  We study quantum information scrambling in a random unitary circuit that
exchanges qubits with an environment at a rate $p$. As a result, initially
localized quantum information not only spreads within the system, but also
spills into the environment. Using the out-of-time-order correlator (OTOC) to
characterize scrambling, we find a nonequilibrium phase transition in the
directed percolation universality class at a critical swap rate $p_c$: for $p <
p_c$ the ensemble-averaged OTOC exhibits ballistic growth with a tunable light
cone velocity, while for $p > p_c$ the OTOC fails to percolate within the
system and vanishes uniformly within a finite timescale, indicating that all
local operators are rapidly swapped into the environment. To elucidate its
information-theoretic consequences, we demonstrate that the transition in
operator spreading coincides with a transition in an observer's ability to
decode the system's initial quantum information from the swapped-out, or
"radiated," qubits. We present a simple decoding scheme which recovers the
system's initial information with perfect fidelity in the nonpercolating phase,
and with continuously decreasing fidelity with decreasing swap rate in the
percolating phase. Depending on the initial state of the swapped-in qubits, we
further observe a corresponding entanglement transition in the coherent
information from the system into the radiated qubits.
