Relative entropy and the multi-variable multi-dimensional moment problem

TL;DR

This paper investigates the use of entropy-like functionals, such as quantum relative entropy, as regularizers for solving multi-variable, multi-dimensional moment problems, providing a homotopy-based method to find and describe all solutions.

Contribution

It introduces a homotopy approach to construct extrema of entropy functionals for moment problems and generalizes solution descriptions via a Riemannian metric renormalization.

Findings

Effective construction of solutions via homotopy methods

Generalization of solution space through Riemannian metric renormalization

Application to inverse power spectrum problems with second-order statistics

Abstract

Entropy-like functionals on operator algebras have been studied since the pioneering work of von Neumann, Umegaki, Lindblad, and Lieb. The most well-known are the von Neumann entropy $trace (\rho\log \rho)$ and a generalization of the Kullback-Leibler distance $trace (\rho \log \rho - \rho \log \sigma)$, refered to as quantum relative entropy and used to quantify distance between states of a quantum system. The purpose of this paper is to explore these as regularizing functionals in seeking solutions to multi-variable and multi-dimensional moment problems. It will be shown that extrema can be effectively constructed via a suitable homotopy. The homotopy approach leads naturally to a further generalization and a description of all the solutions to such moment problems. This is accomplished by a renormalization of a Riemannian metric induced by entropy functionals. As an application we discuss the inverse problem of describing power spectra which are consistent with second-order statistics, which has been the main motivation behind the present work.