AI 中文总结
本文研究有限内存下计数事件的计时精度问题，推导经典d态计数过程的有限内存方差界，识别出违反该界的量子计数过程，确立了时序精度的有限内存量子优势，连接了离散与连续时间计时过程。
AI 中文摘要
计数过程为从光子探测到时钟滴答等随机事件提供了基础描述。核心问题是当仅有有限内存资源可用时，这类事件的计时精度能达到多少。本文在有限维经典与量子计数过程的通用框架下研究该问题。我们推导了每个经典d态计数过程都遵循的严格有限内存方差界，该界是紧的，且被离散Erlang型阶梯过程饱和。通过数值优化，我们识别出违反该经典界的量子计数过程，在相同内存大小和平均滴答时间下，其首次滴答波动比任何经典过程都小。对于量子比特情况，我们进一步在单Kraus无滴答族中推导了解析的大平均界，表明量子优势在该族内渐近存在。优化后的量子过程表现出相干条件动力学，且随平均时间增加趋近于连续时间量子跃迁描述。我们的结果确立了时序精度方面的有限内存量子优势，并将离散时间计数过程与连续时间量子计时联系起来。
英文摘要
Counting processes provide a fundamental description of stochastic events ranging from photon detection to clock ticks. A central question is how accurately such events can be timed when only finite memory resources are available. Here, we investigate this problem within a general framework of finite-dimensional classical and quantum counting processes. We derive a rigorous finite-memory variance bound obeyed by every classical $d$-state counting process, which is tight and saturated by a discrete Erlang-type ladder process. Through numerical optimization, we identify quantum counting processes that violate this classical bound, achieving smaller first-tick fluctuations than any classical process with the same memory size and mean tick time. For the qubit case, we further derive an analytical large-mean bound within a single-Kraus no-tick family, showing that the quantum advantage persists asymptotically within this class. The optimized quantum processes exhibit coherent conditioned dynamics and approach a continuous-time quantum-jump description as the mean increases. Our results establish a finite-memory quantum advantage in temporal precision and connect discrete-time counting processes with continuous-time quantum timekeeping.
Comments9 + 10 pages, 7 figures, comments welcome
