**Part I: Gaussian states from linear complex structures**

In the first part, I will review various physical systems where Gaussian states play an important role. Then I will introduce the structure triangle consisting of a symplectic form, a metric and a complex structure. Based on this, I will explain how the space of linear complex structures can parametrize the class of Gaussian states in the quantum theory. When studying correlations between subsystems, we can use the entanglement entropy that can be directly computed from the restricted complex structure in the subsystem. Quadratic Hamiltonians evolve Gaussian states into Gaussian states which allows us to identify the quantum evolution of the state with the classical evolution of the corresponding complex structure. This enables us to study the time evolution of the entanglement entropy analytically in terms of simple matrix traces. Finally, I will discuss the similarities and differences between bosonic and fermionic Gaussian states.

**Part II: Entanglement production at Instabilities**

In the second part, I will explain how one can compute the asymptotic time dependence of the entanglement entropy when evolving Gaussian states with (possibly time-dependent) unstable quadratic Hamiltonians. This will allow me to proof a new theorem that relates the Lyapunov exponents of the classical system with the slope of the linear asymptotics. The result can be applied to quantum field theory in curved spacetime to find the finite entropy associated to certain subalgebras of observables. The production theorem allows one to find the asymptotic entanglement production during inflation and in the preheating phase. I will conclude the discussion by presenting numerical results supporting our conjecture that the theorem also holds for large classes of non-Gaussian states and certain non-quadratic Hamiltonians.

**28/09/2016**

### Entanglement of Gaussian states

Comprehensive Examination, *Pennsylvania State University*