Entanglement and thermodynamics in non-equilibrium isolated quantum systems

In these lectures, I pedagogically review some recent advances in the study of the non-equilibrium dynamics of isolated quantum systems. In particular I emphasise the role played by the reduced density matrix and by the entanglement entropy in the understanding of the stationary properties after a quantum quench. The idea that the stationary thermodynamic entropy is the entanglement accumulated during the non-equilibrium dynamics is introduced and used to provide quantitative predictions for the time evolution of the entanglement itself. The harmonic chain is studied as an elementary model in which the quench dynamics can be easily and exactly worked out. This example provides a useful playground where general concepts can be simply understood and later applied to more complex and realistic systems.