Nonunitary entanglement dynamics in continuous-variable systems

We construct a random unitary Gaussian circuit for continuous-variable (CV) systems subject to measurements. show that when the measurement rate is nonzero, steady state entanglement entropy saturates an area-law scaling. This different from many-body qubit system, where generic transition widely expected. Due unbounded local Hilbert space, time scale destroy always much shorter than one build it, while balance could be achieved finite space. By same reasoning, absence of should also hold other non-unitary CV dynamics.