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arxiv: 2607.01712 · v1 · pith:P6RQQF5Ynew · submitted 2026-07-02 · 🪐 quant-ph · math-ph· math.MP

Quantum Noncommutativity Uniquely Determines Relative Entropy

Gilad Gour This is my paper

Pith reviewed 2026-07-03 12:47 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords quantum relative entropyUmegaki divergencedistinguishability measuresquantum noncommutativitybinary hypothesis testingadditivityoptimal measurements
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The pith

Quantum noncommutativity forces any additive distinguishability measure matching optimal guessing odds to equal the Umegaki relative entropy.

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

The paper proves that quantum relative entropy is the only additive quantity that exactly reproduces the success probabilities achieved by optimal measurements when an observer tries to discriminate one quantum state from another. This uniqueness holds for any pair of states and requires no large-number limits or extra assumptions about many copies. Classical probability theory allows a continuous family of such measures, but the failure of quantum states to commute eliminates all alternatives. A reader would care because the result selects a single operational quantity from first principles for tasks ranging from communication limits to thermodynamic irreversibility.

Core claim

Any additive measure of distinguishability that respects the odds obtained from optimal binary measurements on pairs of quantum states must coincide exactly with the Umegaki relative entropy. Quantum noncommutativity supplies the rigidity that collapses the freedom present in the classical case, where Rényi divergences and other families remain valid.

What carries the argument

Additivity combined with exact agreement to the success probabilities of optimal measurements in binary guessing games.

If this is right

  • Relative entropy becomes the unique single-shot distinguishability measure compatible with optimal discrimination, without needing asymptotic arguments.
  • All other candidate divergences are ruled out once noncommutativity is present.
  • The data-processing inequality alone does not select the measure; the sharper operational matching condition is required.
  • Classical theory permits continuous families while quantum theory permits only one.

Where Pith is reading between the lines

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

  • Resource theories that rely on monotonicity under free operations could now invoke this uniqueness to fix the quantifier at the single-shot level.
  • Numerical checks on small-dimensional noncommuting pairs could directly test whether proposed alternative measures deviate from optimal guessing odds.
  • The result tightens links between distinguishability and thermodynamic irreversibility without invoking many-copy limits.

Load-bearing premise

The distinguishability measure is required to be additive and to match exactly the guessing odds produced by optimal measurements on pairs of states.

What would settle it

Exhibiting two noncommuting quantum states together with an additive measure different from Umegaki relative entropy that nevertheless yields identical optimal guessing probabilities would disprove the claim.

Figures

Figures reproduced from arXiv: 2607.01712 by Gilad Gour.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Testing regions for qubit pairs. The orange boundary corresponds to a genuinely quantum pair, while the red boundary [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Testing region for a [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The Lorenz distance is the uniform distance between Lorenz curves. [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Classical bracketing of a quantum Lorenz curve. The lower polygonal curve is the chordal interpolation of [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
read the original abstract

Quantum relative entropy is a core concept in physics, governing the limits of communication, thermodynamic irreversibility and quantum resource conversion. However, the requirement that physical processes cannot increase state distinguishability, the data-processing inequality, permits an infinite family of alternative divergence measures. Here we show that quantum relative entropy is uniquely selected by a sharper operational principle. We evaluate distinguishability through binary guessing games, in which an observer discriminates between pairs of quantum states using the optimal measurement. We prove that any additive measure that respects the odds revealed by these optimal measurements must coincide with the Umegaki relative entropy. This rigidity is a purely quantum phenomenon. Whereas classical theory permits a continuous family of valid divergence measures, including R\'enyi divergences, quantum noncommutativity. collapses this mathematical freedom. The result is exact, requiring neither a thermodynamic limit of infinitely many copies nor super-additivity assumptions for correlated states. It establishes quantum relative entropy not merely as an asymptotic quantity, but as the unique additive distinguishability measure compatible with single-shot quantum discrimination.

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 / 1 minor

Summary. The paper claims to prove that the Umegaki relative entropy is the unique additive distinguishability measure compatible with the guessing odds obtained from optimal binary measurements on arbitrary pairs of quantum states. The argument relies on an operational principle of exact matching to these odds plus additivity, and asserts that quantum noncommutativity eliminates the continuous family of classical alternatives (e.g., Rényi divergences) without requiring thermodynamic limits or super-additivity.

Significance. If the uniqueness theorem is correct, the result would be significant: it supplies a single-shot operational foundation for relative entropy that is strictly stronger than the data-processing inequality alone and isolates noncommutativity as the mechanism that removes classical freedom. This would strengthen the status of Umegaki relative entropy in quantum resource theories and thermodynamics.

minor comments (1)
  1. [Abstract] The abstract states the main result clearly but does not indicate the section or theorem number in which the uniqueness proof appears; adding an explicit forward reference would help readers locate the central derivation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for recognizing the potential significance of the result if the uniqueness theorem holds. The referee's description accurately captures the operational principle and the role of quantum noncommutativity. No specific major comments are listed in the report, so we have no point-by-point responses or revisions to propose at this stage. We remain available to address any concrete questions about the proof or its assumptions.

Circularity Check

0 steps flagged

No circularity; uniqueness derived directly from operational axioms

full rationale

The paper states a direct uniqueness theorem: any additive distinguishability measure that exactly reproduces the guessing odds from optimal binary measurements on arbitrary state pairs must equal the Umegaki relative entropy. The abstract and reader's summary describe this as following from the stated operational principle without thermodynamic limits or super-additivity. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations are indicated in the provided material. The derivation is presented as self-contained against the external benchmark of optimal measurement odds, yielding a normal non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two domain assumptions extracted from the abstract: additivity of the measure and exact agreement with optimal guessing odds. No free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption The distinguishability measure must be additive.
    Required for the measure to be compatible with multiple independent uses.
  • domain assumption The measure must respect the odds revealed by optimal measurements in binary guessing games.
    The sharper operational principle used to select among possible measures.

pith-pipeline@v0.9.1-grok · 5702 in / 1187 out tokens · 53911 ms · 2026-07-03T12:47:05.837356+00:00 · methodology

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Reference graph

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