# Photon number and optical tomograms for Gaussian states

## TL;DR

This paper reviews the probability representation of quantum states in optical and photon number tomography, establishing connections between different measurement approaches and deriving new mathematical relations for Gaussian states.

## Contribution

It provides explicit links between photon number and optical tomograms and introduces new integral relations involving Hermite and Laguerre polynomials for Gaussian states.

## Key findings

- Derived explicit connection between photon number and optical tomograms.
- Established new integral relations for Hermite and Laguerre polynomials.
- Analyzed Gaussian photon states such as squeezed and correlated states.

## Abstract

A review of probability representation of quantum states in given for optical and photon number tomography approaches. Explicit connection of photon number tomogram with measurable by homodyne detector optical tomogram is obtained. New integral relations connecting Hermite polynomials of two variables with Laguerre polynomials are found. Examples of generic Gaussian photon states (squeezed and correlated states) are studied in detail.