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arXiv:2204.08162 · Letters in Mathematical Physics

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

Cédric Bény, Jason Crann, Hun Hee Lee, Sang-Jun Park, Sang-Gyun Youn

Published Mar 20, 2025• 1 citation
arXivHTMLPDF

Abstract

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 $$G= F\times \widehat{F}$$ 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 $$d\ge 3$$ 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 (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 $$\mathbb {T}^n\times \mathbb {Z}^n$$ T n × Z n and fermionic/hard-core bosonic systems with phase space $$\mathbb {Z}^{2n}_2$$ Z 2 2 n (which are not 2-regular) and completely characterize their Gaussian states.

DOI
10.1007/s11005-025-01908-1

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