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

TL;DR

The paper addresses entropy-like functionals for normal states on semifinite von Neumann algebras and derives variational representations that unify finite and semifinite settings. It develops a general framework showing $H(\omega)=\sup_{h^*=h\in\mathscr{M}}\{\tau(D_\omega h)-\log\tau(e^h)\}$ for Segal entropy, and extends Jensen-type inequalities to convex/concave functions $f$ to obtain abelian-subalgebra based variational formulas. The results yield explicit, partition-based expressions for Segal and Rényi entropies and connect entropy to the structure of abelian subalgebras and conditional expectations, generalizing known relative-entropy formulas. These variational representations provide tools for analyzing quantum entropies in both finite and semifinite algebras, with implications for quantum information theory.

Abstract

The paper presents variational formulae for entropy-like functionals, including Segal and Rényi entropies, for normal states on semifinite von Neumann algebras. The considered functionals are of the form $τ(f(h))$ where $τ$ is a normal faithful semifinite trace on this algebra, $h$ is a positive selfadjoint operator from $L^1(\M,τ)$, 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.

Theorems & Definitions (13)

• Theorem 1
• proof
• Theorem 2
• Proposition 3
• proof
• Proposition 4
• Theorem 5
• proof
• Theorem 6
• Corollary 7