Iterated nth order nonlinear quantum dynamics with mixed initial states

Nonlinear quantum transformations play an important role in quantum state distillation, aiding quantum communication schemes. The simplest of these involve quadratic rational maps for qubits, higher order versions have not been explored in detail. We consider such iterated nonlinear protocols for qubits, involving a generalized CNOT, a measurement, and a single-qubit Hadamard gate processing n inputs in identical quantum states. To characterize the global properties of these protocols, we determine the fixed points and relevant fixed cycles. We show that a repelling mixed fixed point exists for all orders, marking a phase transition related to the disappearance of the fractalness of borders of different basins of attraction.