This paper studies the stochastic differential equation (SDE) associated with a two-level quantum system (qubit) subject to Hamiltonian evolution as well as unmonitored and monitored decoherence channels. The latter imply a stochastic evolution of the quantum state (density operator), whose associated probability distribution we characterize. We first show that for two sets of typical experimental settings, corresponding either to weak quantum non-demolition measurements or to weak fluorescence measurements, the three Bloch coordinates of the qubit remain confined to a deterministically evolving surface or curve inside the Bloch sphere. We explicitly solve the deterministic evolution, and we provide a closed-form expression for the probability distribution on this surface or curve. Then we relate the existence in general of such deterministically evolving submanifolds to an accessibility question of control theory, 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 weak measurements are essentially the only ones featuring deterministic surfaces or curves.
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We only need the following Itô rule. Let X be a vector of components (X)j and evolving according to , with normalized independent Wiener processes . Consider the change of coordinates (X′)j = Hj(X). Then we have . The last term is the Itô correction term, with respect to the standard (i.e., non-stochastic) calculus.
For instance, if F = 0, m = 1, it is clear that , i.e., the controlled system (A2) would move in one dimension, and the control only determines the speed of evolution in time. The same would be true for the Stratonovich setting (A1), with the Wiener process just randomly varying the speed of evolution. However if this F = 0 and G1 were associated with an Itô setting, then the associated Stratonovich equation might imply motion in several dimensions. Indeed, take, e.g., (G1)k = (xk)2, one computes that . For almost all , this vector field is not parallel to G1.