Parameter estimation of qubit states with unknown phase parameter

TL;DR

This paper investigates optimal parameter estimation for a qubit system with an unknown phase, analyzing bounds and measurement strategies to achieve asymptotic precision limits.

Contribution

It introduces measurement strategies that asymptotically attain the HGM and Holevo bounds for estimating qubit parameters with unknown phase.

Findings

HGM bound can be asymptotically attained with separable measurements.

Holevo bound can be asymptotically attained with collective measurements.

Optimal measurement strategies depend on the trade-offs between different bounds.

Abstract

We discuss a problem of parameter estimation for quantum two-level system, qubit system, in presence of unknown phase parameter. We analyze trade-off relations for mean-square errors when estimating relevant parameters with separable measurements based on known precision bounds; the symmetric logarithmic derivative Cramer-Rao bound and Hayashi-Gill-Massar (HGM) bound. We investigate the optimal measurement which attains the HGM bound and discuss its properties. We show that the HGM bound for relevant parameters can be attained asymptotically by using some fraction of given $n$ quantum states to estimate the phase parameter. We also discuss the Holevo bound which can be attained asymptotically by a collective measurement.