  Continuously monitoring the environment of a quantum many-body system reduces
the entropy of (purifies) the reduced density matrix of the system, conditional
on the outcomes of the measurements. We show that, for mixed initial states, a
balanced competition between measurements and entangling interactions within
the system can result in a dynamical purification phase transition between (i)
a phase that locally purifies at a constant system-size-independent rate, and
(ii) a "mixed" phase where the purification time diverges exponentially in the
system size. The residual entropy density in the mixed phase implies the
existence of a quantum error-protected subspace where quantum information is
reliably encoded against the future non-unitary evolution of the system. We
show that these codes are of potential relevance to fault-tolerant quantum
computation as they are often highly degenerate and satisfy optimal tradeoffs
between encoded information densities and error thresholds. In spatially local
models in 1+1 dimensions, this phase transition for mixed initial states occurs
concurrently with a recently identified class of entanglement phase transitions
for pure initial states. The mutual information of an initially
completely-mixed state in 1+1 dimensions grows sublinearly in time due to the
formation of the error protected subspace. The purification transition studied
here also generalizes to systems with long-range interactions, where
conventional notions of entanglement transitions have to be reformulated.
Purification dynamics is likely a more robust probe of the transition in
experiments, where imperfections generically reduce entanglement and drive the
system towards mixed states. We describe the motivations for studying this
novel class of non-equilibrium quantum dynamics in the context of advanced
quantum computing platforms and fault-tolerant quantum computation.
