Multipartite optimized correlation measures and holography

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.

Phase Transitions of Correlations in Black Hole Geometries

“…The primary goal of the research program of [4,40,42] (inspired by works such as [6,7,39]) is to find new ways to probe the entanglement structure of holographic states using their dual geometries. While simple linear entropic formulas can capture some aspects of the correlations in a quantum state, optimized correlation measures provide access to more detailed information about these correlations.…”

“…For example, using the relationship between E Q and I ss this provides a rare class of non-degradable states for which we can nevertheless evaluate the symmetric side-channel assisted distillable entanglement [45]. We are also lead to new conjectures, such as the possibility that E R is equal to the regularized entanglement of purification [42].…”

“…All these correlation measures are non-negative, a consequence of monotonicity together with being bounded below, and this can be made manifest by writing the objective functions as conic combinations of mutual informations and conditional mutual informations [42]. These five bipartite OCMs can be shown to satisfy a hierarchy of inequalities,…”

“…This vanishing of the objective functions follows from monotonicity, which requires[42] that A and a must appear in f α either as Aa or as S(AX) − S(aY ) with X and Y not including A or a. Given Aa pure and existing in a product state with all other parties, all Aa-dependence cancels out, and the same happens recursively for all other parties, leaving zero.…”