Statistical uncertainty in quantum-optical photodetection measurements

Abstract

We present a complete statistical analysis of quantum-optical measurement schemes based on photodetection. Statistical distributions of quantum observables determined from a finite number of experimental runs are analytically characterized with the help of the generating function, which is derived using the exact statistical description of raw experimental outcomes. We use the developed formalism to point out that the statistical uncertainty results in substantial limitations on the information determined about the quantum state; although a family of observables characterizing the quantum state can be safely evaluated from experimental data, its further use to obtain the expectation value of some operators generates exploding statistical errors. These issues are discussed using the example of phase-insensitive measurements of a single light mode. We study reconstruction of the photon number distribution from photon counting and from random-phase homodyne detection. We show that utilization of the reconstructed distribution to evaluate a simple well behaved observable, namely the parity operator, encounters difficulties owing to accumulation of statistical errors. As the parity operator yields the Wigner function at the phase space origin, this example also demonstrates that transformation between various experimentally determined representations of the quantum state is quite a delicate matter.