us consider the case when the magnetic field is applied on the side spins ($\circ$’s) as shown in Fig. \[fig:q\_circuit\](d). Note that a co-rotating frame with the center spin is taken and accordingly it does not acquire a phase during the period $\tau$. The initial state $\displaystyle \frac{1}{4} | 0 \rangle \langle 0 | \otimes 
\sigma_0 \otimes \sigma_0$ can be decomposed to $$\begin{aligned}
%\hspace{-8ex}
 \frac{1}{4} \Bigl( 
  |000\rangle \langle 000| 
+ |011\rangle \langle 011| 
+ %\left(
|001\rangle \langle 001| + |010\rangle \langle 010| %\right)
\Bigr),\end{aligned}$$ and thus, the final density matrix after the measurement operation is given as $$\begin{aligned}
\label{eq_meas_dm}
\rho_\circ
&=&  
{\small 
\frac{1}{8} \left(
\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & e^{-2i\theta} & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0  \\
 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0  \\
 0 & 0 & 0 & 1 & 0 & 0 & 0 & e^{2i\theta} \\
 e^{2i\theta} & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 