Coherent States and Bayesian Duality

TL;DR

This paper explores how coherent states can incorporate various statistical distributions, revealing a duality akin to Bayesian duality, and highlights the usefulness of nonlinear coherent states in representing standard distributions.

Contribution

It introduces a framework linking coherent states with statistical distributions, demonstrating a duality similar to Bayesian duality, and extends the concept to vector and multidimensional coherent states.

Findings

Coherent states can embed discrete and continuous distributions.

A duality between distributions and parameter families is established.

Nonlinear coherent states encompass many standard statistical distributions.

Abstract

We demonstrate how large classes of discrete and continuous statistical distributions can be incorporated into coherent states, using the concept of a reproducing kernel Hilbert space. Each family of coherent states is shown to contain, in a sort of duality, which resembles an analogous duality in Bayesian statistics, a discrete probability distribution and a discretely parametrized family of continuous distributions. It turns out that nonlinear coherent states, of the type widely studied in quantum optics, are a particularly useful class of coherent states from this point of view, in that they contain many of the standard statistical distributions. We also look at vector coherent states and multidimensional coherent states as carriers of mixtures of probability distributions and joint probability distributions.