Figure 2

The STS weight versus time in a simple quantum network model described in example 1 in the text. (a) Three identical qubits, with coherent coupling J ₁₂ (J ₂₃) between qubit 1 (2) and 2 (3). To simulate the damaged node, we assume qubit 2 suffers a phase damping γ. (b) The blue-solid, black-dashed, and red-dotted curves show the STS weight (ST  SW) of the assemblage \({\{{\sigma }_{a|x}^{{\rm{ST}}}(t)\}}_{a,x}\) of qubit-3 for different dephasing rates of the middle qubit γ/J = 0.01, 1, and 20, respectively. The measurement settings {x} on qubit-1 at time 0 are the Pauli set X, Y, and Z. The initial condition is \(|1\rangle \otimes |0\rangle \otimes |0\rangle \), and \({J}_{12}={J}_{23}\equiv J\). The time t is in units of J ⁻¹. From the figure, we can see that when the dephasing rate increases from 0.01J to 1J, the amplitude of the ST  SW decreases. This means that when dephasing rate increases from 1J to 20J, the dephasing mechanism dominates the dynamics of the system, leading to a disappearance of the oscillatory behavior. Although the dephasing rate is large (e.g., the red-dotted curve), the effect of the measurement on qubit-1 at time 0 can still be transited to qubit-3 via the coherent coupling between the qubits, making the ST  SW gradually increase. For brevity, we are omitting analogous plots for the STS robustness.