# Measuring non-Gaussianity with Correlation

## TL;DR

This paper introduces a correlation-based framework for quantifying quantum non-Gaussianity, linking it to operational measures and providing an experimentally feasible protocol for estimation, advancing both theory and practice.

## Contribution

It presents a novel correlation-based approach to measure quantum non-Gaussianity, including an experimental protocol and theoretical bounds, unifying theory and practical estimation.

## Key findings

- Correlation generation characterizes non-Gaussianity.
- Operational measures are monotonic under Gaussian channels.
- Sample-efficient protocol estimates non-Gaussianity in experiments.

## Abstract

Quantum non-Gaussianity is a key resource for quantum advantage in continuous-variable systems. We introduce a general framework to quantify non-Gaussianity based on correlation generation: two copies of a state become correlated at a $50{:}50$ beam splitter if and only if the state is non-Gaussian, with correlations reducing to entanglement in the pure-state case. This connection enables operational measures of non-Gaussianity, defined through correlation quantifiers such as R\'enyi-$\alpha$ entropy for pure states and R\'enyi-$\alpha$ mutual information for mixed states. We prove that all such measures are monotonic under Gaussian channels. Building on this framework, we propose a sample-efficient experimental protocol to estimate non-Gaussianity using standard optical components, even in the state agnostic setting. Finally, we establish a lower bound on the sample complexity of estimating Wigner negativity, allowing a direct comparison with our protocol. Our results provide both a unifying theoretical framework for non-Gaussianity and a practical route toward its experimental quantification.