Equation of Motion for Estimation Fidelity of Monitored Oscillating Qubits

We study the convergence properties of state estimates of an oscillating qubit being monitored by a sequence of $\mathit{\text{discrete}}$, unsharp measurements. Our method derives a differential equation determining the evolution of the estimation fidelity from a single incremental step. When the oscillation frequency $\mathrm{\Omega }$ is precisely known, the estimation fidelity converges exponentially fast to unity. For imprecise knowledge of $\mathrm{\Omega }$ we derive the asypmtotic estimation fidelity.