Information geometry, Pythagorean-theorem extension, and Euclidean distance behind optical sensing via spectral analysis

Abstract

We present an information-geometric perspective on a generic spectral-analysis task pertaining to a vast class of optical measurements in which a parameter θ needs to be evaluated from θ-dependent spectral features in a measurable optical readout. We show that the spectral shift and line broadening driven by small Δθ variations can be isolated as orthogonal components in a Pythagorean-theorem extension for a Euclidean distance in the space of probability distributions, representing the Δθ-induced information gain, expressible via the relative entropy and the pertinent Fisher information. This result offers important insights into the limits of optical signal analysis, as well as into the ultimate spectral resolution and the limiting sensitivity of a vast class of optical measurements. As one example, we derive a physically transparent closed-form analytical solution for the information-theory bound on the precision of all-optical temperature sensors based on color centers in diamond.