bibkey: chruscinski2022qdeformed authors: D. Chruściński, G. Kimura, H. Ohno, T. Singal year: 2022 title: “Bounding the Frobenius norm of a q-deformed commutator” doi: 10.48550/arXiv.2202.11520 url: https://arxiv.org/abs/2202.11520v2 claim: “For q > 0 the bound ‖AB − qBA‖²_F ≤ (1 + q²)‖A‖²_F‖B‖²_F holds when A or B is normal and can fail in general; Conjecture 1 asserts that it holds and is sharp when A or B is traceless, proved there for n = 2; Conjecture 2 asserts for q ≤ 0, A or B traceless, the sharp bound ‖[A,B]_q‖²_F ≤ max[g(n)(1−q)², 1+q²]‖A‖²_F‖B‖²_F with g(n) = (n²−3n+3)/(n(n−1)).” strata_touched:

• D5/S3/QuantumBounds/QDeformedCommutatorTracelessRefutation
• D5/S3/QuantumBounds/QDeformedCommutatorNegativeQRefutation license: citation-only triage: anchor

Chruściński, Kimura, Ohno and Singal, Frobenius norm of a q-deformed commutator

D. Chruściński, G. Kimura, H. Ohno and T. Singal, Bounding the Frobenius norm of a q-deformed commutator, arXiv:2202.11520v2 (18 March 2022; math.QA, cross-listed math-ph); Linear Algebra Appl. 646 (2022) 95–106.

Verified locator

DOI: 10.48550/arXiv.2202.11520. Primary version: https://arxiv.org/abs/2202.11520v2 (the latest arXiv version). The TeX source arxiv.tex of v2 supplies the definition of the q-deformed commutator (§1), Eq. (7), Proposition 1 (the normal case), Conjecture 1, Proposition 3, Eq. (24) and Conjecture 2 (§3, arxiv.tex lines 357–361, 483–485 and 503–509).

Source statements

Definition (§1): “$[A,B{]}_q:=AB-qBA$ for a real parameter $q$”, with the Frobenius norm $\Vert A{\Vert }_F^2=\mathrm{tr}(A^{†}A)$.

Eq. (7): “\f{[A,B]_q}^2\leq\left(1+q^2 \right)\f{A}^2\f{B}^2”, proved “if either $A$ or $B$ is normal”, and “\eqref{eq:q_commutator_BW} can be violated if neither $A$ nor $B$ are normal.”

Conjecture 1: “For any $q>0$, if $A$ or $B$ is traceless, the inequality \eqref{eq:q_commutator_BW} holds and is sharp.” Followed by “we provide a proof of Conjecture \ref{CON} for $n=2$ for any real $q$.”

Proposition 3: “For $n=2$ and \tr{A}=0, the bound in Eq.~ \eqref{eq:q_commutator_BW} is satisfied and is sharp.”

Eq. (24): “$g(n)=\frac{n^2-3n+3}{n(n-1)}$”.

Conjecture 2: “For any $q\le 0$, if $A$ or $B$ is traceless, the sharp bound is {\f{[A,B]_q}^2} \ \leq \ \max[g(n)(1-q)^2, 1+q^2] \ \f{A}^2 \ \f{B}^2, where $g(n)$ is given by \eqref{eq:gn}.”

Scope

The paper proves Conjecture 1 for $2\times 2$ matrices and supports it numerically in higher dimensions; the counterexample to the general form of (7) is a non-normal $2\times 2$ pair with nonzero traces. The value $g(n)(1-q{)}^2$ in Conjecture 2 is attained by $A=B=\mathrm{diag}(n-1,-1,\dots ,-1)$, both traceless.