Constraints for the spectra of generators of quantum dynamical semigroups

Abstract Motivated by a spectral analysis of the generator of a completely positive trace-preserving semigroup, we analyze the real functional A , B ∈ M n ( C ) → r ( A , B ) = 1 2 ( 〈 [ B , A ] , B A 〉 + 〈 [ B , A ⁎ ] , B A ⁎ 〉 ) ∈ R where 〈 A , B 〉 : = tr ( A ⁎ B ) is the Hilbert-Schmidt inner product, and [ A , B ] : = A B − B A 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 c ± = 1 ± 2 2 . If A is restricted to be traceless, the bounds are further improved to be c ± = 1 ± 2 ( 1 − 1 n ) 2 . 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 Bottcher-Wenzel inequality is also discussed.