Discrete-to-continuous transition in quantum phase estimation
Wojciech Rządkowski, Rafał Demkowicz-Dobrzański · Physical Review A · 2017
We analyze the problem of quantum phase estimation in which the set of allowed phases forms a discrete $N$-element subset of the whole $[0,2\ensuremath{\pi}]$ interval, ${\ensuremath{\varphi}}_{n}=2\ensuremath{\pi}n/N, n=0,\ensuremath{\cdots},N\ensuremath{-}1$, and study the discrete-to-continuous transition $N\ensuremath{\rightarrow}\ensuremath{\infty}$ for various cost functions as well as the mutual information. We also analyze the relation between the problems of phase discrimination and estimation by considering a step cost function of a given width $\ensuremath{\sigma}$ around the true estimated value. We show that in general a direct application of the theory of covariant measurements for a discrete subgroup of the $\text{U}(1)$ group leads to suboptimal strategies due to an implicit requirement of estimating only the phases that appear in the prior distribution. We develop the theory of subcovariant measurements to remedy this situation and demonstrate truly optimal estimation strategies when performing a transition from discrete to continuous phase estimation.