2022/04/18 by Cédric Bény, Jason Crann, Beny, Cedric +7 · 3 citations
Computer Science · Mathematics · Physics and Astronomy · #43A65 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Primary 81P45 #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #Topological and Geometric Data Analysis #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2204.08162
openalex publication_date 2022/04/18 · openalex created_date 2022/04/26 · openalex updated_date 2026/07/28
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×F endowed with a canonical normalized 2-cocycle. This covers, in particular, the case of n-bosonic modes, n-qudit systems with odd d≥ 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 \citeGro to the case of totally disconnected 2-regular LCA groups. We also examine angle-number systems with phase space \mathbbTn×ℤn and fermionic/hard-core bosonic systems with phase space ℤ2n2 (which are not 2-regular), and completely characterize their Gaussian states.