arXiv 2106.080162021-06math-phmath.MP

量子动力学半群生成元谱的约束

Constraints for the spectra of generators of quantum dynamical semigroups

Dariusz Chruscinski, Ryohei Fujii, Gen Kimura, Hiromichi Ohno

备注：: 19 pages
发表信息：: Linear Algebra and its Applications, 630, 293-305 (2021)
DOI：: 10.1016/j.laa.2021.08.012

摘要：

受完全正保迹半群生成元的谱分析的启发，我们分析了实泛函$$ A,B \in M_n(\mathbb{C}) \to r(A,B) = \frac{1}{2}\Bigl(\langle [B,A],BA\rangle + \langle [B,A^\ast],BA^\ast \rangle \Bigr) \in \mathbb{R} $$，其中$\langle A,B\rangle := {\rm tr} (A^\ast B)$是Hilbert-Schmidt内积，$[A,B]:= AB - BA$是交换子.特别是我们讨论的形式$c_- \|A\|^2 \|B\|^2 \le r(A,B) \le c_+ \|A\|^2 \|B\|^2$的上界和下界，其中$\|A\|$是Frobenius范数。我们证明了最优上界和下界由$c_\pm = \frac{1 \pm \sqrt{2}}{2}$给出。如果$A$被限制为无迹的，则边界被进一步改进为$c_\pm = \frac{1 \pm \sqrt{2(1-\frac{1}{n})}}{2}$。有趣的是，这些上界，特别是后者，提供了新的约束松弛速率的量子动力学半群比以前已知的约束在文献中更严格。并讨论了与Böttcher-Wenzel不等式的关系。

英文摘要：

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 Böttcher-Wenzel inequality is also discussed.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文