  The entanglement entropy of subsystems of typical eigenstates of quantum
many-body Hamiltonians has been recently conjectured to be a diagnostic of
quantum chaos and integrability. In quantum chaotic systems it has been found
to behave as in typical pure states, while in integrable systems it has been
found to behave as in typical pure Gaussian states. In this tutorial, we
provide a pedagogical introduction to known results about the entanglement
entropy of subsystems of typical pure states and of typical pure Gaussian
states. They both exhibit a leading term that scales with the volume of the
subsystem, when smaller than one half of the volume of the system, but the
prefactor of the volume law is fundamentally different. It is constant (and
maximal) for typical pure states, and it depends on the ratio between the
volume of the subsystem and of the entire system for typical pure Gaussian
states. Since particle-number conservation plays an important role in many
physical Hamiltonians, we discuss its effect on typical pure states and on
typical pure Gaussian states. We prove that while the behavior of the leading
volume-law terms does not change qualitatively, the nature of the subleading
terms can change. In particular, subleading corrections can appear that depend
on the square root of the volume of the subsystem. We unveil the origin of
those corrections. Finally, we discuss the connection between the entanglement
entropy of typical pure states and analytical results obtained in the context
of random matrix theory, as well as numerical results obtained for physical
Hamiltonians.
