Gaussian quantum information over general quantum kinematical systems I: Gaussian states

TL;DR

This paper develops a comprehensive theory of Gaussian states over general quantum kinematical systems modeled by locally compact abelian groups, extending classical results to new algebraic and topological settings.

Contribution

It introduces a unified framework for Gaussian states on LCA groups, characterizes them explicitly, and generalizes key theorems like Hudson's theorem to broader contexts.

Findings

Characterization of Gaussian states on 2-regular LCA groups.

Identification of topological obstructions to entanglement.

Extension of the Hudson theorem to totally disconnected LCA groups.

Abstract

We develop a theory of Gaussian states over general quantum kinematical systems with finitely many degrees of freedom. The underlying phase space is described by a locally compact abelian (LCA) group $G$ with a symplectic structure determined by a 2-cocycle on $G$. We use the concept of Gaussian distributions on LCA groups in the sense of Bernstein to define Gaussian states and completely characterize Gaussian states over 2-regular LCA groups of the form $G= F\times\hat{F}$ endowed with a canonical normalized 2-cocycle. This covers, in particular, the case of $n$-bosonic modes, $n$-qudit systems with odd $d\ge 3$, and $p$-adic quantum systems. Our characterization reveals a topological obstruction to Gaussian state entanglement when we decompose the quantum kinematical system into the Euclidean part and the remaining part (whose phase space admits a compact open subgroup). We then generalize the discrete Hudson theorem \cite{Gro} to the case of totally disconnected 2-regular LCA groups. We also examine angle-number systems with phase space $\mathbb{T}^n\times\mathbb{Z}^n$ and fermionic/hard-core bosonic systems with phase space $\mathbb{Z}^{2n}_2$ (which are not 2-regular), and completely characterize their Gaussian states.