标题： 量子马尔可夫生成元的松弛率的普适约束：完全正性和超越 显示英文标题

标题： A universal constraint for relaxation rates for quantum Markov generators: complete positivity and beyond

摘要： 松弛率是量子过程的关键特征，因为它们决定了一个量子系统热化、平衡、去相干和耗散的速度。 尽管它们在理论分析中起着至关重要的作用，但松弛率也常常可以通过实验测量直接获得。 最近的研究表明，对于由马尔可夫半群支配的量子过程，松弛率满足一个普适约束：最大速率被希尔伯特空间维数限制在所有速率之和的范围内。 这一界限最初几年前被猜测，直到最近才通过经典李雅普诺夫理论得到证明。 在这项工作中，我们提出了这个约束的一个新的、纯代数的证明方法。 值得注意的是，我们的方法不仅更加直接，还允许自然推广到完全正半群之外的情况。 我们展示了完全正性可以放松到2-正性而不影响约束的有效性。 这揭示了该界限比之前理解的更为微妙：2-正性是必要的，但即使进一步放松到施瓦茨映射，仍然存在一个略微更弱但仍然非平凡的普适约束。 最后，我们探讨了这些界限与量子过程中稳态数量之间的联系，揭示了它们行为背后更深层次的结构。 显示英文摘要

摘要： Relaxation rates are key characteristics of quantum processes, as they determine how quickly a quantum system thermalizes, equilibrates, decoheres, and dissipates. While they play a crucial role in theoretical analyses, relaxation rates are also often directly accessible through experimental measurements. Recently, it was shown that for quantum processes governed by Markovian semigroups, the relaxation rates satisfy a universal constraint: the maximal rate is upper-bounded by the sum of all rates divided by the dimension of the Hilbert space. This bound, initially conjectured a few years ago, was only recently proven using classical Lyapunov theory. In this work, we present a new, purely algebraic proof of this constraint. Remarkably, our approach is not only more direct but also allows for a natural generalization beyond completely positive semigroups. We show that complete positivity can be relaxed to 2-positivity without affecting the validity of the constraint. This reveals that the bound is more subtle than previously understood: 2-positivity is necessary, but even when further relaxed to Schwarz maps, a slightly weaker -- yet still non-trivial -- universal constraint still holds. Finally, we explore the connection between these bounds and the number of steady states in quantum processes, uncovering a deeper structure underlying their behavior.