How two intervals can share a bulk region

Decide how four endpoints may be joined

Draw two intervals along the boundary, each of width $a$ , with positive gap $d$ . Label their endpoints $u_1,v_1,u_2,v_2$ from left to right. Depth is $z$ and the curvature radius is $L$ . We first take the continuum limit, with $a,d$ both much larger than radial cutoff $\epsilon$ .

One pairing joins the two ends of each interval separately. Each arc caps its own bulk region. Another joins the outermost endpoints and also joins the two endpoints bordering the gap. Remove the inner cap from the outer cap: the remaining bulk region again has the original boundary intervals along its edge.

The requirement that the intervals and curves together bound an appropriate bulk region is the homology condition. It identifies the candidates whose lengths must be compared. Arbitrary endpoint pairings can violate that requirement; crossing long arcs cannot be global minimizers here. We minimize total curve length. The shade records the spatial region bounded by those curves.

Source: Matthew Headrick

Find where the two total lengths agree

An interval of width $w$ contributes $2L\log(w/\epsilon)$ at small cutoff. The disconnected candidate uses width $a$ twice. The connected candidate uses the full outer span $2a+d$ and the gap $d$ . These give the two total lengths below.

Both candidates have four endpoints near the boundary, with the same cutoff divergence. Subtract their lengths and that common contribution cancels. A small gap makes the inner connecting arc inexpensive. Increasing the gap eventually favors the separate caps.

\begin{aligned}\mathcal L_{\rm dis}&=4L\log\frac a\epsilon\\\mathcal L_{\rm con}&=2L\log\frac{2a+d}{\epsilon}+2L\log\frac d\epsilon\end{aligned}

These are the small-cutoff expressions. Equal numbers of endpoints produce equal logarithmic divergences.

\mathcal L_{\rm con}-\mathcal L_{\rm dis}=2L\log\frac{d(2a+d)}{a^2}

The connected candidate is shorter when the dimensionless ratio inside the logarithm is smaller than one.

\frac{d_*}{a}=\sqrt2-1,\qquad \eta=\frac{a^2}{(a+d)^2},\qquad \eta_*=\frac12

Equating the lengths gives a quadratic with one positive root. The cross ratio combines the four endpoint positions into a dimensionless number.

Source: Matthew Headrick

Read mutual information from the length difference

Mutual information is $I(A:B)=S_A+S_B-S_{A\cup B}$ . It measures correlations available when the two parts are considered together, including classical correlations and entanglement. At leading classical Einstein-gravity order, entropy is the minimal length divided by $4G_N$ , and $c=3L/(2G_N)$ is the dimensionless central charge. Natural logarithms give entropy in nats.

When the separate caps win, the leading joint entropy equals the sum of the leading individual entropies. The mutual information at this order vanishes. When the connected pairing wins, their length difference gives a positive contribution. The minimum of two smooth functions is continuous at the crossing but can change slope there, producing the corner in the graph.

The leading term grows with $c$ . Its vanishing still allows correlations of order $c^0$ . An exactly zero full mutual information would imply a product joint density matrix. Quantum corrections and finite central charge also modify the sharp classical transition.

\frac{I_{\rm RT}(A:B)}{c}=\frac13\max\!\left(0,\log\frac{a^2}{d(2a+d)}\right)

This is the leading continuum holographic-vacuum prediction obtained from the two pairings. It is not a universal two-interval formula for every two-dimensional quantum theory.

Source: Matthew Headrick

Work through a complete numerical example

Take $a=L$ , $d=0.3L$ and reference cutoff $\epsilon=0.02L$ . The continuum formulas give a disconnected total length of about $15.6481L$ and a connected total of about $14.9060L$ . The connected candidate is shorter by about $0.7421L$ .

Divide that difference by $4G_N$ and then by $c$ , equivalently by $6L$ , to obtain $I_{\rm RT}/c\simeq0.12369$ . Direct substitution of $d(2a+d)/a^2=0.69$ into the logarithmic expression gives the same value.

Keep the widths fixed and increase the gap to $0.6L$ . The ratio becomes $1.56$ , so the connected candidate is longer. The separate pairing now wins and leading mutual information vanishes. Both intervals remain present; their joint minimal curve has changed.

A finite cutoff shifts the crossing

The preceding chapter fixes endpoints on a common $z=\epsilon$ surface and defines width by their horizontal separation there. Its exact length is $f_\epsilon(w)=2L\operatorname{arsinh}(w/(2\epsilon))$ . Compare $2f_\epsilon(a)$ with $f_\epsilon(2a+d)+f_\epsilon(d)$ to keep identical endpoint data for both finite-cutoff candidates.

At $a=L$ and $\epsilon=0.02L$ , a numerical crossing occurs at $d_*/L\simeq0.41365$ , close to the continuum value $0.41421$ . At the larger cutoff $0.2L$ , it moves to about $0.36458$ . These shifts belong to the specified finite-cutoff geometry.

Interpreting this exact finite-cutoff minimum as an entropy of a particular regulated quantum field theory would require a separate theory and dictionary. We retain it as a geometric comparison and display continuum leading RT mutual information separately. Their agreement improves toward the boundary.

