# Quantum State Smoothing for Linear Gaussian Systems

## TL;DR

This paper introduces a simplified quantum state smoothing method for Linear Gaussian systems, providing a closed-form solution that enhances purity and physical validity, with applications demonstrated on optical parametric oscillators.

## Contribution

The authors derive a closed-form quantum smoothing solution for Linear Gaussian systems, improving state purity and physical consistency over previous methods.

## Key findings

- Smoothing yields a more pure quantum state than filtering.
- The method is applicable to systems like optical parametric oscillators.
- Quantum efficiency impacts the effectiveness of purity recovery.

## Abstract

Quantum state smoothing is a technique for assigning a valid quantum state to a partially observed dynamical system, using measurement records both prior and posterior to an estimation time. We show that the technique is greatly simplified for Linear Gaussian quantum systems, which have wide physical applicability. We derive a closed-form solution for the quantum smoothed state, which is more pure than the standard filtered state, whilst still being described by a physical quantum state, unlike other proposed quantum smoothing techniques. We apply the theory to an on-threshold optical parametric oscillator, exploring optimal conditions for purity recovery by smoothing. The role of quantum efficiency is elucidated, in both low and high efficiency limits.