Many real-world tasks include some kind of parameter estimation, i.e., determination of a parameter encoded in a probability distribution. Often, such probability distributions arise from stochastic processes. For a stationary stochastic process with temporal correlations, the random variables that constitute it are identically distributed but not independent. This is the case, for instance, for quantum continuous measurements. In this paper we prove two fundamental results concerning the estimation of parameters encoded in a memoryful stochastic process. First, we show that for processes with finite Markov order, the Fisher information is always asymptotically linear in the number of outcomes, and determined by the conditional distribution of the process' Markov order. Second, we prove with suitable examples that correlations do not necessarily enhance the metrological precision. In fact, we show that unlike for entropic information quantities, in general nothing can be said about the sub- or super-additivity of the joint Fisher information, in the presence of correlations. We discuss how the type of correlations in the process affects the scaling. We then apply these results to the case of thermometry on a spin chain.

翻译：许多实际任务涉及某种参数估计，即确定编码在概率分布中的参数。通常，这类概率分布源自随机过程。对于具有时间相关性的平稳随机过程，构成该过程的随机变量同分布但非独立。例如，量子连续测量即属于此类情况。本文证明了关于记忆性随机过程中参数估计的两个基本结果。首先，我们证明对于有限马尔可夫阶的过程，Fisher信息在渐近意义上始终与观测结果数量呈线性关系，并由过程马尔可夫阶的条件分布决定。其次，我们通过适当的示例证明，相关性不一定会提升计量精度。事实上，我们表明与熵信息量不同，在存在相关性的情况下，通常无法断言联合Fisher信息具有次可加性或超可加性。我们讨论了过程中相关类型如何影响标度行为。最后，我们将这些结果应用于自旋链上的温度测量案例。