Position and momentum tomography

We illustrate the use of the statistical method of moments for determining the position and momentum distributions of a quantum object from the statistics of a single measurement. The method is used for three different, though related, models: the sequential measurement model, the Arthurs-Kelly model, and the eight-port homodyne detection model. In each case, the method of moments gives the position and momentum distributions for a large class of initial states, the relevant condition being the exponential boundedness of the distributions.

Figure 1

The eight-port homodyne detector.