Characterizing resources for multiparameter estimation of SU(2) and SU(1,1) unitaries

TL;DR

This paper analyzes the precision bounds for multiparameter estimation of SU(2) and SU(1,1) unitaries using the method of moments, identifying optimal probe states like twin-Fock states for Heisenberg scaling.

Contribution

It introduces a pragmatic framework for multiparameter estimation using the method of moments and identifies optimal probe states for Heisenberg scaling in resource-limited scenarios.

Findings

Twin-Fock states enable Heisenberg scaling for two parameters.

Optimal states can achieve Heisenberg scaling for three parameters under certain conditions.

Method of moments provides a practical approach for multiparameter quantum estimation.

Abstract

We follow recent works analyzing precision bounds to the estimation of multiple parameters generated by a unitary evolution with non-commuting Hamiltonians that form a closed algebra. We consider the $3$-parameter estimation of SU(2) and SU(1,1) unitaries and investigate the scaling of the precision in a pragmatic framework where the estimation is performed via the so-called method of moments, consisting in the estimation of phases via expectation values of time-evolved observables, which we restrict to be the first two moments of the Hamiltonian generators. We consider optimal or close-to-optimal initial states, that can be obtained by maximizing the quantum Fisher information matrix, and analyze the maximal precision achievable from measuring only the first two moments of the generators. As a result, we find that in our context with limited resources accessible, the twin-Fock state emerges as the best probe state, that allows the estimation of two out of the three parameters with Heisenberg precision scaling. Moreover, in other practical scenarios with less limitations we identify useful probe states that can in principle achieve Heisenberg scaling for the three parameters considered.