  The de Finetti theorem and its extensions concern the structure of
multipartite probability distributions with certain symmetry properties, the
paradigmatic original example being permutation symmetry. These theorems assert
that such symmetric distributions are well approximated by convex combinations
of uncorrelated ones. In this article, we apply de Finetti theorems to quantum
gravity theories, such as the Sachdev-Ye-Kitaev (SYK) model or large-N vector
and gauge theories. For SYK we put recent studies of information/entanglement
dynamics in a general and rigorous basis. For vector and gauge theories, we
find a gauge invariant operator whose expectation value provides the leading
term in the entanglement entropy in all states close enough to a given
classical state. These results can be unified through a generic statement about
the nature of Schmidt decompositions and decoherence in large-N theories. In
the reverse direction, we extend de Finetti theorems in various ways and
provide an independent approach to the theorems only based on the large-N
properties of the gauge invariant coherence group.
