the magnetic impurities [@Iwakura2017]. Moreover, we also emphasize that the nuclear spins of system II are more strongly influenced by the magnetic impurities than the inner ones (system I) since the dipole-dipole interactions are short range [@Binho2019].

![(Color online) Engineered environment. Therein, the Markovian environment indirectly interacts with System I through System II. []{data-label="fig:model"}](Env_model){width="6.cm"}

Sample and Engineered Environment
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The two-step star-topology molecule that we employed in this work is 2-propanol solved in acetone d6 with magnetic impurities of Fe(III). The structure of this molecule is shown in Fig. \[fig\_st\](a). The three $^{13}$C spins corresponds to $\circ-\bigcirc-\circ$ in Fig. \[fig:q\_circuit\](b, c), while the H spins are employed as System II in Fig. \[fig:model\]. We are able to selectively nullify the H spins by dynamical decoupling techniques [@Levitt2008; @RevModPhys.88.041001], as shown in Fig. \[fig\_st\](a, b, c): (a) without decoupling, (b) selective decoupling of the H spin attached on the center $^{13}$C (hereafter, referred to as the selective-decoupling case), and (c) full decoupling of all the H spins (hereafter, referred to as the full-decoupling case). This implies that the behavior of an entangled magnetic sensor under three different engineered noisy environments can be studied. Note that the interaction topology of Fig. \[fig:q\_circuit\](c) 