# On the role of Hermite-like polynomials in the Fock representations of Gaussian states

## TL;DR

This paper explores how Hermite-like polynomials can be used to represent Gaussian states and operators in quantum mechanics, providing explicit formulas that simplify calculations in continuous-variable quantum systems.

## Contribution

It introduces a novel approach using Hermite-like polynomials for Fock representations of Gaussian states, yielding closed-form algebraic results.

## Key findings

- Explicit formulas for Gaussian states in Fock basis
- Simplified evaluation of quantum states and operators
- Fundamental role of Hermite-like polynomials in quantum representations

## Abstract

The expansion of quantum states and operators in terms of Fock states plays a fundamental role in the field of continuous-variable quantum mechanics. In particular, for general single-mode Gaussian operators and Gaussian noisy states, many different approaches have been used in the evaluation of their Fock representation. In this paper a natural approach has been applied using exclusively the operational properties of the Hermite and Hermite-like polynomials and showing their fundamental role in this field. Closed-form results in terms of polynomials, exponentials, and simple algebraic functions are the major contribution of the paper.