Entanglement sudden death as an indicator of fidelity in a four-qubit cluster state

I explore the entanglement evolution of a four-qubit cluster state in a dephasing environment concentrating on the phenomenon of entanglement sudden death (ESD). Specifically, I ask whether the onset of ESD has an effect on the utilization of this cluster state as a means of implementing a single-qubit rotation in the measurement-based cluster state model of quantum computation. To do this, I compare the evolution of the entanglement to the fidelity, a measure of how accurately the desired state (after the measurement-based operations) is achieved. I find that ESD does not cause a change in behavior or discontinuity in the fidelity but may indicate for certain states when the fidelity goes to 0.5.