  In the thermodynamic limit two distinct states of matter cannot be analytic
continuations of each other. Classical phase transitions are characterized by
non-analyticities of the free energy. For quantum phase transitions (QPTs) the
ground state energy often assumes the role of the free energy. But in a number
of important cases this criterion fails to predict a QPT, such as the
three-dimensional metal-insulator transition of non-interacting electrons in a
random potential (Anderson localization). It is therefore essential that we
find alternative criteria that can track fundamental changes in the internal
correlations of the ground state wavefunction. Here we propose that QPTs are
generally accompanied by non-analyticities of the von Neumann (entanglement)
entropy. In particular, the entropy is non-analytic at the Anderson transition,
where it exhibits unusual fractal scaling. We also examine two dissipative
quantum systems of considerable interest to the study of decoherence and find
that non-analyticities occur if and only if the system undergoes a QPT.
