\\
 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\
 0 & 0 & 0 & e^{-2i\theta} & 0 & 0 & 0 & 1 \\
\end{array} \right) }\nonumber \\
&=& \frac{1}{8}
{\small \left( 
\begin{array}{cc}
 1 & e^{-2i\theta}\\
e^{i2\theta} & 1
\end{array} 
\right) }
\otimes 
|00\rangle \langle 00| \nonumber  \\
&+&\frac{1}{8}
{\small \left( 
\begin{array}{cc}
 1 & e^{2i\theta}\\
e^{-i2\theta} & 1
\end{array} 
\right) }
\otimes 
|11\rangle \langle 11| \nonumber \\
&+& \frac{1}{8}
\left( {\small \begin{array}{cc}
 1 & 1\\
 1 & 1
\end{array}} \right) \otimes 
\Bigl(|01\rangle \langle 01| + |10\rangle \langle 10| \Bigr),\end{aligned}$$ where $\circ$ stands for the case that the magnetic field is applied on the side spins. Note that, due to the twice large phase accumulation ($2\theta$, instead of $\theta$) of the initial states of $  |000\rangle \langle 000|$ and $|011\rangle \langle 011|  $, the sensitivity with side spins interacting the magnetic fields becomes twice larger than that of the central spin.

We show the method how to detect the acquired phases in NMR. The state $\rho_k$ ($k = \bigcirc, \circ$) is assumed to develop under the Hamiltonian $$\begin{aligned}
 H&=& \omega_0 \frac{\sigma_z}{2} \otimes \sigma_0 \otimes \sigma_0\nonumber\\
&+&
J \left(\frac{\sigma_z}{2} \otimes \frac{\sigma_z}{2} \otimes \sigma_0 
