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