Fourier Analytic Approach to Quantum Estimation of Group Action

TL;DR

This paper introduces a Fourier analytic method for optimal quantum estimation of various group actions, applicable to both commutative and non-commutative groups, including non-compact and compact groups, under energy constraints.

Contribution

It presents a unified Fourier-based approach for quantum group action estimation, extending to projective representations of diverse groups with optimality guarantees.

Findings

Optimal estimation achieved for R, U(1), SU(2), SO(3), and R^2 groups.

Method applicable to both compact and non-compact groups.

Provides a framework for energy-constrained quantum estimation.

Abstract

This article proposes a unified method to estimation of group action by using the inverse Fourier transform of the input state. The method provides optimal estimation for commutative and non-commutative group with/without energy constraint. The proposed method can be applied to projective representations of non-compact groups as well as of compact groups. This paper addresses the optimal estimation of R, U(1), SU(2), SO(3), and R^2 with Heisenberg representation under a suitable energy constraint.