Quantum Gaussian states on Boson Fock spaces are quantum versions of Gaussian distributions. In this paper, we explore infinite mode quantum Gaussian states. We extend many of the results of Parthasarathy in [ 14 , 16 ] to the infinite mode case, which includes various characterizations, convexity and symmetry properties.
In this article, we study the phase-space distribution of the quantum state as a framework to describe the different properties of quantum systems in continuous-variable systems. The natural approach to quantum systems is given the Gaussian Wigner representation, to unify the description of bosonic and fermionic quantum states, we study the structure of the Kahler space geometry as the geometry generated by three forms under the agreement conditions depended on the nature of the state bosonic or fermionic. Multi-mode light is studied, and we established that the Fock space vacuum corresponds to a certain homogeneous Gaussian state.
The quantum systems with finite-dimensional Hilbert space have several applications and are intensively explored theoretically and experimentally. The mathematical description of these systems follows the analogy with the usual infinite-dimensional case. There exist finite versions for most of the elements used in the continuous case, but here we provide a finite version corresponding to the mixed Gaussian states. Our aim is to fill this gap. The definition we propose for the mixed discrete Gaussian states is based on the explicit formulas we have obtained in the case of pure discrete-variable Gaussian states.
Given a set of ordered quantum states, described by density operators {ρj}j=1n, the Bargmann invariant of {ρj}j=1n is defined as tr(ρ1ρ2, …, ρn). Bargmann invariant serves as a fundamental concept for quantum mechanics and has diverse applications in quantum information science. Bosonic Gaussian states are a class of quantum states on infinite-dimensional Hilbert space, widely used in quantum optics and quantum information science. Bosonic Gaussian states are conveniently and conventionally characterized by their means and covariance matrices. In this work, we provide the expression of Bargmann invariant tr(ρ1ρ2, …, ρn) for any m-mode bosonic Gaussian states {ρj}j=1n in terms of the means and covariance matrices of {ρj}j=1n. We also use this expression to explore the permissible values of Bargmann invariants for bosonic Gaussian states.
We consider the most general Gaussian quantum Markov semigroup on a one-mode Fock space, discuss its construction from the generalized GKSL representation of the generator. We prove the known explicit formula on Weyl operators, characterize irreducibility and its equivalence to a Hörmander type condition on commutators and establish necessary and sufficient conditions for existence and uniqueness of normal invariant states. We illustrate these results by applications to the open quantum oscillator and the quantum Fokker-Planck model.