The proposed noncommuting Tsallis bound order fails

Abstract

A noncommuting positive definite qubit pair reverses the proposed Tsallis bound order.

Definition 1.1 (The deformed logarithm).

$$\forall (r:\mathbb{R}),\forall (x:\mathbb{R}),\mathrm{lnr}\left(r,x\right)=(\frac{(x^r)-(1)}{r})$$

Formalization. D5/S3/Divergence/MaityFuruichiDuttaOrderingRefutation.lnr (✓ std3).

Citation. Srikrishna Maity; Shigeru Furuichi; Supriyo Dutta (2026). r-deformed α-z-Rényi relative entropy. URL: https://arxiv.org/abs/2607.01805v1.

Commentary.

Definition 1, printed p. 2: “The r-logarithm is denoted by ln_r(x) and it is defined by ln_r(x) = (x^r − 1)/r, for r ≠ 0 and x ≥ 0.” The real power is Real.rpow.

Definition 1.2 (The deformed Rényi relative entropy).

$$\forall (n:\mathbb{N}),\forall (rho:\mathrm{Matrix}\left(\mathrm{Fin}\left(n\right),\mathrm{Fin}\left(n\right),\mathbb{C}\right)),\forall (sigma:\mathrm{Matrix}\left(\mathrm{Fin}\left(n\right),\mathrm{Fin}\left(n\right),\mathbb{C}\right)),\forall (alpha:\mathbb{R}),\forall (r:\mathbb{R}),\forall (z:\mathbb{R}),\mathrm{renyiRZ}\left(alpha,r,z,rho,sigma\right)=((\frac{1}{(alpha)-(1)})\cdot (\mathrm{lnr}\left(r,(\mathrm{Matrix}.\mathrm{trace}(\mathrm{cfc}\left(((x:\mathbb{R})\mapsto x^z),((\mathrm{cfc}\left(((x:\mathbb{R})\mapsto x^{\frac{(1)-(alpha)}{(2)\cdot (z)}}),sigma\right))\cdot (\mathrm{cfc}\left(((x:\mathbb{R})\mapsto x^{\frac{alpha}{z}}),rho\right)))\cdot (\mathrm{cfc}\left(((x:\mathbb{R})\mapsto x^{\frac{(1)-(alpha)}{(2)\cdot (z)}}),sigma\right))\right))).\mathrm{re}\right)))$$

Formalization. D5/S3/Divergence/MaityFuruichiDuttaOrderingRefutation.renyiRZ (✓ std3).

Citation. Srikrishna Maity; Shigeru Furuichi; Supriyo Dutta (2026). r-deformed α-z-Rényi relative entropy. URL: https://arxiv.org/abs/2607.01805v1.

Commentary.

Definition 2, printed p. 2: “The r-deformed α-z-Rényi relative entropy for any two positive semi-definite operators ρ and σ with Supp(ρ) ⊆ Supp(σ) is defined by D^{(r)}_{α,z}(ρ||σ) = (1/(α−1)) ln_r Tr[(σ^{(1−α)/(2z)} ρ^{α/z} σ^{(1−α)/(2z)})^z] … where α, r ∈ R and z ∈ R+.” The source calls this quantity B₁. Here n is the dimension, rho and sigma are complex matrices, alpha denotes α, cfc evaluates each real function of a Hermitian matrix, and re takes the real part of its trace.

Definition 1.3 (The Tsallis relative entropy).

$$\forall (n:\mathbb{N}),\forall (rho:\mathrm{Matrix}\left(\mathrm{Fin}\left(n\right),\mathrm{Fin}\left(n\right),\mathbb{C}\right)),\forall (sigma:\mathrm{Matrix}\left(\mathrm{Fin}\left(n\right),\mathrm{Fin}\left(n\right),\mathbb{C}\right)),\forall (alpha:\mathbb{R}),\mathrm{tsallis}\left(alpha,rho,sigma\right)=(\frac{(\mathrm{Matrix}.\mathrm{trace}((rho)-((\mathrm{cfc}\left(((x:\mathbb{R})\mapsto x^{alpha}),rho\right))\cdot (\mathrm{cfc}\left(((x:\mathbb{R})\mapsto x^{(1)-(alpha)}),sigma\right))))).\mathrm{re}}{(1)-(alpha)})$$

Formalization. D5/S3/Divergence/MaityFuruichiDuttaOrderingRefutation.tsallis (✓ std3).

