Holographic mutual information is monogamous

We identify a special information-theoretic property of quantum field theories with holographic duals: the mutual informations among arbitrary disjoint spatial regions $A$, $B$, $C$ obey the inequality $I(A:B\cup C)\ge I(A:B)+I(A:C)$, provided entanglement entropies are given by the Ryu-Takayanagi formula. Inequalities of this type are known as monogamy relations and are characteristic of measures of quantum entanglement. This suggests that correlations in holographic theories arise primarily from entanglement rather than classical correlations. We also show that the Ryu-Takayanagi formula is consistent with all known general inequalities obeyed by the entanglement entropy, including an infinite set recently discovered by Cadney et al.; this constitutes strong evidence in favor of its validity.

Figure 1

Illustration of the proof in ${\mathrm{AdS}}_3/{\mathrm{CFT}}_2$. (i) The three minimal surfaces corresponding to $S(AB)$, $S(BC)$, and $S(AC)$, with the latter consisting of two connected components in the bulk. (ii) The four minimal surfaces corresponding to $S(A)$, $S(B)$, $S(C)$, and $S(ABC)$. In this example, the surfaces for $S(AB)$ and $S(BC)$ each gets decomposed into a purple (left shaded) part and a red (right shaded) part. The purple forms a surface terminating on the boundary of $A$ and the red a surface terminating on the boundary of $C$. Since these new surfaces need not be minimal, their areas form upper bounds on $S(A)$ and $S(C)$, respectively, by the Ryu-Takayanagi formula. The reader is cautioned, however, that these simple cartoons can be misleading. As we will see in Sec. 6a, higher-order versions of the monogamy inequality fail even though naive graphical arguments suggest they should hold.