Non-Gaussianity from superselection rules

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.

Figures (1)

• Figure 1: (Color online) Principles of the inverse stereographic representation of the roots of the Majorana polynomial on the sphere: A root located at position $z$ in the complex plane is mapped onto the sphere by drawing a line from the south pole $S$ through $z$; the intersection of this line with the sphere defines its spherical position (left panel and cross-sectional view on the right). For example, the root $z=0$, corresponding to the state $\lvert N \rangle_a$, is mapped to the north pole $N$. Roots approaching $S$ correspond to $\left\lvert z\right\rvert\to\infty$ in the complex plane. The CV region (red, central region of the sphere and corresponding disk in the plane) is defined by $\left\lvert z\right\rvert^2 \ll \sqrt{N}$, equivalently $\theta \ll 1$. In the CV limit, the spherical measure reduces to the planar Gaussian measure (see main text).