  Recent studies of quantum circuit models have theoretically shown that
frequent measurements induce a transition in a quantum many-body system, which
is characterized by the change of the scaling law of the entanglement entropy
from a volume law to an area law. In order to propose a way for experimentally
observing this measurement-induced transition, we present numerical analyses
using matrix-product states on quench dynamics of a dissipative Bose-Hubbard
model with controllable two-body losses, which has been realized in recent
experiments with ultracold atoms. We find that when the strength of dissipation
increases, there occurs a measurement-induced transition from volume-law
scaling to area-law scaling with a logarithmic correction in a region of
relatively small dissipation. We also find that the strong dissipation leads to
a revival of the volume-law scaling due to a continuous quantum Zeno effect. We
show that dynamics starting with the area-law states exhibits the breaking of
ergodicity, which can be used in experiments for distinguishing them from the
volume-law states.
