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

Abstract

Meunson and Deesuwan (arXiv:2606.31205) define a cumulant-based quantum relative Renyi functional for alpha > 1 and state in the abstract and conclusion that its quantum data-processing inequality under arbitrary CPTP maps remains open. For every alpha > 1 a pair of positive definite qubit states and the complete dephasing channel increase the functional, so the inequality fails at every order above one.

Theorem 1.1 (Functional calculus for a self-adjoint involution).

$$\forall n:\mathrm{Type},[\mathrm{Fintype}\left(n\right)][\mathrm{DecidableEq}\left(n\right)]\forall f:\mathbb{R}\to \mathbb{R},\forall a:\mathbb{R},\forall b:\mathbb{R},\forall H:\mathrm{Matrix}\left(n,n,\mathbb{C}\right),(\mathrm{IsSelfAdjoint}\left(H\right))\Rightarrow ((H\cdot H=1)\Rightarrow (\mathrm{cfc}\left(f,a\cdot 1+b\cdot H\right)=\frac{f\left(a+b\right)+f\left(a-b\right)}{2}\cdot 1+\frac{f\left(a+b\right)-f\left(a-b\right)}{2}\cdot H))$$

Proof. Machine-checked in Lean as D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation.two_point_cfc (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. A. Meunson; T. Deesuwan (2026). Cumulant-based quantum relative Rényi functional. DOI: 10.48550/arXiv.2606.31205. URL: https://arxiv.org/abs/2606.31205v1.

Commentary.

For a self-adjoint complex matrix H with H squared equal to the identity, the spectrum lies in {1, -1}. For every real function f and real a and b, the function of a I + b H is the sum of the mean of f(a+b) and f(a-b) times I and half their difference times H. No continuity hypothesis on f is needed because the spectrum is finite.

Theorem 1.2 (The action of complete dephasing).

$$\forall A:{\mathbb{C}}^{2\times 2},\mathrm{pinchingEnd}\left(A\right)=\mathrm{diagonal}\left(![A\left(0,0\right),A\left(1,1\right)]\right)$$

Proof. Machine-checked in Lean as D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation.dephase_apply (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. A. Meunson; T. Deesuwan (2026). Cumulant-based quantum relative Rényi functional. DOI: 10.48550/arXiv.2606.31205. URL: https://arxiv.org/abs/2606.31205v1.

Commentary.

For every complex two by two matrix A, complete dephasing keeps its two diagonal entries and sets the off-diagonal entries to zero.

Theorem 1.3 (The Kraus representation of complete dephasing).

$$(\mathrm{pinchingEnd}:\mathrm{MatrixMap}\left(\mathrm{Fin}\left(2\right),\mathrm{Fin}\left(2\right),\mathbb{C}\right))=\mathrm{ofKraus}\left((\lambda j:\mathrm{Fin}\left(2\right),\mathrm{single}\left(j,j,1\right)),(\lambda j:\mathrm{Fin}\left(2\right),\mathrm{single}\left(j,j,1\right))\right)$$

Proof. Machine-checked in Lean as D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation.pinching_kraus (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. A. Meunson; T. Deesuwan (2026). Cumulant-based quantum relative Rényi functional. DOI: 10.48550/arXiv.2606.31205. URL: https://arxiv.org/abs/2606.31205v1.

Commentary.

For j in Fin 2, the family K(j)=single(j,j,1) consists of the two computational basis projectors. The linear map pinchingEnd is exactly their finite Kraus map, ofKraus(K,K).

Definition 1.4 (The cumulant-based relative Renyi functional).

$$\forall d:\mathbb{N},\forall \alpha :\mathbb{R},\forall A:{\mathbb{C}}^{d\times d},\forall B:{\mathbb{C}}^{d\times d},\mathrm{cuRenyi}\left(\alpha ,A,B\right)=\frac{1}{(\alpha -1)}\cdot \ln \left(\mathrm{Re}\left(\mathrm{Tr}\left(A\cdot \exp \left((\alpha -1)\cdot (\log \left(A\right)-\log \left(B\right))\right)\right)\right)\right)$$

Formalization. D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation.cuRenyi (✓ std3).

Citation. A. Meunson; T. Deesuwan (2026). Cumulant-based quantum relative Rényi functional. DOI: 10.48550/arXiv.2606.31205. URL: https://arxiv.org/abs/2606.31205v1.

Commentary.

Definition 3 of the paper, on faithful inputs: for complex d by d matrices A and B and real alpha, the value is 1/(alpha-1) times the real logarithm of the real part of Tr(A exp((alpha-1)(log A - log B))). Log of a matrix is continuous functional calculus for the real logarithm and exp is the matrix exponential, as in the frozen alpha-zero refutation for the same paper. On positive definite A and B the trace is a positive real number, and the formula is the paper’s.

