&=& 
\sum_{j} \omega_0^{(j)} \frac{\sigma_z^{(j)}}{2}
 + \sum_{j<k} J^{(j,k)} \frac{\sigma_z^{(j)}\otimes\sigma_z^{(k)}}{4},\end{aligned}$$ because $|\omega_{0}^{(j)} -\omega_{0}^{(k)}| \gg J^{(j,k)}$, or weak coupling approximation [@Levitt2008], is satisfied here. $\omega_{0}^{(j)}$ is the isotropic chemically shifted Larmor frequency of the $j$’th spin, and $J^{(j,k)}$ the interaction strength between the $j$’th and $k$’th spins [@Levitt2008; @Kondo2018]. $\omega_{0}^{(j)}$ and $J^{(j,k)}$ were measured from the spectra of the sample without magnetic impurities and are summarized in Table \[H\_parameters\].

  ------------------------------------------------------------------------
          CC             CSs                  HC                HSs
  ----- ------ ----------------------- ----------------- -----------------
   CC    62.6   $J^{\rm (CC, CSs)} =    $2\pi\cdot 140$   $2\pi\cdot 4.4$
                  2\pi \cdot 38.4$                       

   CSs                  25.5                  NR          $2\pi \cdot124$

   HC                                        3.78               NR

   HSs                                                         1.21
  ------------------------------------------------------------------------

  : \[H\_parameters\] $\omega_{0}^{(j)}$ and $J^{(j,k)}$ are summarized. We label the spins as CC (the center $^{13}$C in the C spin chain), CSs (the two side $^{13}$C’s in the C spin chain), HC (the H spin attached on CC) and HSs (the H spins attached on CS). Diagonal elements are $\omega_{0}^{(j)}$’s in ppm, while the off-diagonal elements are those of $J^{(j,k)}$ in rad/s. NR implies that these are too small to be measured reliably.

Simulation of Entangled Sensor
------------------------------

We implemented the quantum circuit shown in Fig. \[fig:q\_circuit\](d) with a standard NMR pulse sequence [@liquidNMRQC]. The rotation operations applied on CC 