  We examine the growth of entanglement under a quantum quench at point
contacts of simple fractional quantum Hall fluids and its relation with the
measurement of local observables. Recently Klich and Levitov proposed that the
noise generated from a local quantum quench provides a measure of the
entanglement entropy. Their methods were specific to non-interacting electrons
and the generalization to interacting systems was left as an open question. In
this work, we generalize their result to the Laughlin states. We investigate
the noise generated in the current along the edge of a fractional quantum Hall
state at filling factors $\nu=1/m$, when a quantum point contact, initially
closed, is fully opened at some initial time $t_0=0$. We find that local
quenching in these systems gives time dependent correlation functions that have
universal behavior on sufficiently long time and length scales. We calculate
the noise and full counting statistics for $\nu=1/m$ and find that in general,
the entanglement entropy and noise generated are unrelated quantities. We also
discuss a generalization of this problem to the critical quantum Ising spin
chain.
