Abstract
We study the conjectured holographic duality between entanglement of purification and the entanglement wedge cross-section. We generalize both quantities and prove several information theoretic inequalities involving them. These include upper bounds on conditional mutual information and tripartite information, as well as a lower bound for tripartite information. These inequalities are proven both holographically and for general quantum states. In addition, we use the cyclic entropy inequalities to derive a new holographic inequality for the entanglement wedge cross-section, and provide numerical evidence that the corresponding inequality for the entanglement of purification may be true in general. Finally, we use intuition from bit threads to extend the conjecture to holographic duals of suboptimal purifications.
Highlights
Theoretic concept of the entanglement of purification Ep
We study the conjectured holographic duality between entanglement of purification and the entanglement wedge cross-section
We will study and generalize the relationship between EW, Ep, and the holographic entanglement entropy inequalities in three ways: first, we investigate whether EW can nontrivially bound combinations of entanglement entropies that appear in the holographic entropy inequalities; second, we check whether Ep provably provides the same type of bounds to these objects; lastly, we ask whether one can extend the EW = Ep conjecture to suboptimal purifications and cuts of the entanglement wedge
Summary
Let us define both the entanglement wedge cross section EW , and the entanglement of purification Ep. First, we define holographic states to be quantum states of the boundary conformal field theory that are dual to a well defined classical bulk gravitational theory in AdS/CFT. The entanglement wedge cross-section is defined for any two regions of time reversal symmetric slices (though the generalization to the fully covariant case exists in [13]) as EW (A : B) = min{Area(Γ ); Γ ⊂ rAB splits A and B}. EW is the minimal area of a surface Γ that splits rAB into two regions, one of which is bounded by A but not B, and other by B but not A. We refer interchangeably to the surface and the area thereof as the entanglement wedge cross-section, but the meaning should be clear from context. These coinciding bounds for Ep and EW is what motivated the conjecture of [12, 13] that EW is the holographic dual of Ep
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