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

New Variational Formulae for Entropy‑Like Functionals

A recent article in Quantum (Vol 10, 2266, 2026) by Andrzej Łuczak, Hanna Podsędkowska, and Rafał Wieczorek introduces a family of variational expressions for entropy‑like quantities defined on normal states of semifinite von Neumann algebras. The work builds on classic constructions such as Segal’s entropy and Rényi entropies, but it does so in a far more general setting that accommodates both finite and semifinite algebras.

From Traces to Convex Functions

The authors consider functionals of the form

$$\tau\bigl(f(h)\bigr),$$

where \(\tau\) is a normal, faithful, semifinite trace on a von Neumann algebra \(\mathscr{M}\), \(h\) is a positive self‑adjoint operator belonging to \(L^1(\mathscr{M},\tau)\), and \(f\) is a convex (or concave) function satisfying suitable growth conditions. By exploiting the duality between \(L^1\) and \(L^\infty\) spaces associated with \(\tau\), they derive a variational principle that expresses \(\tau(f(h))\) as a supremum (or infimum) over a class of auxiliary operators. This mirrors the well‑known variational characterisation of relative entropy due to Petz, but the new formulas apply to a broader class of entropic functionals.

Generalising Known Results

The paper shows that classic cases—Segal’s entropy, the quantum Rényi entropies, and the relative entropy—appear as special instances of the general framework. In particular, the authors recover Petz’s variational expression for relative entropy when \(f(t)=t\log t\). They also demonstrate how the new formulae simplify when the underlying algebra is finite, thereby connecting with earlier results on matrix algebras.

Links to Quantum Information Theory

Beyond the mathematical development, the authors highlight a conceptual bridge to quantum information. By examining the structure of abelian sub‑algebras within \(\mathscr{M}\), they provide fresh interpretations of entropy‑like quantities as measures of information loss under coarse‑graining. This perspective may prove useful for studying quantum channels, resource theories, and the thermodynamics of infinite‑dimensional systems.

Outlook

The variational approach opens several avenues for future work. One promising direction is the numerical implementation of the supremum/infimum problems, which could yield practical algorithms for estimating entropic quantities in large‑scale quantum systems. Another is the exploration of operational meanings for the newly defined functionals, especially in contexts where semifinite traces naturally arise (e.g., quantum field theory on curved spacetimes). The authors’ results thus enrich the toolbox available to both mathematicians and physicists working at the interface of operator algebras and quantum information.

For readers interested in the technical details, the full article is available via the DOI link below, and a pre‑print can be consulted on arXiv.