  The ability to measure single quanta has allowed complete characterization of
small quantum systems such as quantum dots in terms of statistics of detected
signals known as full-counting statistics. Quantum gas microscopy enables one
to observe many-body systems at the single-atom precision. We extend the idea
of full-counting statistics to nonequilibrium open many-particle dynamics and
apply it to discuss the quench dynamics. By way of illustration, we consider an
exactly solvable model to demonstrate the emergence of unique phenomena such as
nonlocal and chiral propagation of correlations, leading to a concomitant
oscillatory entanglement growth. We find that correlations can propagate beyond
the conventional maximal speed, known as the Lieb-Robinson bound, at the cost
of probabilistic nature of quantum measurement. These features become most
prominent at the real-to-complex spectrum transition point of an underlying
parity-time-symmetric effective non-Hermitian Hamiltonian. A possible
experimental realization with quantum gas microscopy is discussed.
