Phase Transitions in the Classical Simulability of Open Quantum Systems

We study the evolution of an open quantum system using a Langevin unravelling of the density matrix evolution over matrix product states. As the strength of coupling to and temperature of the environment is increased, we find a transition where the entanglement of the individual trajectories saturates, permitting a classical simulation of the system for all times. This is the Hamiltonian open system counterpart of the saturation in entanglement found in random circuits with projective or weak measurements. If a system is open, there is a limit to the advantage in simulating its behaviour on a quantum computer, even when that evolution harbours important quantum effects.