Inefficient eight-port homodyne detection and covariant phase space observables

TL;DR

This paper rigorously analyzes the eight-port homodyne detection scheme with inefficient detectors, showing that the measured observable is a smeared covariant phase space observable, and that detector inefficiencies do not affect the state's distinguishability.

Contribution

It provides a rigorous quantum mechanical proof that detector inefficiencies lead to a smeared phase space observable without impacting state discrimination capabilities.

Findings

Measured observable is a smeared covariant phase space observable.

Detector inefficiencies do not affect the state distinguishing power.

The proof applies to arbitrary parameter fields.

Abstract

We consider the quantum optical eight-port homodyne detection scheme in the case that each of the associated photon detectors is assigned with a different quantum efficiency. We give a mathematically rigorous and strictly quantum mechanical proof of the fact that the measured observable (positive operator measure) in the high-amplitude limit is a smearing of the covariant phase space observable related to the ideal measurement. The result is proved for an arbitary parameter field. Furthermore, we investigate some properties of the measured observable. In particular, we show that the state distinguishing power of the observable is not affected by detector inefficiencies.