We construct random unitary Gaussian circuits 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 nonunitary Gaussian CV dynamics.

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• Continuous phase transition
• Entanglement entropy
• Quantum optics