\[fig:q\_circuit\]. Therefore, we simulated the magnetic field by applying a $Z_k(\theta)$-rotation, $$\begin{aligned}
&\textbf{field on CC:}\\
&\left(R_{\rm C}(-\frac{\pi}{2}, \frac{\pi}{2})-Z_{\rm C}(\theta)\right)-
{\rm CNOT}-(\frac{\tau}{2}-R_{\rm S}(0,\pi)-\frac{\tau}{2})-{\rm CNOT}, \\
&\textbf{field on CSs:}\\
%{\rm  field \hspace{1ex} on  \hspace{1ex} CSs } \\
&R_{\rm C}(-\frac{\pi}{2},\frac{\pi}{2})-\left({\rm CNOT}-Z_{\rm S}(\theta)\right)-
(\frac{\tau}{2}-R_{\rm S}(0,\pi)-\frac{\tau}{2}) - {\rm CNOT}, \end{aligned}$$ where $(\tau/2-R_{\rm S}(0,\pi)-\tau/2)$ is the period when the entangled sensor acquires the phase $\theta = \gamma_{\rm G} B \tau$ in real measurements. In our simulations, the “entangled magnetic sensor” (CSs) was under the engineered noisy environment during only this period. $R_{\rm S}(0, \pi)$ in the middle of this period was added for nullifying the time development caused by the interaction between CC and CSs. We started from the thermal state and observed only CC in our simulations.

We first demonstrate the “entanglement-enhanced” phase sensitivity [@Jones1166; @Liu:2015aa; @Knott:2016aa] in the full-decoupling case (Fig. \[fig\_sp\](c)). The sample of 0.41 M 2-propanol solved in acetone-d6 with 12 mM of Fe(acac) as a magnetic impurity was used. $T_1$’s of $^{13}$C of this sample were measured to be 1.3 s, while $T_1$’s of all H spins were about 100 ms. Figure \[fig:p\_meas\] shows the spectra of CC as a function of $\theta$ (the “field strength”) at $\tau = 3.4$ ms. 