Mutual Information for Disjoint Regions

Mutual information for disjoint QFT regions is a finite correlation measure whose dependence on separation is controlled by operator content, geometry, and state. In a CFT it becomes a function of conformal cross ratios, but an OPE approximation is reliable only inside its convergence regime and can fail near contact.

Required background. Use Mutual Information and Regulator-Independent Correlations. Helpful background. Interval, Sphere, and Cylinder Entanglement in CFT fixes the single-region conformal formulas.

Cross ratios and replica correlators

For two intervals $A=[u_1,v_1]$ and $B=[u_2,v_2]$ ordered as $u_1<v_1<u_2<v_2$, one convenient cross ratio is

$$x=\frac{(v_1-u_1)(v_2-u_2)} {(u_2-u_1)(v_2-v_1)}, \qquad 0<x<1.$$

The Rényi mutual information is obtained from a four-point function of twist fields and depends on the full replicated-theory operator content, not only on the central charge. Analytic continuation to $n=1$ gives $I(A{:}B)$ when justified. At small $x$, the twist OPE organizes the result by the lowest-dimension exchanged operators.

The structure figure distinguishes this disjoint-correlation branch from mixed-state entanglement and multipartite constraints.

Positive separation removes local shared-boundary divergences and exposes cross-ratio and OPE dependence. Mutual information still measures total correlation, not specifically quantum entanglement. Schematic.

Small separation and large separation

For $x\ll1$, the leading nonidentity exchange controls the decay, with a power set by its scaling dimension and a coefficient set by OPE data. This reproduces clustering while quantifying the total correlation missed by any one correlator. Cardy 2013, §§2–3 develops the large-separation expansion for disjoint compact regions.

As $x\to1$, the intervals approach contact and the mutual information diverges with local short-distance structure. A truncated small-$x$ OPE cannot be extrapolated there. In some approximations—especially large-central-charge or holographic limits—a sharp-looking change of dominant saddle can appear. Generic finite QFT answers need not have a nonanalytic transition.

A controlled calculation uses the same regulator for $S(A)$, $S(B)$, and $S(AB)$, checks cutoff stability at fixed physical separation, and compares successive OPE truncations within their convergence range.

Algebra and approximation limits

The validity map records the choices that remain despite ultraviolet cancellation.

The separated combination can be regulator independent, but changing centers, state, global geometry, or replica continuation changes the result. Contact and use of an OPE outside its convergence domain are distinct failure modes. Schematic.

Do not infer logarithmic negativity, entanglement of purification, or channel capacity from mutual information alone. Each needs its own construction and operation class.

References

• Cardy, John. “Some Results on the Mutual Information of Disjoint Regions in Higher Dimensions.” Journal of Physics A 46 (2013): 285402. DOI. Open preprint.

Further reading

• Calabrese, Pasquale, John Cardy, and Erik Tonni. “Entanglement Entropy of Two Disjoint Intervals in Conformal Field Theory.” Journal of Statistical Mechanics (2009): P11001. DOI. Open preprint.