Constraints for the spectra of generators of quantum dynamical semigroups

Motivated by a spectral analysis of the generator completely positive trace-preserving semigroup, we analyze real functionalA,B∈Mn(C)→r(A,B)=12(〈[B,A],BA〉+〈[B,A⁎],BA⁎〉)∈R where 〈A,B〉:=tr(A⁎B) is Hilbert-Schmidt inner product, and [A,B]:=AB−BA commutator. In particular discuss upper lower bounds form c−‖A‖2‖B‖2≤r(A,B)≤c+‖A‖2‖B‖2 ‖A‖ Frobenius norm. We prove that optimal are given c±=1±22. If A restricted to be traceless, further improved c±=1±2(1−1n)2. Interestingly, these bounds, especially latter one, provide new constraints on relaxation rates for quantum dynamical semigroup tighter than previously known in literature. relation with Böttcher-Wenzel inequality also discussed.