Parameter estimation with limited access of measurements

TL;DR

This paper develops a theoretical framework for quantum parameter estimation when measurement access is limited, analyzing how non-optimal measurements affect precision and proposing optimization strategies to approach optimal bounds.

Contribution

It introduces a method to quantify and optimize measurement observables under limited access, enhancing estimation precision in quantum systems.

Findings

Shorter Euclidean distance to optimal observables improves estimation accuracy.

Optimizing accessible observables can approach the precision of fully accessible measurements.

The framework is demonstrated with examples involving a driven qubit and an NV-center system.

Abstract

Quantum parameter estimation holds the promise of quantum technologies, in which physical parameters can be measured with much greater precision than what is achieved with classical technologies. However, how to obtain a best precision when the optimal measurement is not accessible is still an open problem. In this work, we present a theoretical framework to explore the parameter estimation with limited access of measurements by analyzing the effect of non-optimal measurement on the estimation precision. We define a quantity to characterize the effect and illustrate how to optimize observables to attain a bound with limited accessibility of observables. On the other side, we introduce the minimum Euclidean distance to quantify the difference between an observable and the optimal ones in terms of Frobenius norm and find that the measurement with a shorter distance to the optimal ones benefits the estimation. Two examples are presented to show our theory. In the first, we analyze the effect of non-optimal measurement on the estimation precision of the transition frequency for a driven qubit. While in the second example, we consider a bipartite system, in which one of them is measurement inaccessible. To be specific, we take a toy model, the NV-center in diamond as the bipartite system, where the NV-center electronic spin interacts with a single nucleus via the dipole-dipole interaction. We achieve a precise estimation for the nuclear Larmor frequency by optimizing only the observables of the electronic spin. In these two examples, the minimum Euclidean distance between an observable and the optimal ones is analyzed and the results show that the observable closed to the optimal ones better the estimation precision.