Achieving the Heisenberg limit of metrology via measurement on an ancillary qubit

TL;DR

This paper proposes a quantum metrology protocol using ancillary qubits that achieves Heisenberg-limited precision through optimized joint evolution and measurement, without requiring entangled states or nonlinear Hamiltonians.

Contribution

It introduces a measurement-based protocol on an ancillary qubit that attains Heisenberg scaling in quantum metrology without entanglement or nonlinear resources.

Findings

Achieves Heisenberg scaling $N^2$ in quantum Fisher information.

Quadratic scaling is robust against control imprecision and decoherence.

Thermal states can reach asymptotic quadratic scaling without GHZ-like states.

Abstract

In the scenario of the probe-ancilla interaction, we propose a quantum metrology protocol by the unconditional measurement on the ancillary qubit after an optimized period of joint evolution from product state. Its key element is the construction of two parallel evolution paths by the measurement that can transform the probe system (a spin ensemble) from an eigenstate of a collective angular momentum operator $|j,m\rangle$ to a superposed state $(|j,m\rangle+|j,-m\rangle)/\sqrt2$. With synchronous parametric encoding and qubit measurement, the quantum Fisher information about the phase encoded in the probe system with optimized initial states can exactly attain the Heisenberg scaling $N^2$ with respect to the probe size (spin number) $N$. The quadratic scaling behavior is not sensitive to the imprecise control over the joint evolution time, the time delay between encoding and measurement, and the coherence in the probe ensemble or the ancillary system that would be degraded by local dephasing. The classical Fisher information of the spin ensemble is found to saturate with its quantum counterpart, irrespective of the idle joint evolution after the parametric encoding. We suggest that both Greenberger-Horne-Zeilinger (GHZ)-like states and nonlinear Hamiltonian are {\em not} necessary resources for exceeding the standard quantum limit in metrology precision since in our protocol even thermal states can hold an asymptotic quadratic scaling.