Non-demolition measurements of observables with general spectra

It has recently been established that, in a non-demolition measurement of an observable $mathcal{N}$ with a finite point spectrum, the density matrix of the system approaches an eigenstate of $mathcal{N}$, i.e., it purifies over the spectrum of $mathcal{N}$. We extend this result to observables with general spectra. It is shown that the spectral density of the state of the system converges to a delta function exponentially fast, in an appropriate sense. Furthermore, for observables with absolutely continuous spectra, we show that the spectral density approaches a Gaussian distribution over the spectrum of $mathcal{N}$. Our methods highlight the connection between the theory of non-demolition measurements and classical estimation theory.