Optimal quantum estimation in spin systems at criticality

TL;DR

This paper investigates how quantum criticality in the 1D quantum Ising model can be exploited to optimize the estimation of coupling constants, achieving enhanced precision at critical points.

Contribution

It derives the optimal quantum estimator for the coupling constant in the quantum Ising model, revealing maximal Fisher information at criticality and the effectiveness of magnetization measurements.

Findings

Optimal external field maximizes quantum Fisher information at criticality.

Precision improves proportionally with system size at the critical point.

Total magnetization measurement is optimal above a temperature-dependent threshold.

Abstract

It is a general fact that the coupling constant of an interacting many-body Hamiltonian do not correspond to any observable and one has to infer its value by an indirect measurement. For this purpose, quantum systems at criticality can be considered as a resource to improve the ultimate quantum limits to precision of the estimation procedure. In this paper, we consider the one-dimensional quantum Ising model as a paradigmatic example of many-body system exhibiting criticality, and derive the optimal quantum estimator of the coupling constant varying size and temperature. We find the optimal external field, which maximizes the quantum Fisher information of the coupling constant, both for few spins and in the thermodynamic limit, and show that at the critical point a precision improvement of order $L$ is achieved. We also show that the measurement of the total magnetization provides optimal estimation for couplings larger than a threshold value, which itself decreases with temperature.