(untitled) (Statistics, Optimization & Information Computing, 2023)
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.