Motivated by a spectral analysis of the generator of a completely positive trace-preserving semigroup, we analyze the real functional [Formula presented] where 〈A,B〉:=tr(A⁎B) is the Hilbert-Schmidt inner product, and [A,B]:=AB−BA is the commutator. In particular we discuss upper and lower bounds of the form c−‖A‖2‖B‖2≤r(A,B)≤c+‖A‖2‖B‖2 where ‖A‖ is the Frobenius norm. We prove that the optimal upper and lower bounds are given by [Formula presented]. If A is restricted to be traceless, the bounds are further improved to be [Formula presented]. Interestingly, these upper bounds, especially the latter one, provide new constraints on relaxation rates for the quantum dynamical semigroup tighter than previously known constraints in the literature. A relation with the Böttcher-Wenzel inequality is also discussed.
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