Optimal parameter estimation with a fixed rate of abstention

TL;DR

This paper explores optimal quantum parameter estimation strategies that incorporate abstention, demonstrating significant improvements and achieving the Heisenberg limit in some scenarios through a comprehensive mathematical framework and numerical methods.

Contribution

It introduces a general framework for quantum parameter estimation with abstention, providing analytical results and efficient numerical optimization techniques.

Findings

Abstention schemes can reach the Heisenberg limit in some cases.

A mathematical framework for large quantum systems is developed.

Parameter estimation with abstention is formulated as a semidefinite programming problem.

Abstract

The problems of optimally estimating a phase, a direction, and the orientation of a Cartesian frame (or trihedron) with general pure states are addressed. Special emphasis is put on estimation schemes that allow for inconclusive answers or abstention. It is shown that such schemes enable drastic improvements, up to the extent of attaining the Heisenberg limit in some cases, and the required amount of abstention is quantified. A general mathematical framework to deal with the asymptotic limit of many qubits or large angular momentum is introduced and used to obtain analytical results for all the relevant cases under consideration. Parameter estimation with abstention is also formulated as a semidefinite programming problem, for which very efficient numerical optimization techniques exist.