On the analytical approach to infinite-mode Boson-Gaussian states

Abstract

We develop an analytical approach to quantum Gaussian states in infinite-mode representation of the Canonical Commutation Relations (CCR's), using Yosida approximations to define integrability of possibly unbounded observables with respect to a state $\rho$ ($\rho$-integrability). It turns out that all elements of the commutative $\ast$-algebra generated by a possibly unbounded $\rho$-integrable observable $A$, denoted by $⟨A⟩$, are normal and $\rho$-integrable. Besides, $⟨A⟩$ can be endowed with the well-defined norm $\Vert \cdot {\Vert }_{\rho }:=\mathrm{tr}(\rho |\cdot |)$. Our approach allows us to rigorously establish fundamental properties and derive key formulae for the mean value vector and the covariance operator. We additionally show that the covariance operator $S$ of any Gaussian state is real, bounded, positive, and invertible, with the property that $S-iJ\ge 0$, being $J$ the multiplication operator by $-i$ on ${\ell }_2(\mathbb{N})$.

arXiv comment16 pages