GAUSSIAN QUANTUM INFORMATION OVER GENERAL QUANTUM KINEMATICAL SYSTEMS I: GAUSSIAN STATES

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=FxF<^>\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G= F\times \widehat{F}$$\end{document} endowed with a canonical normalized 2-cocycle. This covers, in particular, the case of n-bosonic modes, n-qudit systems with odd d >= 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 3$$\end{document}, 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 TnxZn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {T}<^>n\times \mathbb {Z}<^>n$$\end{document} and fermionic/hard-core bosonic systems with phase space Z22n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}<^>{2n}_2$$\end{document} (which are not 2-regular) and completely characterize their Gaussian states.