slug: meunson-deesuwan-2026-cumulant-renyi-qdpi
bibkey: meunson2026cumulant
doi: 10.48550/arXiv.2606.31205
url: https://arxiv.org/abs/2606.31205v1
triage: theorem
motivation_gids:
  - D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation.result

Data processing fails for the cumulant-based Rényi functional at every order above one

Problem

A. Meunson and T. Deesuwan, Cumulant-based quantum relative Rényi functional, arXiv:2606.31205v1, define for $\alpha >1$ (Definition 3, Section IV)

$$S_{\alpha }^Q(\rho \Vert \sigma )=\frac{1}{\alpha -1}\ln \mathrm{Tr}\left(\rho e^{(\alpha -1)(\ln \rho -\ln \sigma )}\right)$$

on pairs with $\mathrm{supp}\rho \subseteq \mathrm{supp}\sigma$. The abstract and the conclusion state that the quantum data-processing inequality (QDPI) for arbitrary CPTP maps remains open. The verbatim sentences and Definition 3 are in the literature note. This question differs from Conjecture 15 of the same paper ($\alpha =0$, commutativity-preserving channels), settled in the earlier dossier.

Issue #13439 fixes the reading. QDPI α asserts $S_{\alpha }(\mathcal{N}\rho \Vert \mathcal{N}\sigma )\le S_{\alpha }(\rho \Vert \sigma )$ for every dimension $d$, every pair of positive definite density matrices and every CPTP map $\mathcal{N}$ with positive definite outputs. Matrix logarithms are continuous functional calculus, as in the frozen $\alpha =0$ module. claim asserts that some $\alpha >1$ satisfies QDPI α. Faithful inputs meet the support condition, so refuting this restricted claim refutes QDPI for the paper’s functional at every $\alpha >1$.

Motivation

Data processing is the property that makes a Rényi-type quantity a divergence: Petz Rényi divergences satisfy it for $\alpha \in (0,2]$ and sandwiched ones for $\alpha \ge 1/2$, as the paper recalls in Section III. The paper offers the functional as a candidate quantum Rényi divergence; its Section VIII.C (PDF p. 21) reports a preliminary study that found no violation for $\alpha \in (0.9,1.3)$ in $10^4$ random qubit pairs per run. D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation.result shows that the inequality fails at every $\alpha >1$.

Gap

Issue #13439 records the literature check before any Lean. The paper has a single version; the abstract and conclusion state the question as open. The earlier study (J. Phys.: Conf. Ser. 3168 (2025) 012014) is numerical only. The repository contains only the $\alpha =0$ refutation for this paper. The search seat’s bounded checks of INSPIRE, Google Scholar, MathDB and formal-conjectures found no settlement; the Semantic Scholar citation API was rate limited. not-found-in-searched-scope.

Route

Fix $\alpha >1$, put $t=\alpha -1$, $x=16^{1+1/t}$, $r=(x-1)/(x+1)$ and $a=\frac{1}{2}\ln x$.

1. Witness. $N=\frac{1}{2}\left(\begin{array}{cc}1& \sqrt{3}\\ \sqrt{3}& -1\end{array}\right)$ and $Z=\mathrm{diag}(1,-1)$ are self-adjoint involutions, and so is $B=N-Z$. The states $\rho =(I+rN)/2$ and $\sigma =(I+rZ)/2$ are positive definite with trace one and spectrum $\{x/(x+1),1/(x+1)\}$. Here $Z$ is the frozen qubitZ. The channel is complete dephasing, the frozen standard-basis pinching pinchingEnd, which equals the Kraus map with Kraus operators $\mathrm{diag}(1,0)$ and $\mathrm{diag}(0,1)$.
2. Input. Two-point functional calculus gives $\ln \rho =cI+aN$ and $\ln \sigma =cI+aZ$ with $c=\frac{1}{2}\ln x-\ln (x+1)$. Hence $\ln \rho -\ln \sigma =aB$ and $e^{taB}=\cosh (ta)I+\sinh (ta)B$, so $$F_{\mathrm{in}}=\cosh (ta)+\frac{r}{2}\sinh (ta)<e^{ta}=x^{t/2}=4^{t+1}.$$
3. Output. The outputs are $\mathrm{diag}(p_{+},p_{-})$ and $\mathrm{diag}(q_{+},q_{-})$ with $p_{-}=(x+3)/(4(x+1))>1/4$ and $q_{-}=1/(x+1)$, both positive definite. So $$F_{\mathrm{out}}=p_{+}^{t+1}{q}_{+}^{-t}+p_{-}^{t+1}{q}_{-}^{-t}>\frac{1}{4}{\left(\frac{x+3}{4}\right)}^t>4^{-t-1}{x}^t=4^{t+1}>F_{\mathrm{in}}.$$
4. Strict monotonicity of $\ln$ and $1/t>0$ give $S_{\alpha }(\Phi \rho \Vert \Phi \sigma )>S_{\alpha }(\rho \Vert \sigma )$, contradicting QDPI α.

Falsifier

The proof would fail if $B=N-Z$ were not an involution, if the dephasing map were not CPTP, or if the scalar comparison $t\ln x=2(t+1)\ln 4$ did not place $F_{\mathrm{in}}$ below $4^{t+1}$.

