  Using the mapping of the Fokker-Planck description of classical stochastic
dynamics onto a quantum Hamiltonian, we argue that a dynamical glass transition
in the former must have a precise definition in terms of a quantum phase
transition in the latter. At the dynamical level, the transition corresponds to
a collapse of the excitation spectrum at a critical point. At the static level,
the transition affects the ground state wavefunction: while in some cases it
could be picked up by the expectation value of a local operator, in others the
order may be non-local, and impossible to be determined with any local probe.
Here we propose instead to use concepts from quantum information theory that
are not centered around local order parameters, such as fidelity and
entanglement measures. We show that for systems derived from the mapping of
classical stochastic dynamics, singularities in the fidelity susceptibility
translate directly into singularities in the heat capacity of the classical
system. In classical glassy systems with an extensive number of metastable
states, we find that the prefactor of the area law term in the entanglement
entropy jumps across the transition. We also discuss how entanglement measures
can be used to detect a growing correlation length that diverges at the
transition. Finally, we illustrate how static order can be hidden in systems
with a macroscopically large number of degenerate equilibrium states by
constructing a three dimensional lattice gauge model with only short-range
interactions but with a finite temperature continuous phase transition into a
massively degenerate phase.
