B\"ottcher-Wenzel inequality for weighted Frobenius norms and its application to quantum physics

By employing a weighted Frobenius norm with a positive matrix ω, we introduce natural generalizations of the famous B\"ottcher-Wenzel (BW) inequality. Specifically, we explore six types of bounds, labeled (i) through (vi), on the norms of the commutator $[A,B]:= AB - BA$, based on the combination of the weighted Frobenius norm \|A\|_ω := √ tr(A^ A ω) and the usual Frobenius norm \|A\| := √ tr(A^ A). While the tight bound for the case (vi) corresponds to the BW inequality itself, we establish the tight bounds for cases (iii) and (v), and propose conjectures for the tight bounds of cases (i) and (ii), with the tight bound for case (iv) presented as a corollary of case (i). Conversely, all these bounds (i)-(v) serve as generalizations of the BW inequality. The conjectured bounds for cases (i) and (ii) are numerically supported for matrices up to size $n=15$, and we provide proofs for 2× 2 matrices. Additionally, we present applications of these bounds in quantum physics, particularly in the contexts of the uncertainty relation and open quantum dynamics.