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 \(G= F\times \widehat{F}\) 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 (Gross in J Math Phys 47(12):122107, 2006) 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.
