Linear Algebra and its Applications ( IF 1.0 ) Pub Date : 2021-08-18 , DOI: 10.1016/j.laa.2021.08.012 Dariusz Chruściński 1 , Ryohei Fujii 2 , Gen Kimura 2 , Hiromichi Ohno 3
Motivated by a spectral analysis of the generator of a completely positive trace-preserving semigroup, we analyze the real functional where is the Hilbert-Schmidt inner product, and is the commutator. In particular we discuss upper and lower bounds of the form where is the Frobenius norm. We prove that the optimal upper and lower bounds are given by . If A is restricted to be traceless, the bounds are further improved to be . 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.
中文翻译:
量子动力学半群发生器谱的约束
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 在哪里 是希尔伯特-施密特内积,并且 是换向器。我们特别讨论形式的上限和下限 在哪里 是 Frobenius 范数。我们证明最优上下界由下式给出. 如果A被限制为无迹,则边界进一步改进为. 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。