Non-demolition measurements of observables with general spectra

TL;DR

This paper extends the understanding of non-demolition measurements by showing how the spectral density of a quantum system's state converges to specific distributions, including delta functions and Gaussians, depending on the spectrum type.

Contribution

It generalizes previous results to observables with arbitrary spectra, demonstrating exponential convergence to eigenstates or Gaussian distributions.

Findings

Spectral density converges exponentially to a delta function for general spectra.

For absolutely continuous spectra, spectral density approaches a Gaussian distribution.

Highlights the link between non-demolition measurement theory and classical estimation theory.

Abstract

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.