  In this paper we utilize superoperator formalism to explore the entanglement
evolution of four-qubit cluster states in a number of decohering environments.
A four-qubit cluster state is a resource for the performance of an arbitrary
single logical qubit rotation via measurement based cluster state quantum
computation. We are specifically interested in the relationship between
entanglement evolution and the fidelity with which the arbitrary single logical
qubit rotation can be implemented in the presence of decoherence as this will
have important experimental ramifications. We also note the exhibition of
entanglement sudden death (ESD) and ask how severely its onset affects the
utilization of the cluster state as a means of implementing an arbitrary single
logical qubit rotation.
