Nonlinear qubit transformations

We generalize our previous results of universal linear manipulations [Phys. Rev. A 63, 032304 (2001)] to investigate three types of nonlinear qubit transformations using measurement and quantum-based schemes. First, nonlinear rotations are studied. We rotate different parts of a Bloch sphere in opposite directions about the z axis. The second transformation is a map that sends a qubit to its orthogonal state. We consider the case in which the orthogonal state is applied to only a partial area of a Bloch sphere. We also study nonlinear general transformation, i.e., $(\vartheta ,\phi )\to (\vartheta -\alpha ,\phi ),$ again applied only to part of the Bloch sphere. In order to achieve these three operations, we consider different measurement preparations and derive the optimal average (instead of universal) quantum unitary transformations. We also introduce a simple method for a qubit measurement and its application to other cases.