  Measurements are able to fundamentally affect quantum dynamics. We here show
that a continuously measured quantum many-body system can undergo a spontaneous
transition from asynchronous stochastic dynamics to noise-free stable
synchronization at the level of single trajectories. We formulate general
criteria for this quantum phenomenon to occur, and demonstrate that the number
of synchronized realizations can be controlled from none to all. We
additionally find that ergodicity is typically broken, since time and ensemble
averages may exhibit radically different synchronization behavior. We further
introduce a quantum type of multiplexing that involves individual trajectories
with distinct synchronization frequencies. Measurement-induced synchronization
appears as a genuine nonclassical form of synchrony that exploits quantum
superpositions.
