# Discrete-to-continuous transition in quantum phase estimation

## TL;DR

This paper investigates the transition from discrete to continuous quantum phase estimation, developing optimal measurement strategies and analyzing the impact of cost functions and prior distributions on estimation accuracy.

## Contribution

It introduces the theory of sub-covariant measurements to achieve truly optimal strategies during the discrete-to-continuous phase transition.

## Key findings

- Optimal measurement strategies are developed for the discrete-to-continuous transition.
- Sub-covariant measurements outperform covariant ones in this regime.
- The analysis reveals the influence of prior distributions on estimation performance.

## Abstract

We analyze the problem of quantum phase estimation where the set of allowed phases forms a discrete $N$ element subset of the whole $[0,2\pi]$ interval, $\varphi_n = 2\pi n/N$, $n=0,\dots N-1$ and study the discrete-to-continuous transition $N\rightarrow\infty$ 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 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.