Mitigating Detuning-Induced Systematic Errors in Entanglement-Enhanced Metrology

TL;DR

This work addresses detuning-induced coherent errors in GHZ-based metrology and demonstrates how such detuning can prevent reaching the Heisenberg limit. It analyzes a frequency-selective GHZ preparation protocol, derives how detuning biases the estimator via $\Delta=\sum_i \delta_i$, and shows the resulting impact on measurement probabilities. To combat this, the authors design a composite-pulse sequence that cancels first-order detuning, restoring near-Heisenberg scaling and proving robust performance in both homogeneous and inhomogeneous detuning scenarios under realistic time budgets. The results provide a practical route to fault-tolerant entanglement-enhanced sensing and outline avenues for further robustness through tunable-pulse-strength control and alternative GHZ-generation methods.

Abstract

Quantum sensing leverages non-classical resources to enhance precision. In particular, Greenberger-Horne-Zeilinger (GHZ) states can, in principle, attain the Heisenberg limit that surpasses the standard quantum limit. While many studies have examined how open-system noise-typically modeled with Lindblad master equations-degrades GHZ-based metrology, coherent control imperfections during state preparation and readout have received less attention. Here, we analyze the effect of detuning between actual and nominal spin frequencies in a GHZ-state preparation scheme employing a frequency selective pulse. We show that detuning induces coherent, systematic error that prevents GHZ sensing from reaching the Heisenberg limit. To mitigate this effect, we design a composite-pulse protocol that compensates for detuning-induced errors and improves the sensitivity under the effect of coherent error.

Figures (2)

• Figure 1: The RSD with respect to the number of spins $N$. In both panels, the blue thin line represents the RSD of the normal protocol under detuning while the red thick line represents the RSD for the case where we employ the composite pulse sequence to prepare the GHZ state. The orange dashed thick line indicates the RSD for the case where we use a composite pulse with changeable pulse strength, which is explained in Appendix \ref{['sec:app']}. The green dashed line shows the Heisenberg scaling as a reference. We set the parameters as $M=T/\tau=10^{6}$, $\tau=100\pi$, $\Omega=0.00001$. The detuning is set to $\delta_{i}=\delta=~(a)~0.00001=\Omega,~(b)~0.000001=0.1\Omega$ for all $i$. All parameters are normalized by $\lambda_{\rm max}$.
• Figure 2: The RSD with respect to the number of spins $N$. In both panels, the blue thin line represents the RSD of the normal protocol under detuning while the red thick line represents the RSD for the case where we employ the composite pulse sequence to prepare the GHZ state. The orange dashed thick line is the RSD for the case where we use a composite pulse with changeable pulse strength, which is explained in Appendix \ref{['sec:app']}. The parameters are set to $M=T/\tau=10^{6}$, $\tau=100\pi$, $\Omega=0.00001$. The detuning $\delta_{i}$ is randomly sampled from the intervals: $(a)~[0,\Omega]=[0,0.00001],~(b)~[0,0.1\Omega]=[0,0.000001]$ for all $i$. All parameters are normalized by $\lambda_{\rm max}$.