  We construct 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 many-body qubit system, where a generic
entanglement transition is widely expected. The absence of entanglement phase
transition is attributed to the diverging entangling time for the local degrees
of freedom under unitary evolution, in contrast to the order one time to
disentangle by local measurement. For the same reasoning, the absence of
transition should also hold for other non-unitary Gaussian CV dynamics.
