We consider a class of correlation measures for quantum states called optimized correlation measures defined as a minimization of a linear combination of von Neumann entropies over purifications of a given state. Examples include the entanglement of purification E P and squashed entanglement E sq . We show that when evaluating such measures on "nice" holographic states in the large-N limit, the optimal purification has a semiclassical geometric dual. We then apply this result to confirm several holographic dual proposals, including the n-party squashed entanglement. Moreover, our result suggests two new techniques for determining holographic duals: holographic entropy inequalities and direct optimization of the dual geometry.NEWTON CHENG PHYS. REV. D 101, 066009 (2020)Note that ð P J ∈R α J ÞS mix ≥ 0 as both terms are nonnegative. By the minimality of our choice of jψi UU 0 , we must also have the upper bound NEWTON CHENG