Definition 1.5 (The data-processing inequality at alpha).

$$\forall \alpha :\mathbb{R},\mathrm{QDPI}\left(\alpha \right)\iff (\forall d:\mathbb{N},\forall \rho :{\mathbb{C}}^{d\times d},\forall \sigma :{\mathbb{C}}^{d\times d},\forall N:\mathrm{MatrixMap}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),(\mathrm{IsDensity}\left(\rho \right))\Rightarrow ((\mathrm{IsDensity}\left(\sigma \right))\Rightarrow ((\mathrm{PosDef}\left(\rho \right))\Rightarrow ((\mathrm{PosDef}\left(\sigma \right))\Rightarrow ((\mathrm{IsCPTP}\left(N\right))\Rightarrow ((\mathrm{PosDef}\left(N\left(\rho \right)\right))\Rightarrow ((\mathrm{PosDef}\left(N\left(\sigma \right)\right))\Rightarrow (\mathrm{cuRenyi}\left(\alpha ,N\left(\rho \right),N\left(\sigma \right)\right)\le \mathrm{cuRenyi}\left(\alpha ,\rho ,\sigma \right)))))))))$$

Formalization. D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation.QDPI (✓ std3).

Citation. A. Meunson; T. Deesuwan (2026). Cumulant-based quantum relative Rényi functional. DOI: 10.48550/arXiv.2606.31205. URL: https://arxiv.org/abs/2606.31205v1.

Commentary.

Section III of the paper states data processing as S_alpha(rho||sigma) >= S_alpha(N(rho)||N(sigma)) for every CPTP map N and states with supp(rho) contained in supp(sigma). QDPI(alpha) states it for density matrices rho and sigma on C^d of every dimension d that are positive definite, for every CPTP map N on d by d complex matrices with positive definite outputs. IsDensity and IsCPTP are the frozen definitions of the alpha-zero refutation for the same paper. Faithful inputs satisfy the support condition, and faithful outputs need no convention for the logarithm of a singular matrix.

Definition 1.6 (Data processing for some order above one).

$$claim\iff (\exists \alpha :\mathbb{R},(1<\alpha )\wedge (\mathrm{QDPI}\left(\alpha \right)))$$

Formalization. D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation.claim (✓ std3).

Citation. A. Meunson; T. Deesuwan (2026). Cumulant-based quantum relative Rényi functional. DOI: 10.48550/arXiv.2606.31205. URL: https://arxiv.org/abs/2606.31205v1.

Commentary.

The open question of the abstract and conclusion, in its weakest form: some alpha > 1 satisfies QDPI(alpha). Its negation refutes data processing at every order above one, already for faithful inputs and outputs.

Theorem 1.7 (No order above one satisfies data processing).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation.result (✓ std3). ∎

Resolves. Problems/meunson-deesuwan-2026-cumulant-renyi-qdpi (refuted) by D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation.result.

Source. Repository-derived.

Acknowledgement. A. Meunson; T. Deesuwan (2026). Cumulant-based quantum relative Rényi functional. DOI: 10.48550/arXiv.2606.31205. URL: https://arxiv.org/abs/2606.31205v1.

Commentary.

Fix alpha > 1, put t = alpha - 1, x = 16^(1 + 1/t), r = (x - 1)/(x + 1) and a = (ln x)/2. Let N be the symmetric involution with rows (1/2, sqrt(3)/2) and (sqrt(3)/2, -1/2), and Z = diag(1, -1); then B = N - Z is also a self-adjoint involution. The states rho = (I + r N)/2 and sigma = (I + r Z)/2 are positive definite with trace one and common spectrum {x/(x+1), 1/(x+1)}. Two-point functional calculus gives log rho - log sigma = a B and exp(t a B) = cosh(t a) I + sinh(t a) B, so the input trace is cosh(t a) + (r/2) sinh(t a), which is less than exp(t a) = 4^(t+1). The complete dephasing channel with Kraus operators diag(1, 0) and diag(0, 1) is CPTP and sends rho and sigma to the positive definite diagonal matrices diag(p+, p-) and diag(q+, q-) with p- = (x+3)/(4(x+1)) > 1/4 and q- = 1/(x+1). The output trace is p+^(t+1) q+^(-t) + p-^(t+1) q-^(-t) > (1/4) ((x+3)/4)^t > (1/4) (x/4)^t = 4^(t+1). By strict monotonicity of the logarithm and 1/t > 0 the functional increases under the channel, contradicting QDPI(alpha).

References

• Truth anchor: D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation.QDPI
• Truth anchor: D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation.claim
• Truth anchor: D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation.cuRenyi
• Truth anchor: D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation.dephase_apply
• Truth anchor: D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation.pinching_kraus
• Truth anchor: D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation.result
• Truth anchor: D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation.two_point_cfc
• Dependency: D5/S3/Quantum/Dynamics/EntropyProductionCoherenceDeletionIdentity
• Dependency: D5/S3/Quantum/Matrix/TwoPointFunctionalCalculus
• Dependency: D5/S3/QuantumChannels/CoPRelativeQuantumnessRefutation
• Dependency: D5/S3/QuantumChannels/ProjectionDiagnostics/StaticDynamicScalarSeparation