Scrambling Transition in a Radiative Random Unitary Circuit

While states and observables in generic isolated many-body systems are expected to grow in complexity indefinitely, interactions with external degrees of freedom can dramatically modify the system's dynamics. To determine whether sharp thresholds in the flow of quantum information can exist in systems with loss processes, we study quantum information scrambling in a random unitary circuit that exchanges qubits with an environment. As a result, initially localized quantum information not only spreads within the system, but spills into the environment. Using the out-of-time-order correlator (OTOC) to characterize scrambling, we find a phase transition in the directed percolation universality class at a critical swap rate: below the threshold the average OTOC exhibits ballistic growth with a tunable light cone velocity, while above it the OTOC fails to percolate within the system and vanishes uniformly after a finite time, indicating that all local operators are rapidly swapped into the environment. The transition additionally manifests in the ability to decode the system's initial quantum information from the swapped-out 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.