  The dynamics of entanglement has recently been realized as a useful probe in
studying ergodicity and its breakdown in quantum many-body systems. In this
paper, we study theoretically the growth of entanglement in quantum many-body
systems and propose a method to measure it experimentally. We show that
entanglement growth is related to the spreading of local operators in real
space. We present a simple toy model for ergodic systems in which linear
spreading of operators results in a universal, linear in time growth of
entanglement for initial product states, in contrast with the logarithmic
growth of entanglement in many-body localized (MBL) systems. Furthermore, we
show that entanglement growth is directly related to the decay of the Loschmidt
echo in a composite system comprised of several copies of the original system,
in which connections are controlled by a quantum switch (two-level system). By
measuring only the switch's dynamics, the growth of the R\'enyi entropies can
be extracted. Our work provides a way of understanding entanglement dynamics in
many-body systems, and to directly measure its growth in time via a single
local measurement.
