  We introduce a class of quantum channels with correlations acting on pairs of
qubits, where the correlation takes the form of a shift operator onto a
maximally entangled state. We optimise the output purity and show that below a
certain threshold the optimum is achieved by partially entangled states whose
degree of entanglement increases monotonically with the correlation parameter.
Above this threshold, the optimum is achieved by the maximally entangled state
characterizing the shift. Although, a full analysis can only be done for the
2-norm, both numerical and heuristic arguments indicate that this behavior and
the optimal inputs are independent of p>1 when the optimal output purity is
measured using the p-norm.
