Fisher information of correlated stochastic processes

TL;DR

This paper investigates how correlations in stationary stochastic processes influence the Fisher information for parameter estimation, revealing that correlations do not always improve precision and providing fundamental scaling results.

Contribution

It proves that Fisher information scales linearly with outcomes for finite Markov order processes and shows correlations do not necessarily enhance estimation precision.

Findings

Fisher information is asymptotically linear for finite Markov order processes.

Correlations do not guarantee improved metrological precision.

Application to thermometry on a spin chain demonstrates practical relevance.

Abstract

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.