Non-Gaussianity from superselection rules

TL;DR

This paper reinterprets non-Gaussianity in quantum electromagnetic states by linking it to superselection rules, showing it as a witness of particle entanglement and computational resources, with implications for quantum advantage.

Contribution

It provides a physical interpretation of non-Gaussianity, connects stellar rank to superselection rules, and generalizes stellar rank as a basis-dependent resource indicator for quantum advantage.

Findings

Quadrature non-Gaussianity and stellar rank indicate particle entanglement when superselection rules are considered.

Stellar rank's relation to quantum resources is basis-dependent, tied to quadrature eigenstates.

Generalization of stellar rank to arbitrary bases as a resource witness for quantum advantage.

Abstract

The quantum theory of the electromagnetic field enables the description of multiphoton states exhibiting nonclassical statistical properties, often reflected in non-Gaussian phase-space distributions. While non-Gaussianity alone does not fully characterize quantum states, several classifications have been proposed to hierarchize non-Gaussian states according to physically or informationally relevant resources. Here, we provide a physical interpretation of non-Gaussianity and connect it to a computational perspective by showing how a prominent classification-the stellar rank-emerges as a limiting case of the roots of polynomials that univocally represent bosonic states defined with a quantized phase reference, namely the Majorana polynomials. A direct consequence of our results is a revised interpretation of both the stellar rank and non-Gaussianity itself: when superselection rules are properly taken into account, quadrature non-Gaussianity - and nonzero stellar rank - act as witnesses of particle entanglement, rather than being linked with photon addition to Gaussian states as previously assumed. In addition, we show that because the stellar rank depends on a specific choice of coherent states, its relation to computational resources and potential quantum advantage is inherently basis-dependent, being naturally tied to quadrature eigenstates as the computational basis. Motivated by this observation, we generalize the notion of stellar rank to arbitrary computational bases, thereby establishing it as a genuine witness of bosonic resources that may enable quantum advantage.