Quantum tomography, phase space observables, and generalized Markov kernels

TL;DR

This paper develops a generalized Markov kernel to convert homodyne tomography observables into phase space observables, with examples including Schrödinger cat states and Cahill-Glauber distributions, highlighting cases where such kernels cannot be constructed.

Contribution

It introduces a method to transform homodyne tomography observables into phase space observables using a generalized Markov kernel, with explicit examples and limitations.

Findings

Successfully constructs a generalized Markov kernel for certain states.

Provides examples with Schrödinger cat and Cahill-Glauber distributions.

Identifies cases where the kernel cannot be constructed.

Abstract

We construct a generalized Markov kernel which transforms the observable associated with the homodyne tomography into a covariant phase space observable with a regular kernel state. Illustrative examples are given in the cases of a 'Schrodinger cat' kernel state and the Cahill-Glauber s-parametrized distributions. Also we consider an example of a kernel state when the generalized Markov kernel cannot be constructed.