Quantum metrology for the Ising Hamiltonian with transverse magnetic field

TL;DR

This paper analytically investigates quantum metrology for the Ising model with a transverse magnetic field, demonstrating Heisenberg-limited precision with specific quantum states and contrasting it with standard quantum limit scaling.

Contribution

It provides an analytical solution showing Heisenberg-limited precision for magnetic field estimation in the Ising model, identifying optimal states and scaling regimes.

Findings

Heisenberg scaling achieved with vacuum and fermion states

GHZ states exhibit Heisenberg scaling across parameters

Product states reach standard quantum limit

Abstract

We consider quantum metrology for unitary evolutions generated by parameter-dependent Hamiltonians. We focus on the unitary evolutions generated by the Ising Hamiltonian that describes the dynamics of a one-dimensional chain of spins with nearest-neighbour interactions and in the pres- ence of a global, transverse, magnetic field. We analytically solve the problem and show that the precision with which one can estimate the magnetic field (interaction strength) given one knows the interaction strength (magnetic field) scales at the Heisenberg limit, and can be achieved by a linear superposition of the vacuum and N free fermion states. In addition, we show that GHZ-type states exhibit Heisenberg scaling in precision throughout the entire regime of parameters. Moreover, we numerically observe that the optimal precision using a product input state scales at the standard quantum limit.