Controllability properties of a continuously monitored qubit

We consider a stochastic dynamics describing the evolution of a qubit controlled by an external field and subject to continuous-time measurements. Motivated by stabilization techniques recently developed in, e.g., [10], [11], we investigate the support of the corresponding solution, which is a random variable taking values on the space of two-by-two density matrices. By making use of the Strook-Varadhan support theorem and by classical geometric control arguments we compute the support for two possible choices of the measurement and Hamiltonian operators. In one case we show that, in Bloch coordinates, the support is always contained inside an ellipsoid depending on the physical parameters of the system. More precisely, a solution starting from the ellipsoid never exit it, with probability one, and every open subset of the ellipsoid is visited with nonzero probability for some choice of the control function. In the second case the support coincides with (the interior of) the Bloch ball: every open subset of the Bloch ball is visited with nonzero probability up to suitably choosing the control function.