Entanglement Wedges, Nesting, and Information Inequalities

Entanglement-wedge nesting says that inclusion of boundary domains implies inclusion of their semiclassical bulk entanglement wedges. At leading classical order, maximin and null focusing keep the relevant HRT surfaces from crossing. This geometric ordering supports entropy inequalities and later reconstruction consistency, but it is conditional on correct global saddle selection and the same causal hypotheses as maximin. At quantum order, area nesting is replaced by a generalized-entropy/QES statement; one cannot keep a classical surface while appending bulk entropy afterward.

Required background. HRT supplies the extremal surfaces, and maximin supplies the existence and focusing argument.

Helpful background. Data processing supplies the boundary information ordering, and the QES definition supplies the quantum replacement.

Wedges and the nesting statement

For an admissible HRT surface $\gamma_A$, choose a bulk achronal region $R_A$ satisfying

$$\partial R_A=A\cup\gamma_A.$$

The entanglement wedge is its bulk domain of dependence,

$$\mathcal E_W[A]=D_{\rm bulk}[R_A].$$

If the boundary domains obey $D[A]\subseteq D[B]$, classical entanglement-wedge nesting is

$$\mathcal E_W[A]\subseteq\mathcal E_W[B].$$

For boundary regions on one Cauchy slice, maximin permits the relevant minimal surfaces to be compared on a common bulk slice. If they crossed, exchanging segments at an intersection would produce homologous competitors whose total area is no larger and, after smoothing a transverse crossing, strictly smaller. Stable global minima can therefore be chosen without crossing. In time-dependent settings, focused null representatives transport the comparison to a common slice; this is where the null-curvature condition enters (Wall 2014, §§3–4).

Nested balls in vacuum AdS

In Poincaré $\mathrm{AdS}_{d+1}$, take two concentric boundary balls at $t=0$,

$$A=\{r<R_A\}, \qquad B=\{r<R_B\}, \qquad R_A<R_B.$$

Their RT surfaces are hemispheres

$$\gamma_A:\ z^2+r^2=R_A^2, \qquad \gamma_B:\ z^2+r^2=R_B^2.$$

On the reflection-symmetric slice, the homology regions are

$$R_A^{\rm bulk}=\{z>0:\ z^2+r^2<R_A^2\}, \qquad R_B^{\rm bulk}=\{z>0:\ z^2+r^2<R_B^2\}.$$

The radius inequality gives the pointwise inclusion

$$R_A^{\rm bulk}\subset R_B^{\rm bulk},$$

and taking bulk domains of dependence gives $\mathcal E_W[A]\subset\mathcal E_W[B]$. The surfaces never intersect. This direct construction verifies the first application and also distinguishes the wedge from the causal wedge: the two coincide for these symmetric vacuum balls but need not do so after changing state or shape.

The same ordering is visible infinitesimally. A radial deformation $R\to R+\delta R$, $\delta R>0$, moves every point of the hemisphere outward along its normal, rather than creating alternating crossings. In a general state that monotonic response follows from the geometric theorem, not from the explicit equation.

Cut-and-paste entropy inequalities

On a common static slice, let $m_{AB}$ and $m_{BC}$ be minimal surfaces for overlapping regions $AB$ and $BC$. Cut them at intersections and reglue the pieces into admissible—though not necessarily minimal—surfaces $\widetilde m_B$ and $\widetilde m_{ABC}$. Areas are additive under the cut:

$$\operatorname{Area}(m_{AB})+ \operatorname{Area}(m_{BC}) =\operatorname{Area}(\widetilde m_B)+ \operatorname{Area}(\widetilde m_{ABC}).$$

Minimality gives

$$\operatorname{Area}(m_B)\leq\operatorname{Area}(\widetilde m_B), \qquad \operatorname{Area}(m_{ABC})\leq \operatorname{Area}(\widetilde m_{ABC}),$$

and hence

$$S(AB)+S(BC)\geq S(B)+S(ABC),$$

the strong-subadditivity inequality at leading RT order. Headrick and Takayanagi gave this geometric proof (Headrick and Takayanagi 2007, §2). In Lorentzian settings maximin supplies a common comparison slice under its hypotheses.

Universal quantum entropy already satisfies strong subadditivity; the geometric proof shows that leading holographic areas reproduce it. More restrictive inequalities, such as monogamy of mutual information, characterize a smaller holographic entropy cone and are treated on the next page.

Saddle-transition adversarial check

For disconnected or thermal regions, several homologous extrema compete. Near a transition their generalized areas can cross. If $A\subset B$ but one chooses a subdominant connected surface for $A$ and a subdominant disconnected surface for $B$, the resulting “wedges” can cross. That is not a counterexample to nesting; it is evidence that at least one candidate was not selected by the global prescription. The control is to enumerate all branches, include horizon components, regulate them identically, and compare before testing inclusion.

A genuine boundary of the classical theorem arises if null focusing fails. With sufficient violation of $R_{ab}k^ak^b\geq0$, null representatives can increase area and the maximin comparison no longer proves noncrossing. At order $G_N^0$, ordinary matter can violate the classical pointwise condition even when a quantum focusing principle may control generalized entropy. The revised claim must use renormalized QES and an appropriate quantum nesting theorem; classical area nesting cannot simply be asserted.

Degenerate transitions also require care. At the exact crossing, more than one surface has the same entropy. The entropy is continuous but the wedge may be nonunique at leading order; subleading corrections or a specified algebra select the physical branch. Any reconstruction claim at that point must record this ambiguity.

Handoff

The classical result is strongest for globally hyperbolic asymptotically AdS Einstein geometries satisfying the null-curvature condition, with stable maximin surfaces and global homology/saddle comparison. Entropy cones develops the inequalities special to classical holographic cut functions. Reflected entropy tests a mixed-state cross-section proposal whose connectivity changes precisely at wedge transitions.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Headrick, M., and Takayanagi, T. (2007). “A holographic proof of the strong subadditivity of entanglement entropy.” Physical Review D 76, 106013. DOI.
• Wall, A. C. (2014). “Maximin surfaces, and the strong subadditivity of the covariant holographic entanglement entropy.” Classical and Quantum Gravity 31, 225007. DOI.