Evidence

The canonical source is D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation.lean. Its public declarations are cuRenyi, QDPI, claim and result; the witness and twenty-one supporting identities are private. Its direct frozen dependencies are IsDensity, IsCPTP and pauli of D5/S3/QuantumChannels/CoPRelativeQuantumnessRefutation, qubitZ and qubit_weyl_star of D5/S3/Quantum/FiniteDimensional, pinching_apply of D5/S3/QuantumChannels/Pinching, pinchingEnd of D5/S3/QuantumChannels/ProjectionDiagnostics/StaticDynamicScalarSeparation, cfc_log_diagonal of D5/S3/Quantum/Dynamics/EntropyProductionCoherenceDeletionIdentity, and MatrixMap, MatrixMap.of_kraus and MatrixMap.of_kraus_isCompletelyPositive of D5/S3/Quantum/Foundation/FiniteKrausChannel. The proof uses only the standard axioms propext, Classical.choice and Quot.sound; no sorry, native_decide, or new axiom. make lean builds the module in 12 s; lake env lean on it peaks at 5,852,708,864 bytes resident.

Numerical check (mpmath, 60 digits): at $x=16^{1+1/t}$ the increase $S_{\mathrm{out}}-S_{\mathrm{in}}$ is 8.58, 1.57, 0.342 and 0.213 at $\alpha =1.05$, 1.25, 2 and 3. Positive control: the commuting pair $\rho =(I+Z/3)/2$, $\sigma =(I-Z/3)/2$ at $\alpha =3/2$ under the same channel gives equal values (0.328504066972 before and after).

Triage

Tier 1 question stated in the abstract and conclusion of a 2026 paper, preregistered in issue #13439 before any Lean. theorem; resolution refuted.

The twenty-one private theorems are bind-only and each is used on the proof path of result (CLAUDE.md §3.2 「有消费的辅助声明」). Utility is certified-instance with a typed refutes edge to claim; there is no digestion atom.

What the settlement shows

Proved by result: QDPI fails at every $\alpha >1$.

Established inside the proof of result. The proof applies QDPI α at dimension two to positive definite density matrices and the standard-basis pinching channel, with positive definite outputs (the application of the hypothesis at the end of result); so restricting to qubits or to faithful states and outputs does not restore the inequality. The private lemmas of the module prove $\ln \rho -\ln \sigma =\frac{1}{2}\ln xB$ with $B=N-Z$ a self-adjoint involution (log_state, involutions, trace_input), and the scalar bounds $F_{\mathrm{in}}<x^{t/2}=4^{t+1}<F_{\mathrm{out}}$ (scalar_gap_general).

Argued, not formalized.

• Mechanism. The input states have the same spectrum and Bloch directions $60^{\circ }$ apart, so the large parts of $\ln \rho$ and $\ln \sigma$ cancel: the involution $B$ has operator norm 1, so the logarithm difference has norm $\frac{1}{2}\ln x$. Dephasing keeps weight above $1/4$ of $\Phi \rho$ on the basis vector where $\Phi \sigma$ has weight $1/(x+1)$, so the classical output value grows like $x^t$. The functional measures the logarithm difference, which noncommutativity shrinks, while the channel output sees the eigenvalue ratio itself.
• Complete dephasing fixes the identity matrix, so requiring unital channels does not restore the inequality.
• Complete dephasing sends every pair to diagonal matrices, which commute, so it is commutativity preserving. The analogue of the paper’s CoP-QDPI for $\alpha >1$ therefore fails on the same witness.
• For a commuting input pair whose outputs commute, both sides reduce to classical Rényi divergences of distributions related by one stochastic matrix. The classical inequality for $\alpha >1$ is the frozen D5/S3/RenyiDivergence/DataProcessingAboveOne.renyi_divergence_channel_le_of_one_lt_of_ac, so a violation with commuting outputs needs noncommuting inputs, as here. The reduction step is not formalized.

Computed (mpmath, 60 digits, grid $x=10^{k/40}$). Within this family the first violating eigenvalue ratio is about $7.1\cdot 10^9$ at $\alpha =1.05$, $3.5\cdot 10^6$ at $1.1$, $9.4\cdot 10^3$ at $1.25$, $631$ at $1.5$, $119$ at $2$ and $40$ at $3$. At $x=3$ and $\alpha =1.25$ there is no violation ($S_{\mathrm{out}}=0.048<S_{\mathrm{in}}=0.172$). This is consistent with the random sampling of Section VIII.C, whose typical qubit states have moderate eigenvalue ratios. The same formula extended to $\alpha <1$ also increases under the channel in this family: the first violating ratio is about $30$ at $\alpha =0.5$, $7.1\cdot 10^4$ at $0.9$ and $6.0\cdot 10^7$ at $0.95$.

Open. Whether violations exist near $\alpha =1$ with bounded eigenvalue ratio, a characterization of channels or state pairs on which the functional contracts, and a proof for $\alpha <1$ are not established here.

Effect on the paper. The open question of the abstract and conclusion is answered negatively for every $\alpha >1$, so the functional is not a quantum Rényi divergence in the sense of data processing at any $\alpha >1$. The proved positivity, classical reduction, additivity, unitary invariance, continuity and monotonicity in $\alpha$ do not depend on QDPI and are not affected. The $\alpha =0$ conjecture was refuted separately.

ASSUMED-UNVERIFIED

The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent proof. The search seat’s database readings are seat-reported.