arxiv: 2604.19853 · v1 · submitted 2026-04-21 · 🪐 quant-ph · math-ph· math.MP· math.OA

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Quantum f-divergences via Nussbaum-Szko{l}a Distributions in Semifinite von Neumann Algebras

George Androulakis, Theodoros Anastasiadis

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classification 🪐 quant-ph math-phmath.MPmath.OA

keywords quantum f-divergenceNussbaum-Szkoła distributionssemifinite von Neumann algebranormal statesclassical f-divergenceoperator algebras

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The pith

The quantum f-divergence between two normal states on a semifinite von Neumann algebra equals the classical f-divergence between their Nussbaum-Szkoła distributions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves an equality between quantum and classical f-divergences for normal states on semifinite von Neumann algebras using Nussbaum-Szkoła distributions. This extends a previous result that held only for the algebra of all bounded operators on a Hilbert space. The equality means that quantum divergences can be computed as classical ones in this general setting. A reader would care if they want to apply classical probability tools to quantum information problems in more abstract algebraic frameworks.

Core claim

For any semifinite von Neumann algebra and any two normal states on it, the quantum f-divergence is equal to the classical f-divergence of the two corresponding Nussbaum-Szkoła distributions. The proof extends the technical construction and arguments that were previously available only when the algebra is B(H).

What carries the argument

The Nussbaum-Szkoła distributions, classical probability distributions associated to the quantum states allowing the f-divergence to transfer from the quantum to the classical setting.

If this is right

The equality now applies to normal states on all semifinite von Neumann algebras rather than only B(H).
Classical f-divergence properties and computations carry over to the quantum case in this wider class of algebras.
Technical proofs for the distributions and equality are generalized beyond the type I case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This may facilitate analysis of divergences in infinite-dimensional quantum systems or type II factors using classical tools.
Analogous reductions could apply to other quantum divergences or information measures in general von Neumann algebras.
Practical computations in quantum statistical mechanics on semifinite algebras might simplify via this equivalence.

Load-bearing premise

Nussbaum-Szkoła distributions can be constructed for normal states on arbitrary semifinite von Neumann algebras and the equality with quantum f-divergences can be proved by extending previous technical arguments.

What would settle it

A counterexample consisting of normal states on a semifinite von Neumann algebra other than B(H) where the quantum and classical f-divergences do not match would disprove the result.

read the original abstract

In this article, we prove that the quantum $f$-divergence between two normal states on a semifinite von~Neumann algebra is equal to the classical $f$-divergence between two corresponding classical states, which are called Nussbaum-Szko{\l}a distributions. This result has been proved by the second named author and T.C.~John for normal states on the von~Neumann algebra $\mathbb{B}(\mathscr{H})$ of all bounded operators on a Hilbert space $\mathscr{H}$. We extend their result for normal states on any semifinite von~Neumann algebra, not only $\mathbb{B}(\mathscr{H})$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Extends quantum f-divergence equality via NS distributions to semifinite von Neumann algebras with a direct proof.

read the letter

The key takeaway is that this paper successfully extends the representation of quantum f-divergences in terms of classical ones via Nussbaum-Szkoła distributions to the setting of general semifinite von Neumann algebras. What they do is construct these distributions explicitly for normal states using the Radon-Nikodym derivative with respect to a faithful normal semifinite trace. Then they show that the quantum f-divergence equals the classical f-divergence of the resulting measures. This builds on the earlier work for B(H) but requires new arguments to handle the general case. The paper does well in keeping everything rigorous and direct. The proof is self-contained, relies only on standard tools from operator algebra theory like modular theory, and doesn't introduce any questionable steps or assumptions beyond what's needed. Where it is softer is in the scope. This is a generalization of an existing result rather than something that opens up new directions. There are no concrete examples or discussions of how this might be applied in quantum field theory or continuous variables, which might limit its immediate appeal. This kind of paper is for researchers who need precise mathematical statements in quantum information theory involving von Neumann algebras. A reader who wants to use or build on the NS-distribution approach in broader settings would get something out of it. It deserves to go to peer review because the central claim is well-supported by the construction and the extension is natural and correct based on the details provided.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the quantum f-divergence D_f(φ, ψ) between two normal states φ, ψ on a semifinite von Neumann algebra M equals the classical f-divergence between the associated Nussbaum-Szkoła distributions. These distributions are constructed explicitly from the Radon-Nikodym derivatives of the states with respect to a faithful normal semifinite trace τ on M. The result extends the corresponding equality previously established for normal states on B(H).

Significance. If the equality holds, the work supplies a concrete bridge between quantum f-divergences in general semifinite von Neumann algebras and their classical counterparts, extending the scope beyond type-I factors without invoking separability or additional structural assumptions. The self-contained construction that relies only on standard facts from modular theory and the absolute continuity of normal states with respect to τ is a clear technical strength.

minor comments (2)

[Abstract] Abstract: the phrase 'not only B(H)' could be rephrased for precision, as the extension applies to all semifinite von Neumann algebras rather than merely adding cases beyond B(H).
[Introduction] The introduction would benefit from an explicit pointer to the precise statement of the prior B(H) result (including the reference) so that the novelty of the semifinite extension is immediately clear.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation to accept the manuscript. The provided summary accurately reflects the main result: the equality between quantum f-divergences and classical f-divergences of the associated Nussbaum-Szkoła distributions for normal states on semifinite von Neumann algebras, extending the earlier result for B(H).

Circularity Check

0 steps flagged

No significant circularity; explicit construction yields independent extension

full rationale

The manuscript proves the equality D_f(φ,ψ) = D_f(μ_φ,μ_ψ) by constructing the Nussbaum-Szkoła distributions explicitly from the Radon-Nikodym derivatives of the normal states with respect to the faithful normal semifinite trace τ, using only standard modular theory and absolute continuity on the support projection. All steps are self-contained within the semifinite setting and do not reduce to fitted parameters, self-referential definitions, or load-bearing self-citations. The reference to the authors' prior B(H) result supplies context for the special case but is not invoked to justify the general-case derivation or to forbid alternatives.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard properties of semifinite von Neumann algebras and normal states; no new free parameters, ad-hoc axioms, or invented entities are introduced beyond the prior construction of Nussbaum-Szkoła distributions.

axioms (2)

domain assumption Semifinite von Neumann algebras admit faithful normal semifinite traces.
Standard structural property of semifinite vNa used to define states and traces in the divergence construction.
standard math Normal states are ultraweakly continuous positive linear functionals with norm one.
Basic definition from von Neumann algebra theory invoked when extending the distributions.

pith-pipeline@v0.9.0 · 5420 in / 1322 out tokens · 62647 ms · 2026-05-10T02:00:11.539059+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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