  We consider the variation of von Neumann entropy of subsystem reduced states
of general many- body lattice spin systems due to local quantum quenches. We
obtain Lieb-Robinson-like bounds that are independent of the subsystem volume.
The main assumptions are that the Hamiltonian satisfies a Lieb-Robinson bound
and that the volume of spheres on the lattice grows at most exponentially with
their radius. More specifically, the bound exponentially increases with time
but exponentially decreases with the distance between the subsystem and the
region where the quench takes place. The fact that the bound is independent of
the subsystem volume leads to stronger constraints (than previously known) on
the propagation of information throughout many-body systems. In particular, it
shows that bipartite entanglement satisfies an effective "light cone",
regardless of system size. Further implications to t density-matrix
renormalization-group simulations of quantum spin chains and limitations to the
propagation of information are discussed.
