A traceless counterexample to the q-deformed commutator bound

Abstract

Two traceless complex matrices of size five violate the proposed q-deformed commutator inequality at q = 2.

Definition 1.1 (The inequality in Conjecture 1).

$$claim⟺\forall n:\mathbb{N},\forall q:\mathbb{R},0<q\to \forall AB:Matrix(Finn)(Finn)\mathbb{C},(\mathrm{trace}\left(A\right)=0\vee \mathrm{trace}\left(B\right)=0)\to \mathrm{mass}\left(AB-q\cdot (BA)\right)\le (1+q^2)\cdot \mathrm{mass}\left(A\right)\cdot \mathrm{mass}\left(B\right)$$

Formalization. D5/S3/QuantumBounds/QDeformedCommutatorTracelessRefutation.claim (✓ std3).

Citation. D. Chruściński, G. Kimura, H. Ohno, T. Singal (2022). Bounding the Frobenius norm of a q-deformed commutator. DOI: 10.48550/arXiv.2202.11520. URL: https://arxiv.org/abs/2202.11520v2.

Commentary.

Chruściński, Kimura, Ohno and Singal define the q-deformed commutator by [A,B]q = AB − qBA. Conjecture 1 in Section 3 states: “For any q > 0, if A or B is traceless, the inequality (7) holds and is sharp.” Here mass M = ∑ᵢ ∑ⱼ |M i j|² is the squared Frobenius norm, using the existing definition from the Moreau–Yosida module. Fin n indexes the rows and columns by 0,…,n−1. The real scalar q is embedded in C for scalar multiplication of matrices. The displayed claim is the inequality clause for every dimension n, every positive real q and every pair of complex matrices satisfying the disjunction of trace conditions. A violation of this clause also refutes the conjunction that the inequality holds and is sharp.

Theorem 1.2 (A five-dimensional counterexample).

$$\neg claim$$

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

Resolves. Problems/chruscinski-2022-qdeformed-commutator-traceless-refutation (refuted) by D5/S3/QuantumBounds/QDeformedCommutatorTracelessRefutation.result.

Source. Repository-derived.

Acknowledgement. D. Chruściński, G. Kimura, H. Ohno, T. Singal (2022). Bounding the Frobenius norm of a q-deformed commutator. DOI: 10.48550/arXiv.2202.11520. URL: https://arxiv.org/abs/2202.11520v2.

Commentary.

Take q = 2 and n = 5. Let A = [6,42; 0,−3] ⊕ (−I₃) and B = [6,0; −42,−3] ⊕ (−I₃), where the bracketed arrays are two-by-two matrices and I₃ is the three-by-three identity. Both traces are 6 − 3 − 1 − 1 − 1 = 0. Their squared Frobenius norms are mass A = mass B = 1812. Direct multiplication gives AB − 2BA = [−1800,−630; 630,3519] ⊕ (−I₃), with mass (AB − 2BA) = 16417164. The proposed upper bound is (1 + 2²) · mass A · mass B = 5 · 1812² = 16416720. Thus 16417164 > 16416720, exceeding the bound by 444 and contradicting claim.

References

• Truth anchor: D5/S3/QuantumBounds/QDeformedCommutatorTracelessRefutation.claim
• Truth anchor: D5/S3/QuantumBounds/QDeformedCommutatorTracelessRefutation.result
• Dependency: D5/S3/Quantum/Entanglement/MoreauYosidaFormationSelectiveLoccRefutation