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 signi cance for the symmetric polynomials in quantum information
theory. Using the symmetric polynomials we also derive a series of further properties
of H and Q.This is the accepted manuscript. The final version is available at http://scitation.aip.org/content/aip/journal/jmp/56/6/10.1063/1.4922317
