Constraints for the spectra of generators of quantum dynamical semigroups

Abstract

Motivated by a spectral analysis of the generator of completely positive trace-preserving semigroup, we analyze a real functional $$A,B\in M_n(\mathbb{C})\to r(A,B)=\frac{1}{2}(⟨[B,A],BA⟩+⟨[B,A^{\ast }],B{A}^{\ast }⟩)\in \mathbb{R}$$ where $⟨A,B⟩:=\mathrm{tr}(A^{\ast }B)$ is the Hilbert-Schmidt inner product, and $[A,B]:=AB-BA$ is the commutator. In particular we discuss the upper and lower bounds of the form $c_{-}\Vert A{\Vert }^2\Vert B{\Vert }^2\le r(A,B)\le c_{+}\Vert A{\Vert }^2\Vert B{\Vert }^2$ where $\Vert A\Vert$ is the Frobenius norm. We prove that the optimal upper and lower bounds are given by $c_{\pm }=\frac{1\pm \sqrt{2}}{2}$. If $A$ is restricted to be traceless, the bounds are further improved to be $c_{\pm }=\frac{1\pm \sqrt{2(1-\frac{1}{n})}}{2}$. 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 Böttcher-Wenzel inequality is also discussed.

19 pages

References in corpus (1)

• Universal Constraints on Relaxation Times for d-level GKLS master equations