Motivated by a spectral analysis of the generator of a completely positive trace-preserving semigroup, we analyze the real functional$A,B\in M_n(\mathbb{C})\to r(A,B)=\frac{1}{2}(〈[B,A],BA〉+〈[B,A^{⁎}],B{A}^{⁎}〉)\in \mathbb{R}$ where $〈A,B〉:=\mathrm{tr}(A^{⁎}B)$ 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 $c_{-}{\Vert A\Vert }^2{\Vert B\Vert }^2\le r(A,B)\le c_{+}{\Vert A\Vert }^2{\Vert B\Vert }^2$ 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 the Böttcher-Wenzel inequality is also discussed.

受完全正迹保留半群发生器的谱分析的启发，我们分析了真实泛函$\mathrm{一种},乙\in {米}_n(\mathbb{C})\to r(\mathrm{一种},乙)=\frac{1}{2}(〈[乙,\mathrm{一种}],乙\mathrm{一种}〉+〈[乙,{\mathrm{一种}}^{⁎}],乙{\mathrm{一种}}^{⁎}〉)\in \mathbb{电阻}$ 在哪里 $〈\mathrm{一种},乙〉：=\mathrm{tr}({\mathrm{一种}}^{⁎}乙)$ 是希尔伯特-施密特内积，并且 $[\mathrm{一种},乙]：=\mathrm{一种}乙-乙\mathrm{一种}$是换向器。我们特别讨论形式的上限和下限$C_{-}{\Vert \mathrm{一种}\Vert }^2{\Vert 乙\Vert }^2\le r(\mathrm{一种},乙)\le C_{+}{\Vert \mathrm{一种}\Vert }^2{\Vert 乙\Vert }^2$ 在哪里 $\Vert \mathrm{一种}\Vert$是 Frobenius 范数。我们证明最优上下界由下式给出$C_{\pm }=\frac{1\pm \sqrt{2}}{2}$. 如果A被限制为无迹，则边界进一步改进为$C_{\pm }=\frac{1\pm \sqrt{2(1-\frac{1}{n})}}{2}$. 有趣的是，这些上限，尤其是后者，为量子动力学半群的弛豫率提供了新的约束，比以前文献中已知的约束更严格。还讨论了与 Böttcher-Wenzel 不等式的关系。