This paper studies the stochastic differential equation (SDE) associated to atwo-level quantum system (qubit) subject to Hamiltonian evolution as well asunmonitored and monitored decoherence channels. The latter imply a stochasticevolution of the quantum state (density operator), whose associated probabilitydistribution we characterize. We first show that for two sets of typicalexperimental settings, corresponding either to weak quantum non demolitionmeasurements or to weak fluorescence measurements, the three Bloch coordinatesof the qubit remain confined to a deterministically evolving surface or curveinside the Bloch sphere. We explicitly solve the deterministic evolution, andwe provide a closed-form expression for the probability distribution on thissurface or curve. Then we relate the existence in general of suchdeterministically evolving submanifolds to an accessibility question of controltheory, which can be answered with an explicit algebraic criterion on the SDE.This allows us to show that, for a qubit, the above two sets of weakmeasurements are essentially the only ones featuring deterministic surfaces orcurves.
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