We analyze the problem of quantum phase estimation where the set of allowed phases forms a discrete $N$ element subset of the whole $[0,2pi]$ interval, $varphi_n = 2pi n/N$, $n=0,dots N-1$ and study the discrete-to-continuous transition $Nrightarrowinfty$ 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 functions of a given width $sigma$ 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 $U(1)$ 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 sub-covariant measurements to remedy this situation and demonstrate truly optimal estimation strategies when performing transition from a discrete to the continuous phase estimation regime.
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