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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.04537 |
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| _version_ | 1866916780655509504 |
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| author | Bolaños-Servín, Jorge R. Quezada, Roberto Rios-Cangas, Josué I. |
| author_facet | Bolaños-Servín, Jorge R. Quezada, Roberto Rios-Cangas, Josué I. |
| contents | 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 $ρ$ ($ρ$-integrability). It turns out that all elements of the commutative $*$-algebra generated by a possibly unbounded $ρ$-integrable observable $A$, denoted by $\langle A\rangle$, are normal and $ρ\, $-integrable. Besides, $\langle A\rangle$ can be endowed with the well-defined norm $\|\cdot\|_ρ:= {\rm tr}\,(ρ|\cdot| )$. 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 $S-iJ\geq 0$, being $J$ the multiplication operator by $-i$ on $\ell_2({\mathbb N})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04537 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the analytical approach to infinite-mode Boson-Gaussian states Bolaños-Servín, Jorge R. Quezada, Roberto Rios-Cangas, Josué I. Mathematical Physics Quantum Physics Primary 81S05, Secondary 78M05, 60B15 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 $ρ$ ($ρ$-integrability). It turns out that all elements of the commutative $*$-algebra generated by a possibly unbounded $ρ$-integrable observable $A$, denoted by $\langle A\rangle$, are normal and $ρ\, $-integrable. Besides, $\langle A\rangle$ can be endowed with the well-defined norm $\|\cdot\|_ρ:= {\rm tr}\,(ρ|\cdot| )$. 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 $S-iJ\geq 0$, being $J$ the multiplication operator by $-i$ on $\ell_2({\mathbb N})$. |
| title | On the analytical approach to infinite-mode Boson-Gaussian states |
| topic | Mathematical Physics Quantum Physics Primary 81S05, Secondary 78M05, 60B15 |
| url | https://arxiv.org/abs/2506.04537 |