Symmetric polynomials in information theory: entropy and subentropy

TL;DR

This paper explores entropy and subentropy as functions of elementary symmetric polynomials, revealing their complete monotonicity, Bernstein function properties, and connections to quantum information theory.

Contribution

It introduces a novel approach of expressing entropy and subentropy through symmetric polynomials, uncovering their mathematical properties and implications for quantum information.

Findings

Derivatives of entropy and subentropy satisfy complete monotonicity.

Entropy and subentropy are multivariate Bernstein functions with explicit Levy-Khintchine densities.

H and Q are shown to be Pick functions in each symmetric polynomial variable.

Abstract

Entropy and other fundamental quantities of information theory are customarily expressed and manipulated as functions of probabilities. Here we study the entropy H and subentropy Q as functions of the elementary symmetric polynomials in the probabilities, and reveal a series of remarkable properties. Derivatives of all orders are shown to satisfy a complete monotonicity property. H and Q themselves become multivariate Bernstein functions and we derive the density functions of their Levy-Khintchine representations. We also show that H and Q are Pick functions in each symmetric polynomial variable separately. Furthermore we see that H and the intrinsically quantum informational quantity Q become surprisingly closely related in functional form, suggesting a special significance for the symmetric polynomials in quantum information theory. Using the symmetric polynomials we also derive a series of further properties of H and Q.