酉量子信道中希尔伯特 - 施密特速度的压缩性：见证非马尔可夫性以及区分酉与非酉马尔可夫动力学的基础
Contractivity of the Hilbert--Schmidt Speed and Unitality--Divisibility-Based Witnesses in Finite-Dimensional Quantum Dynamics
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中文总结 AI 辅助
研究酉量子信道中希尔伯特 - 施密特速度（HSS），证明其在酉CPTP映射下压缩，为相关物理环境用HSS见证非马尔可夫性提供基础，同时指出非酉动力学中HSS可能增加，明确其诊断范围及酉性对其有效性的关键作用。
中文摘要 AI 辅助
我们研究了希尔伯特 - 施密特速度（HSS），它是通过量子态参数化族的切向量的希尔伯特 - 施密特范数定义的几何指标，处于一般开放系统动力学下。在有限维、与参数无关的完全正定保迹（CPTP）演化框架中，参数仅编码在初始状态，我们证明了HSS在每个酉CPTP映射下是压缩的。因此，对于任何中间传播子为酉的CP可分演化，HSS在时间上单调非增。然后我们建立了由具有厄米林德布拉德算子的戈里尼 - 科萨克夫斯基 - 苏达山 - 林德布拉德（GKSL）主方程所支配的马尔可夫动力学的生成器层面的对应物，得出了HSS平方的时间导数的明确非正表达式。这些结果为在相关CP可分马尔可夫动力学已知先验为酉的物理环境中，使用HSS回流作为非马尔可夫性的充分见证提供了严格基础。相反，我们通过一个明确的三量子比特反例表明，即使在完全马尔可夫但非酉的动力学中HSS也可能增加，这表明一般来说，除非保证酉性，HSS的非单调性不是记忆效应的可靠指标。我们的发现阐明了基于HSS诊断的精确范围，并确定酉性是其有效性背后的关键结构要素。
英文摘要
We investigate the Hilbert--Schmidt speed as a witness of non-Markovianity in finite-dimensional quantum dynamics. For a differentiable one-parameter family of quantum states, the Hilbert--Schmidt speed is defined, up to a conventional factor, as the Hilbert--Schmidt norm of the corresponding Hermitian traceless tangent operator. We show that, for unital dynamics in arbitrary finite dimension, P-divisibility implies the monotonic decrease of the Hilbert--Schmidt speed. Hence, any increase in this quantity signals the breakdown of P-divisibility, and therefore also excludes CP-divisibility. For qubit systems, we establish a stronger result: every P-divisible evolution decreases the Hilbert--Schmidt speed, independently of unitality. Thus, in dimension two, HSS growth is a valid witness of non-Markovian behaviour for arbitrary positive divisible dynamics. This conclusion is dimension dependent. In dimensions $d\geq3$, non-unitality can generate HSS growth even under CP-divisible dynamics. We demonstrate this by constructing an explicit non-unital CP-divisible qutrit semigroup for which the Hilbert--Schmidt speed strictly increases. Finally, for unital GKSL dynamics, we derive a generator-level dissipation identity explaining the monotonic decay of the Hilbert--Schmidt speed. These results specify the regimes in which HSS growth can be interpreted as evidence of the failure of divisibility and clarify its limitation for higher-dimensional non-unital evolutions.
