bibkey: mayumi2024weightedbw authors: Aina Mayumi; Gen Kimura; Hiromichi Ohno; Dariusz Chruściński year: 2024 title: “Böttcher-Wenzel inequality for weighted Frobenius norms and its application to quantum physics” doi: 10.1016/j.laa.2024.07.013 url: https://arxiv.org/abs/2403.04199v2 claim: “Conjecture 1, case (ii), equation (15), bounds the weighted commutator norm by sqrt((lambda_min + lambda_max)/lambda_min) times the weighted norm of A and the Frobenius norm of B.” strata_touched:

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Böttcher-Wenzel inequality for weighted Frobenius norms and its application to quantum physics

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DOI: https://doi.org/10.1016/j.laa.2024.07.013

Source: https://arxiv.org/abs/2403.04199v2

The weighted norm is defined on page 2, equation (2):

In what follows we call ω-weighted Frobenius norm

$$\Vert A{\Vert }_{\omega }:=\sqrt{\mathrm{tr}(A^{\ast }A\omega )}.$$

Conjecture 1 (page 6; cases (i) and (ii)):

Conjecture 1. For any matrices $A,B\in M_n(\mathbb{C})$,

$$\Vert [A,B]{\Vert }_{\omega }\le \sqrt{\frac{{\lambda }_m+{\lambda }_{sm}}{{\lambda }_m{\lambda }_{sm}}}\Vert A{\Vert }_{\omega }\Vert B{\Vert }_{\omega },\text{\hspace{1em}(14)}$$

$$\Vert [A,B]{\Vert }_{\omega }\le \sqrt{\frac{{\lambda }_m+{\lambda }_M}{{\lambda }_m}}\Vert A{\Vert }_{\omega }\Vert B\Vert ,\text{\hspace{1em}(15)}$$

and on page 7:

and (as a corollary of (14))

$$\Vert [A,B]\Vert \le \sqrt{\frac{{\lambda }_m+{\lambda }_{sm}}{{\lambda }_m^2{\lambda }_{sm}}}\Vert A{\Vert }_{\omega }\Vert B{\Vert }_{\omega }.\text{\hspace{1em}(16)}$$

All bounds are tight, i.e., there are non-zero matrices A and B that attain the equalities.

Here $\omega$ is positive definite, ${\lambda }_m$ and ${\lambda }_M$ are the smallest and largest eigenvalues, and ${\lambda }_{sm}$ is the second-smallest eigenvalue. Equation (15) is case (ii); cases (i) and (iv) are equations (14) and (16). The settling claim is equation (15) squared. Its sharpness example is $A=|{\lambda }_M⟩⟨{\lambda }_m|$, $B=A^{\ast }$ (page 7). The relaxation-rate application is equation (28) in Section 3.2, conditional there on equation (15).