Constraints for the spectra of generators of quantum dynamical semigroups

TL;DR

This paper establishes optimal bounds for a specific real functional related to quantum generator spectra, revealing new constraints on relaxation rates in quantum dynamical semigroups and connecting to known inequalities.

Contribution

It derives the exact optimal bounds for a functional involving commutators, improving understanding of spectral constraints in quantum dynamical semigroups.

Findings

Optimal bounds c_± = (1 ± √2)/2 for the functional r(A,B).

Improved bounds when A is traceless, c_± = (1 ± √{2(1-1/n)})/2.

New constraints on relaxation rates tighter than previous results.

Abstract

Motivated by a spectral analysis of the generator of completely positive trace-preserving semigroup, we analyze a real functional $$ A,B \in M_n(\mathbb{C}) \to r(A,B) = \frac{1}{2}\Bigl(\langle [B,A],BA\rangle + \langle [B,A^\ast],BA^\ast \rangle \Bigr) \in \mathbb{R} $$ where $\langle A,B\rangle := {\rm tr} (A^\ast B)$ 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 $c_- \|A\|^2 \|B\|^2 \le r(A,B) \le c_+ \|A\|^2 \|B\|^2$ where $\|A\|$ 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 B\"{o}ttcher-Wenzel inequality is also discussed.