  We explore ways to quantify multipartite correlations, in quantum information
and in holography. We focus on optimized correlation measures, linear
combinations of entropies minimized over all possible purifications of a state
that satisfy monotonicity conditions. These contain far more information about
correlations than entanglement entropy alone. We present a procedure to derive
such quantities, and construct a menagerie of symmetric optimized correlation
measures on three parties. These include tripartite generalizations of the
entanglement of purification, the squashed entanglement, and the recently
introduced Q-correlation and R-correlation. Some correlation measures vanish
only on product states, and thus quantify both classical and quantum
correlations; others vanish on any separable state, capturing quantum
correlations alone. We then use a procedure motivated by the surface-state
correspondence to construct holographic duals for the correlation measures as
linear combinations of bulk surfaces. The geometry of the surfaces can
preserve, partially break, or fully break the symmetry of the correlation
measure. The optimal purification is encoded in the locations of certain
points, whose locations are fixed by constraints on the areas of combinations
of surfaces. This gives a new concrete connection between information theoretic
quantities evaluated on a boundary state and detailed geometric properties of
its dual.