Citation. Srikrishna Maity; Shigeru Furuichi; Supriyo Dutta (2026). r-deformed α-z-Rényi relative entropy. URL: https://arxiv.org/abs/2607.01805v1.

Commentary.

Equation (6), printed p. 2, gives D_α(ρ||σ) = Tr[ρ − ρ^α σ^{1−α}]/(1 − α). The displayed expression uses cfc for the two matrix powers and the real part of the trace.

Definition 1.4 (The second upper bound).

$$\forall (n:\mathbb{N}),\forall (rho:\mathrm{Matrix}\left(\mathrm{Fin}\left(n\right),\mathrm{Fin}\left(n\right),\mathbb{C}\right)),\forall (sigma:\mathrm{Matrix}\left(\mathrm{Fin}\left(n\right),\mathrm{Fin}\left(n\right),\mathbb{C}\right)),\forall (alpha:\mathbb{R}),\forall (q:\mathbb{R}),B2\left(alpha,q,rho,sigma\right)=(-((\mathrm{Matrix}.\mathrm{trace}((rho)\cdot (\mathrm{cfc}\left(((x:\mathbb{R})\mapsto \mathrm{lnr}\left((1)-(alpha),x\right)),\mathrm{cfc}\left(((x:\mathbb{R})\mapsto x^{\frac{1}{q}}),((\mathrm{cfc}\left(((x:\mathbb{R})\mapsto x^{-(\frac{q}{2})}),rho\right))\cdot (\mathrm{cfc}\left(((x:\mathbb{R})\mapsto x^q),sigma\right)))\cdot (\mathrm{cfc}\left(((x:\mathbb{R})\mapsto x^{-(\frac{q}{2})}),rho\right))\right)\right)))).\mathrm{re}))$$

Formalization. D5/S3/Divergence/MaityFuruichiDuttaOrderingRefutation.B2 (✓ std3).

Citation. Srikrishna Maity; Shigeru Furuichi; Supriyo Dutta (2026). r-deformed α-z-Rényi relative entropy. URL: https://arxiv.org/abs/2607.01805v1.

Commentary.

Equations (8) and (47), printed pp. 2 and 11, give B₂(ρ||σ) = −Tr[ρ ln_{1−α}((ρ^{−q/2} σ^q ρ^{−q/2})^{1/q})]. Each real matrix function is evaluated by cfc, with the negative sign outside the real trace.

Definition 1.5 (The source’s proposed ordering).

$$claim\iff (\forall (n:\mathbb{N}),\forall (rho:\mathrm{Matrix}\left(\mathrm{Fin}\left(n\right),\mathrm{Fin}\left(n\right),\mathbb{C}\right)),\forall (sigma:\mathrm{Matrix}\left(\mathrm{Fin}\left(n\right),\mathrm{Fin}\left(n\right),\mathbb{C}\right)),((rho).\mathrm{PosDef})\Rightarrow (((rho).\mathrm{trace}=(1))\Rightarrow (((sigma).\mathrm{PosDef})\Rightarrow (((sigma).\mathrm{trace}=(1))\Rightarrow (((rho)\cdot (sigma)\ne (sigma)\cdot (rho))\Rightarrow (\forall (alpha:\mathbb{R}),\forall (r:\mathbb{R}),\forall (z:\mathbb{R}),\forall (q:\mathbb{R}),(0\le alpha)\Rightarrow ((alpha<1)\Rightarrow ((r\ne 0)\Rightarrow ((0<z)\Rightarrow (((1)-(alpha)<q)\Rightarrow ((\mathrm{tsallis}\left(alpha,rho,sigma\right)\le \mathrm{renyiRZ}\left(alpha,r,z,rho,sigma\right))\wedge (\mathrm{renyiRZ}\left(alpha,r,z,rho,sigma\right)\le B2\left(alpha,q,rho,sigma\right)))))))))))))$$

Formalization. D5/S3/Divergence/MaityFuruichiDuttaOrderingRefutation.claim (✓ std3).

Citation. Srikrishna Maity; Shigeru Furuichi; Supriyo Dutta (2026). r-deformed α-z-Rényi relative entropy. URL: https://arxiv.org/abs/2607.01805v1.

Commentary.

