Memory is the fundamental form of temporal complexity: when present but uncontrollable, it manifests as non-Markovian noise; conversely, if controllable, memory can be a powerful resource for information processing. Memory effects arise from/are transmitted via interactions between a system and its environment; as such, they can be either classical or quantum. From a practical standpoint, quantum processes with classical memory promise near-term applicability: they are more powerful than their memoryless counterpart, yet at the same time can be controlled over significant timeframes without being spoiled by decoherence. However, despite practical and foundational value, apart from simple two-time scenarios, the distinction between quantum and classical memory remains unexplored. Here, we analyse multi-time quantum processes with memory mechanisms that transmit only classical information forward in time. Complementing this analysis, we also study two related – but simpler to characterise – sets of processes that could also be considered to have classical memory from a structural perspective, and demonstrate that these lead to remarkably distinct phenomena in the multi-time setting. Subsequently, we systematically stratify the full hierarchy of memory effects in quantum mechanics, many levels of which collapse in the two-time setting, making our results genuinely multi-time phenomena.

中文翻译：

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用经典记忆表征多时间量子过程的层次结构

记忆是时间复杂性的基本形式：当存在但不可控时，它表现为非马尔可夫噪声；相反，如果可控，内存可以成为信息处理的强大资源。记忆效应由系统与其环境之间的相互作用产生/传播；因此，它们可以是经典的，也可以是量子的。从实用的角度来看，具有经典记忆的量子过程有望在短期内适用：它们比无记忆的对应物更强大，但同时可以在重要的时间范围内进行控制，而不会被退相干破坏。然而，尽管具有实用和基础价值，除了简单的两次场景之外，量子记忆和经典记忆之间的区别仍未被探索。在这里，我们分析了具有仅及时向前传输经典信息的记忆机制的多时间量子过程。作为对这一分析的补充，我们还研究了两组相关的（但更容易表征）过程，从结构的角度来看，它们也可以被认为具有经典记忆，并证明这些过程会在多时间环境中导致明显不同的现象。随后，我们系统地对量子力学中记忆效应的完整层次进行了分层，其中许多层次在两次设置中崩溃，使我们的结果真正成为多时间现象。