Quantum Homodyne Tomography as an Informationally Complete Positive Operator Valued Measure

TL;DR

This paper introduces a new POVM for quantum homodyne tomography that captures measurements of randomly sampled quadratures, and analyzes its relation to the Wigner transform of quantum states.

Contribution

It defines an informationally complete POVM for quantum homodyne measurements and explores its mathematical relationship with the Wigner function.

Findings

Defined a POVM on [0,2π]×R for quadrature sampling

Analyzed the probabilistic properties of the measurement

Established the relation between the POVM description and the Wigner transform

Abstract

We define a positive operator valued measure $E$ on $[0,2\pi]\times R$ describing the measurement of randomly sampled quadratures in quantum homodyne tomography, and we study its probabilistic properties. Moreover, we give a mathematical analysis of the relation between the description of a state in terms of $E$ and the description provided by its Wigner transform.