We develop unifying mathematical methods to describe bosonic and fermionic states in systems with finitely and infinitely many degrees of freedom. Our approach is based on the triplet of Kähler structures, consisting of a positive definite metric, a symplectic form and a linear complex structure that are all defined on the classical phase space of the quantum system under consideration. We explain how the space of linear complex structures, either compatible with the symplectic form (bosons) or the metric (fermions), can be identified with the set of pure Gaussian states. Time evolution by quadratic Hamiltonians is described by the action of the Hamiltonian flow onto linear complex structures as phase space function. Reducing Gaussian states to subsystems corresponds to restricting their linear complex structure to sub phase spaces. These restricted complex structures are objects in their own right and can be identified with mixed Gaussian states. In particular, we can compute the von Neumann entanglement entropy and all Rényi entropies from their eigenvalues. Remarkably, we find identical expressions for these entropies in terms of their respective linear complex structures. Applying these methods we are able to study the properties of Gaussian states such as their time evolution, entanglement and complexity and we present a range of results related to understanding both bosonic and fermionic systems.

First, we prove a theorem on entanglement production in systems with instabilities. The rate of entropy production in a classical dynamical system is characterized by the Kolmogorov-Sinai entropy rate given by the sum of all positive Lyapunov exponents of the system. We prove a quantum version of this result valid for bosonic systems with unstable quadratic Hamiltonian. We show that the entanglement entropy of a Gaussian state grows linearly for large times in unstable systems, with a rate determined by the Lyapunov exponents and the choice of the subsystem. We conjecture that the same rate appears in the entanglement growth of chaotic quantum systems prepared in a semiclassical state.

Second, we show that the typical entanglement entropy of energy eigenstates can be vastly different from the typical entanglement entropy of general states in the Hilbert space. It is well known that typical pure states are maximally entangled with respect to any system decomposition in the thermodynamic limit. We develop tools to compute the entanglement entropy averaged over all eigenstates of quadratic fermionic Hamiltonians. In particular, we derive exact bounds for the most general translationally invariant models. We use this to prove that if the subsystem size is a finite fraction of the system size then the average over eigenstates of the Hamiltonian departs from the result for typical pure states. Furthermore, in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal. Based on numerical evidence, we conjecture that the average entanglement entropy of translationally invariant systems is universal and only depends on the subsystem fraction.

Third, we provide two alternative definitions of bosonic and fermionic Gaussian circuit complexity and show their equivalence. The circuit complexity associated to a quantum state quantifies the difficulty of reaching this target state by applying a sequence of unitary operations to a specified reference state. Defining circuit complexity in field theories is an important question of current research as it is expected to be part of a new duality in holography. In approach A, we equip the Lie group of Gaussian transformations with a right-invariant positive definite metric and define circuit complexity as the minimal geodesic connecting the identity with a group element that prepares a given target state from the specified reference state. In approach B, we directly compute the geodesic distance between reference and target state on the Gaussian state manifold equipped with the canonical Fubini-Study metric. We prove that complexity computed in the two approaches are equivalent up to a relative normalization constant.