Quantum metrology for a general Hamiltonian parameter

TL;DR

This paper investigates quantum metrology for estimating general Hamiltonian parameters, revealing optimal scaling limits, deriving bounds on quantum Fisher information, and illustrating with magnetic field examples.

Contribution

It provides a comprehensive analysis of quantum parameter estimation for general Hamiltonians, including bounds and time-scaling behavior, extending beyond the case of multiplicative factors.

Findings

Quantum Fisher information can be divided into quadratic and oscillatory parts.

The Heisenberg limit can be achieved for the number scaling of estimation precision.

When Hamiltonian eigenvalues are parameter-independent, the quantum Fisher information is bounded.

Abstract

Quantum metrology enhances the sensitivity of parameter estimation using the distinctive resources of quantum mechanics such as entanglement. It has been shown that the precision of estimating an overall multiplicative factor of a Hamiltonian can be increased to exceed the classical limit, yet little is known about estimating a general Hamiltonian parameter. In this paper, we study this problem in detail. We find that the scaling of the estimation precision with the number of systems can always be optimized to the Heisenberg limit, while the time scaling can be quite different from that of estimating an overall multiplicative factor. We derive the generator of local parameter translation on the unitary evolution operator of the Hamiltonian, and use it to evaluate the estimation precision of the parameter and establish a general upper bound on the quantum Fisher information. The results indicate that the quantum Fisher information generally can be divided into two parts: one is quadratic in time, while the other oscillates with time. When the eigenvalues of the Hamiltonian do not depend on the parameter, the quadratic term vanishes, and the quantum Fisher information will be bounded in this case. To illustrate the results, we give an example of estimating a parameter of a magnetic field by measuring a spin-$\frac{1}{2}$ particle, and compare the results for estimating the amplitude and the direction of the magnetic field.