Source: Shinsei Ryu and Tadashi Takayanagi · Matthew Headrick

Turn a spatial region into a wedge

An equal-time picture contains only a spatial region. After shading the region between the selected curves and boundary intervals, we must extend the construction through time. Initial data on a spatial region determine events in its domain of dependence: every inextendible causal curve through such an event must meet that region. The boundary theory has its own domain of dependence.

Write the shaded bulk region as $r_A$ . Its bulk domain of dependence $D_{\rm bulk}(r_A)$ defines the classical entanglement wedge. A causal wedge asks a different operational question: which bulk events can both receive a signal from and send a signal back to the boundary domain $D_{\partial}(A)$ ? One construction uses the extremal surface and homology; the other uses light cones.

For a single vacuum interval in empty AdS, symmetry makes the wedges equal. The boundary domain is a diamond with tips at $t=\pm a/2$ in units with light speed one. Light cones from those tips intersect the equal-time slice on the radius- $a/2$ geodesic. More general regions or states allow the wedges to differ. For nearby disjoint intervals, the joint entanglement wedge can connect even though the causal wedges obtained from signal round trips remain separate.

\mathcal E_A=D_{\rm bulk}(r_A),\qquad \mathcal C_A=J^+_{\rm bulk}\!\left(D_{\partial}(A)\right)\cap J^-_{\rm bulk}\!\left(D_{\partial}(A)\right)

A domain of dependence describes which events are fixed by initial data. Causal future and past describe signal accessibility. The spatial shade is only one slice of the entanglement wedge.

Source: Xi Dong

Why a bulk region leads to a reconstruction question

If the reduced state of a boundary region retains the observable information of a bulk region, operators supported on that boundary region should reproduce the relevant bulk operations. This is reconstruction: specify the allowed states and operators, then compare their actions and measurement statistics. The geometric region identifies where to ask that information question.

Dong, Harlow and Wall formulate it as a quantum-information statement on a code subspace in $1601.05416v2$ . States must stay within a controlled semiclassical setting. The reconstruction argument uses relative-entropy and generalized-entropy assumptions, with established expansion orders distinguished from higher-order assumptions. The next chapter uses a finite quantum code to calculate how operations can remain available after a subsystem is erased.

Source: Xi Dong

Try the reasoning

Predict the pairing before comparing the calculations

Choose A shared interior. Record the interval width and multiply it by the square root of two minus one to predict the continuum transition gap.
Increase Gap between intervals slowly. Compare the change of selected finite-cutoff curves with continuum leading mutual information in the right plot.
Open the cutoff settings and try small and large radial cutoffs. Explain why the scene crossing moves but the continuum reference does not. Save the calculation and compare both candidate totals.

Exercises and explanations

Multiply both interval widths and the gap by three. What happens to continuum $I_{\rm RT}/c$ ? Is finite-cutoff geometry also unchanged?
Read the explanation
Leading mutual information is unchanged because the logarithm contains only a ratio. For exact finite-cutoff lengths to remain unchanged, multiply the cutoff depth by three too. Moving only horizontal endpoints changes their scale relative to the cutoff.
Solve $d(2a+d)=a^2$ for the positive transition gap. Why reject the other root?
Read the explanation
Set $q=d/a$ to obtain $q^2+2q-1=0$ , with roots $-1\pm\sqrt2$ . Positive separation selects $q=\sqrt2-1$ . The negative root contradicts the assumed disjoint intervals.
Can disconnected spatial caps establish that every correlation between the two intervals vanishes?
Read the explanation
The caps are selected by classical minimal curves and control only the leading entropy. Full mutual information also includes quantum corrections. Exact factorization would require proving that the complete mutual information vanishes.
The shade is a two-dimensional spatial region. What remains to define its spacetime entanglement wedge?
Read the explanation
Identify the region bounded by the boundary intervals and their appropriate extremal curves, then take its bulk domain of dependence. A general construction needs the full spacetime and the applicable extremal-surface prescription. A spatial picture alone supplies no temporal boundary.

Sources and recalculated scope

The course derives its worked examples and draws its own schematics. The numerical references recalculate specified analytic geometry and a finite five-qubit encoding. The source locations below distinguish those calculations from theoretical reading topics.

Shinsei Ryu and Tadashi Takayanagi, Holographic Derivation of Entanglement Entropy from AdS/CFT, hep-th/0603001v2; PRL 96, 181602 (2006)
Equation (5), static area prescription; equations (6)–(10) and Figure 1, vacuum AdS3 interval geometry and the circle result. We independently integrate the Poincaré geodesic and compare regulator limits. Our equal-height finite-cutoff anchors are a separately defined geometric diagnostic. Multiple-interval minima use Headrick, not equation (12) of this early paper.
Matthew Headrick, Entanglement Rényi entropies in holographic theories, arXiv:1006.0047v3; PRD 82, 126010 (2010)
Sections 3.1.3 and 4.2, especially Figure 1: the two homologous pairings for two vacuum intervals, their transition and leading mutual information. We recalculate the equal-length specialization as the gap is varied; no numerical replica saddles or source artwork are reproduced.
Xi Dong, Daniel Harlow and Aron C. Wall, Bulk Reconstruction in the Entanglement Wedge in AdS/CFT, arXiv:1601.05416v2; PRL 117, 021601 (2016)
Sections I–II and Figure 1 define causal and entanglement wedges and a perturbative code subspace. Sections III–IV state the relative-entropy assumptions behind reconstruction. We explain these assumptions; the finite code checks are a separate exact model.