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arxiv: 1906.10412 · v1 · pith:XAHSAOHYnew · submitted 2019-06-25 · 🧮 math.OA · math.FA

Quantum R\'enyi relative entropies on density spaces of C^*-algebras: their symmetries and their essential difference

Lajos Moln\'ar This is my paper

Pith reviewed 2026-05-25 16:10 UTC · model grok-4.3

classification 🧮 math.OA math.FA
keywords quantum Rényi relative entropyC*-algebrasdensity spacessurjective transformationssymmetry groupsnon-commutative algebrasrelative entropy
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The pith

Different types of quantum Rényi relative entropies on C*-algebra density spaces cannot be mapped to each other by surjective transformations.

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

The paper extends definitions of several quantum Rényi relative entropies to the density spaces of C*-algebras. It proves that these quantities, which are the same in the commutative case, are essentially different in the non-commutative case because no surjective transformation between the density spaces can turn one into another. The symmetry groups of the density spaces for each of these entropies are shown to be the same. The same conclusions hold for the Umegaki and Belavkin-Staszewski relative entropies.

Core claim

Extending the quantum Rényi relative entropies to density spaces of C*-algebras reveals that they are essentially different on non-commutative algebras, since none can be obtained from another via any surjective transformation between density spaces, although their symmetry groups are identical.

What carries the argument

Surjective transformations between density spaces, which are used to test whether one relative entropy can be transformed into another.

If this is right

  • The different Rényi relative entropies define inequivalent structures on non-commutative density spaces.
  • The symmetry groups of the density spaces are the same for all considered relative entropies.
  • Similar essential differences hold for the Umegaki and Belavkin-Staszewski relative entropies.

Where Pith is reading between the lines

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

  • The distinction implies that multiple inequivalent measures of relative entropy are required for quantum systems modeled by non-commutative C*-algebras.
  • Classical coincidence of the entropies relies on the ability to interchange them via maps that exist only when algebras commute.

Load-bearing premise

The definitions of the quantum Rényi relative entropies extend to density spaces of C*-algebras such that surjective transformations between those spaces can be considered.

What would settle it

A surjective transformation between the density spaces of some non-commutative C*-algebra that converts one type of quantum Rényi relative entropy into another would show they are not essentially different.

read the original abstract

We extend the definitions of different types of quantum R\'enyi relative entropy from the finite dimensional setting of density matrices to density spaces of $C^*$-algebras. We show that those quantities (which trivially coincide in the classical commutative case) are essentially different on non-commutative algebras in the sense that none of them can be transformed to another one by any surjective transformation between density spaces. Besides, we determine the symmetry groups of density spaces corresponding to each of those quantum R\'enyi relative entropies and find that they are identical. Similar results concerning the Umegaki and the Belavkin-Staszewksi relative entropies are also presented.

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.

Referee Report

0 major / 3 minor

Summary. The paper extends several quantum Rényi relative entropies (and the Umegaki and Belavkin-Staszewski relative entropies) from finite-dimensional density matrices to density spaces of general C*-algebras. It proves that the resulting functionals are inequivalent in the non-commutative setting: no surjective map between density spaces can intertwine any pair of them. At the same time, the symmetry groups of the density spaces with respect to each entropy coincide.

Significance. The result supplies a precise, transformation-invariant distinction among quantum relative entropies that is invisible in the commutative case. The explicit computation of the common symmetry groups and the inequivalence proofs constitute a concrete advance for the infinite-dimensional theory; they furnish invariants that can be used to classify entropy functionals on C*-algebras and may inform the choice of divergence in non-commutative information theory.

minor comments (3)
  1. [§2] §2, Definition 2.3: the extension of the Rényi functional to general C*-algebras is stated via a supremum over finite-rank approximations; it would help to record explicitly that this supremum is attained on the support projection of the pair (ρ,σ).
  2. [Theorem 4.7] Theorem 4.7 and Corollary 4.8: the proof that the symmetry groups coincide relies on the fact that every symmetry preserves the support lattice; a one-sentence reminder of this lattice preservation would make the argument self-contained for readers who have not yet reached the appendix.
  3. [Notation] Notation: the symbol D_α is reused for both the finite-dimensional and the C*-algebraic versions; a brief parenthetical distinction (e.g., D_α^fd vs. D_α^C*) in the first two sections would prevent momentary confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper extends finite-dimensional definitions of quantum Rényi relative entropies to density spaces of C*-algebras via suitable functionals on positive elements, then derives inequivalence under surjective transformations by comparing invariants such as monotonicity and value ranges, while showing identical symmetry groups. These steps rely on direct mathematical properties of the extensions rather than any self-definition, fitted inputs renamed as predictions, or load-bearing self-citations. The commutative coincidence is explicitly treated as trivial background, and the non-commutative distinction follows from the extended definitions without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard mathematical framework of C*-algebras and density spaces without introducing new free parameters or entities.

axioms (1)
  • standard math Standard properties of C*-algebras and their states/density spaces
    The paper extends definitions using the established theory of C*-algebras.

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Reference graph

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