Equations (8)–(9), printed p. 2: “For 0 ≤ α < 1 and q > (1 − α), we compare two upper bounds B₁(ρ||σ) = D^{(r)}{α,z}(ρ||σ) and B₂(ρ||σ) = −Tr[ρ ln{1−α}((ρ^{−q/2} σ^q ρ^{−q/2})^{1/q})] of the Tsallis relative entropy, numerically. We observe that for non-commutative density operators ρ and σ, D_α(ρ||σ) ≤ B₁(ρ||σ) ≤ B₂(ρ||σ).” Section 4, Case 5, printed p. 12: “Let Tr(ρ) = 1 and Tr(σ) = 1, that is both ρ and σ are density operators. Here, we numerically observe that for any two operators ρ and σ, B₁(ρ||σ) is a tighter bound of Tsallis relative entropy than B₂(ρ||σ).” The formula ranges over every finite dimension. Positive definite matrices with unit trace satisfy the support condition and admit negative powers. The parameters r≠0 and z>0 come from Definitions 1–2. The function renyiRZ is precisely the source’s B₁.

Theorem 1.6 (A noncommuting qubit counterexample).

$$\neg (\forall (n:\mathbb{N}),\forall (rho:\mathrm{Matrix}\left(\mathrm{Fin}\left(n\right),\mathrm{Fin}\left(n\right),\mathbb{C}\right)),\forall (sigma:\mathrm{Matrix}\left(\mathrm{Fin}\left(n\right),\mathrm{Fin}\left(n\right),\mathbb{C}\right)),((rho).\mathrm{PosDef})\Rightarrow (((rho).\mathrm{trace}=(1))\Rightarrow (((sigma).\mathrm{PosDef})\Rightarrow (((sigma).\mathrm{trace}=(1))\Rightarrow (((rho)\cdot (sigma)\ne (sigma)\cdot (rho))\Rightarrow (\forall (alpha:\mathbb{R}),\forall (r:\mathbb{R}),\forall (z:\mathbb{R}),\forall (q:\mathbb{R}),(0\le alpha)\Rightarrow ((alpha<1)\Rightarrow ((r\ne 0)\Rightarrow ((0<z)\Rightarrow (((1)-(alpha)<q)\Rightarrow ((\mathrm{tsallis}\left(alpha,rho,sigma\right)\le \mathrm{renyiRZ}\left(alpha,r,z,rho,sigma\right))\wedge (\mathrm{renyiRZ}\left(alpha,r,z,rho,sigma\right)\le B2\left(alpha,q,rho,sigma\right)))))))))))))$$

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

Resolves. Problems/maity-furuichi-dutta-2026-tsallis-ordering-case-5 (refuted) by D5/S3/Divergence/MaityFuruichiDuttaOrderingRefutation.result.

Source. Repository-derived.

Acknowledgement. Srikrishna Maity; Shigeru Furuichi; Supriyo Dutta (2026). r-deformed α-z-Rényi relative entropy. URL: https://arxiv.org/abs/2607.01805v1.

Commentary.

Take rho = diag(4/5,1/5) and sigma = (1/125)[[28,6√6],[6√6,97]]. Both matrices are positive definite with unit trace, and they do not commute. At alpha = r = 1/2, z = 2 and q = 1, the proof gives 2/5 < renyiRZ(alpha,r,z,rho,sigma) and B2(alpha,q,rho,sigma) < 2/5. For the first value the trace inside the deformed logarithm is (2117+288√2)/3125, so B₁ = 4(1−√((2117+288√2)/3125)). For the second, put A = (1/25)[[7,3√6],[3√6,97]]. Its positive square root is (5/√154)(A+1), giving B₂ = 2−20/√154. These inequalities contradict the second conjunct of the proposed order. The inequality tsallis≤renyiRZ is not refuted here.

References

• Truth anchor: D5/S3/Divergence/MaityFuruichiDuttaOrderingRefutation.B2
• Truth anchor: D5/S3/Divergence/MaityFuruichiDuttaOrderingRefutation.claim
• Truth anchor: D5/S3/Divergence/MaityFuruichiDuttaOrderingRefutation.lnr
• Truth anchor: D5/S3/Divergence/MaityFuruichiDuttaOrderingRefutation.renyiRZ
• Truth anchor: D5/S3/Divergence/MaityFuruichiDuttaOrderingRefutation.result
• Truth anchor: D5/S3/Divergence/MaityFuruichiDuttaOrderingRefutation.tsallis
• Dependency: D5/S3/Quantum/Matrix/TwoPointFunctionalCalculus