structure for an entangled magnetic sensor simulation under an engineered noisy environment. The large open circle ($\bigcirc$) is called the center spin, while the small open ones ($\circ$’s) are the side spins and play the role of an entangled sensor. The six surrounding solid circles ($\bullet$’s) are introduced in order to generate a time inhomogeneous noisy environment acting on the entangled sensor. We call them environment spins. (d) a basic quantum circuit for a measurement. Because of the symmetry of the interaction structure, all $\circ$ spins can be accessed globally. []{data-label="fig:q_circuit"}](quantum_circuit){width="8.6cm"}

We take the initial density matrix $\displaystyle |0\rangle \langle 0| 
\otimes \frac{\sigma_0}{2}  \otimes \frac{\sigma_0}{2} $ [@kondo2016], see also Appendix: The center spin is $|0\rangle \langle 0|$, while the two side spins are fully mixed state. When the magnetic field is applied only on the center spin ($\bigcirc$), i.e., the star-topology structure is not effective, the density matrix becomes $$\begin{aligned}
\rho_\bigcirc &=& 
 \frac{1}{8}
{\small \left( 
\begin{array}{cc}
 1 & e^{-i\theta}\\
e^{i\theta} & 1
\end{array} 
\right) }
\otimes \sigma_0 \otimes \sigma_0,\end{aligned}$$ after the magnetic field is applied. Here, $\theta = \gamma_{\rm G}B \tau$ and the subscript $\bigcirc$ stands for the case that the center spin is exposed to the field.

Next, let 