Quantum walker as a probe for its coin parameter

Abstract: In discrete-time quantum walk (DTQW) the walker's coin space entangles with the position space after the very first step of the evolution. This phenomenon may be exploited to obtain the value of the coin parameter $\theta$ by performing measurements on the sole position space of the walker. In this paper, we evaluate the ultimate quantum limits to precision for this class of estimation protocols, and use this result to assess measurement schemes having limited access to the position space of the walker in one dimension. We find that the quantum Fisher information (QFI) of the walker's position space $H_w(\theta)$ increases with $\theta$ and with time which, in turn, may be seen as a metrological resource. We also find a difference in the QFI of {\em bounded} and {\em unbounded} DTQWs, and provide an interpretation of the different behaviors in terms of interference in the position space. Finally, we compare $H_w(\theta)$ to the full QFI $H_f(\theta)$, i.e., the QFI of the walkers position plus coin state, and find that their ratio is dependent on $\theta$, but saturates to a constant value, meaning that the walker may probe its coin parameter quite faithfully.