Abstract
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.
Highlights
Finding ways to quantify the correlations between multiple parties in a quantum system is of natural importance
VII we review the general method for assigning a holographic dual to an optimized correlation measure and the bipartite holographic results of [46], which we utilize in Sec
Any other configuration is equivalent by a conformal transformation to one in this class and will calculate the same numerical value of the correlation measures. This use of conformal symmetry requires that the correlation measures are cutoff independent, as we demonstrate in the subsection
Summary
Finding ways to quantify the correlations between multiple parties in a quantum system is of natural importance. In the case of pure AdS3 dual to CFT2, geometric duals for all the bipartite correlation measures were obtained in [46], associating both EP and ER with the EWCS, Esq with half the mutual information as in [42], and EQ with a novel combination of bulk surfaces; the same bulk dual for EQ was independently proposed and studied in [48]. Beyond bipartite correlations in AdS=CFT is the observation by [50] that if the EWCS is dual to either EP or the reflected entropy, holographic CFT states must have significant amounts of tripartite entanglement in the limit of classical geometry. Building on the bipartite cases, we take the optimal purification to lie along the boundary of the entanglement wedge, while allowing the optimization points dividing the ancilla degrees of freedom to vary, and use the generalization of the RT formula suitable to the surface-state correspondence to calculate the relevant entropies and minimize the measure. IX with concluding remarks and some comments on future work
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