  Generic many-body systems coupled to an environment lose their quantum
entanglement due to decoherence and evolve to a mixed state with only classical
correlations. Here, we show that measurements can stabilize quantum
entanglement within open quantum systems. Specifically, in random unitary
circuits with dephasing at the boundary, we find both numerically and
analytically that projective measurements performed at a small nonvanishing
rate results in a steady state with an $L^{1/3}$ power-law scaling entanglement
negativity within the system. Using an analytical mapping to a statistical
mechanics model of directed polymers in a random environment, we show that the
power-law negativity scaling can be understood as Kardar-Parisi-Zhang
fluctuations due to the random measurement locations. Further increasing the
measurement rate leads to a phase transition into an area-law negativity phase,
which is of the same universality as the entanglement transition in monitored
random circuits without decoherence.
