Constraints for the spectra of generators of quantum dynamical semigroups

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.