Dissipative ‘Groups’ and the Bloch Ball

Recent developments in quantum computing have emphasized the need for a realistic analysis of dissipation in systems which have the potential for use as qubits. In this note we discuss the effects of control and dissipation on a two-level system. For a single qubit pure state, it is well known that the unitary evolution may be visualised as the movement of a vector, the Bloch vector, on the surface of a 2-sphere, the Bloch sphere. In this note we extend the idea to a two-level mixed state. For this system, unitary evolution is on a spherical shell within the Bloch Sphere. Dissipation causes more general motion within the Bloch ball. This motion corresponds to the action of a certain semi-group. We also show that the effects of dissipation may not be compensated by the interaction with the external control. However, taking into account the effects of measurement, which may be modelled by certain projection operators, the dissipative effects may indeed be modified, allowing more effective control of the system.