Detecting quantum non-Gaussianity with a single quadrature

TL;DR

This paper demonstrates that a single quadrature measurement can effectively certify quantum non-Gaussianity in bosonic systems, simplifying experimental requirements for quantum advantage applications.

Contribution

It introduces a method to detect non-Gaussianity using only one quadrature measurement, based on a new understanding of zeros in homodyne distributions as signatures of non-Gaussianity.

Findings

Single quadrature measurement witnesses non-Gaussianity.

Zeros in homodyne distribution indicate quantum non-Gaussianity.

Method is robust and feasible for experimental implementation.

Abstract

Full reconstruction of quantum states from measurement samples is often a prohibitively complex task, both in terms of the experimental setup and the scaling of the sample size with the system. This motivates the relatively easier task of certifying application-specific quantities using measurements that are not tomographically complete, i.e. that provide only partial information about the state related to the application of interest. Here, we focus on simplifying the measurements needed to certify non-Gaussianity in bosonic systems, a resource related to quantum advantage in various information processing tasks. We show that the statistics of a single quadrature measurement, corresponding to standard homodyne detection in quantum optics, can witness arbitrary degrees of non-Gaussianity as quantified by stellar rank. Our results are based on a version of Hudson's theorem for wavefunctions, proved in a companion paper [arXiv:2507.23468], revealing that the zeros in a homodyne distribution are signatures of quantum non-Gaussianity and higher stellar ranks. The validity of our witnesses is supported by a technical result showing that sets of states with bounded energy and finite stellar rank are compact. We provide an analysis of sample complexity, noise robustness, and experimental prospects. Our work drastically simplifies the setup required to detect quantum non-Gaussianity in bosonic quantum states.