Variational formulae for entropy-like functionals for states in von Neumann algebras

Abstract

The paper presents variational formulae for entropy-like functionals, including Segal and R\'enyi entropies, for normal states on semifinite von Neumann algebras. The considered functionals are of the form $\tau(f(h))$ where $\tau$ is a normal faithful semifinite trace on this algebra, $h$ is a positive selfadjoint operator from $L^1(\M,\tau)$, and $f$ is an appropriate convex or concave function. The results cover both finite and semifinite algebras, and the obtained formulae generalise known results, in particular, those concerning relative entropy. Moreover, the connection between quantum entropies and the structure of abelian subalgebras is highlighted, providing new interpretations in the context of quantum information theory.