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We show that bosonic and fermionic Gaussian states (also known as``squeezed coherent states’’) can be uniquely characterized by theirlinear complex structure J J which is a linear map on the classical phase space. This extendsconventional Gaussian methods based on covariance matrices and providesa unified framework to treat bosons and fermions simultaneously. PureGaussian states can be identified with the triple (G,\Omega,J) ( G , Ω , J ) of compatible Kähler structures, consisting of a positive definitemetric G G ,a symplectic form \Omega Ω and a linear complex structure J J with J^2=-\mathbb{1} J 2 = − 1 .Mixed Gaussian states can also be identified with such a triple, butwith J^2\neq -\mathbb{1} J 2 ≠ − 1 .We apply these methods to show how computations involving Gaussianstates can be reduced to algebraic operations of these objects, leadingto many known and some unknown identities. We apply these methods to thestudy of (A) entanglement and complexity, (B) dynamics of stablesystems, (C) dynamics of driven systems. From this, we compile acomprehensive list of mathematical structures and formulas to comparebosonic and fermionic Gaussian states side-by-side. 

We present a systematic geometric framework to study closed quantum systems based on suitably chosen variational families. For the purpose of (A) real time evolution, (B) excitation spectra, (C) spectral functions and (D) imaginary time evolution, we show how the geometric approach highlights the necessity to distinguish between two classes of manifolds: Kähler and non-Kähler. Traditional variational methods typically require the variational family to be a Kähler manifold, where multiplication by the imaginary unit preserves the tangent spaces. This covers the vast majority of cases studied in the literature. However, recently proposed classes of generalized Gaussian states make it necessary to also include the non-Kähler case, which has already been encountered occasionally. We illustrate our approach in detail with a range of concrete examples where the geometric structures of the considered manifolds are particularly relevant. These go from Gaussian states and group theoretic coherent states to generalized Gaussian states.