Fisher information, disorder, and optical signal estimation

Abstract

Optical signals may be regarded as probability laws (on photon events). The problem of optical signal recovery may therefore be viewed as a problem in estimating a probability law p(x). Maximum entropy has, in the past, been used for this purpose. Here, we discuss instead the use of minimum Fisher information I = ∫ d x [ p ′ ( x ) ] 2 / p ( x ) . Consider a physical system consisting of a random aggregate of particles or pho tons. Consider the experiment of measuring one particle's coordinate y, and from this, best estimating the mean coordinate θ of the ensemble, where y = θ + x. As time passes, the particles randomly become more spread out, so that the error in the estimate should increase. But the error goes inversely as I. Therefore, I should statistically decrease with time. With no constraint on the system, as t → ∞, I → 0, defining a maximally random law p(x) = constant. However, in the presence of a physical constraint, I should approach a finite value, obeying I = minimum. When the constraint is linear in the mean kinetic energy of the system, the solution p(x) obeys the stationary differential equation for the system. In this way, the Schrodinger (energy) wave equation, Helmholtz wave equation, diffusion equation, and Maxwell-Boltzmann law may be derived from one classical principle of disorder.