  The dynamics of quantum entanglement plays a central role in explaining the
emergence of thermal equilibrium in isolated many-body systems. However,
entanglement is notoriously hard to measure, and can in fact be forged: recent
works have introduced a notion of pseudoentanglement describing ensembles of
many-body states that, while only weakly entangled, cannot be efficiently
distinguished from states with much higher entanglement, such as random states
in the Hilbert space. In this work we initiate the study of the dynamical
generation and propagation of pseudoentanglement. As generic quantum dynamics
tends to maximize actual entanglement, we consider constrained models of time
evolution: automaton (i.e. reversible classical) circuits that, when fed
suitable input states, provably produce the "standard models" of
pseudoentangled ensembles--uniformly random subset(-phase) states--at late
times, a phenomenon we name 'pseudothermalization'. We examine (i) how a
pseudoentangled ensemble on a small subsystem spreads to the whole system as a
function of time, and (ii) how a pseudoentangled ensemble is generated from an
initial product state. We map the above problems onto a family of classical
Markov chains on subsets of the computational basis. The mixing times of such
Markov chains are related to the time scales at which the states produced from
the dynamics become indistinguishable from Haar-random states at the level of
each statistical moment (or number of copies). Based on a combination of
rigorous bounds and conjectures supported by numerics, we argue that each
Markov chain's relaxation time and mixing time have different asymptotic
behavior in the limit of large system size. This is a necessary condition for a
cutoff phenomenon: an abrupt dynamical transition to equilibrium. We thus
conjecture that our random circuits give rise to asymptotically sharp
pseudothermalization transitions.
