltzmann law. This describes the ideal gas scenario and gives it a new significance.

© 1991 Optical Society of America