FR09-B · How far does an interval reach?

AdS Interval Entanglement

How can the size of a boundary region select a curve inside AdS? A wider interval selects a curve that reaches deeper into AdS. For two nearby intervals, a different pairing of endpoints can have less total length; the leading holographic entropy follows the shorter pairing.

Pull the interval endpoints apart and watch the shortest curves reach deeper. Bring two intervals together and watch the shorter pairing change the enclosed bulk region.

Selected regulated length $14.7L$

Deepest point $2.28L$

One interval minus unit interval $1.38L$

Continuum leading mutual information $I/c=0.157$

Model assumptions

The scene is a constant-time Poincaré slice of empty AdS in three spacetime dimensions, with curvature radius set to one. Endpoints are fixed on a common finite radial cutoff. The selected curves minimize regulated hyperbolic length among the two homology-compatible pairings; the shaded region is their spatial homology region. The separately labeled continuum entropy and mutual information use the leading classical Ryu–Takayanagi relation for a holographic vacuum. They omit bulk quantum corrections, backreaction and finite central-charge effects. A finite-cutoff length is a geometric diagnostic, not an exact regulated CFT entropy. The picture is a coordinate map with depth increasing downward; it is a theoretical calculation.

Calculated geometry · analytic reference

What survives when the cutoff moves?

Exact finite-cutoff geometryContinuum analytic limitIndependent metric integral

\left|\mathcal L_{\mathrm{int}}-\mathcal L_{\mathrm{exact}}\right|/L<10^{-6}

Read the comparison

The left plot subtracts the length for a unit-width interval at the same cutoff. The divergence cancels in the continuum limit; a finite cutoff leaves small corrections. The right plot shows the continuum leading RT result. A vanishing leading term can coexist with quantum correlations at the next order.

\frac{\mathcal L_{\epsilon}(\ell)}{L}=2\operatorname{arsinh}\!\frac{\ell}{2\epsilon}

\lim_{\epsilon\to0}\frac{\mathcal L_{\epsilon}(\ell)-\mathcal L_{\epsilon}(L)}{L}=2\log\frac{\ell}{L}

\frac{I_{\mathrm{RT}}}{c}=\frac13\max\!\left(0,\log\frac{\ell^2}{d(2\ell+d)}\right)

The first equation uses endpoints fixed at the finite cutoff. Ryu and Takayanagi’s boundary-anchored circle is instead truncated after anchoring at the boundary; both conventions approach the same logarithm. The two-interval minimization follows Headrick’s homology prescription.

Ryu–Takayanagi, equations 15–16 Headrick, equations 38–42 Read the course

Physics tutorial

AdS Interval Entanglement

BackgroundThe shortest path in a curved space can look bent on a coordinate map. In this AdS slice, motion near the boundary costs more proper length than the same coordinate motion deeper inside. A curve can save length by dipping inward before returning to its endpoint.

Why it mattersFor a holographic vacuum with a classical gravity description, the Ryu–Takayanagi relation turns the length of a suitable minimal curve into the entropy of a boundary interval. Comparing curves then tells us how the entropy changes when regions grow or approach each other.

Start with the essentials

Focus question: How can the size of a boundary region select a curve inside AdS?
One-sentence intuition: The minimal curve is selected by the curved-space metric and the endpoint condition; its apparent Euclidean shape alone does not determine its length.

Core mathematical model

Lengths on a fixed-time AdS slice

\begin{aligned}ds^2&=\frac{L^2}{z^2}(dx^2+dz^2)\\(x-x_0)^2+z^2&=R^2\end{aligned}

Depth is positive. The curvature radius is L; x and z have units of length. Geodesics joining two points at the same positive depth are arcs of circles centered on the boundary line.

Fix the cutoff before comparing lengths

\begin{aligned}R^2&=(\ell/2)^2+\epsilon^2\\\mathcal L_\epsilon(\ell)/L&=2\operatorname{arsinh}\!\left(\frac{\ell}{2\epsilon}\right)\\&=2\log(\ell/\epsilon)+O(\epsilon^2/\ell^2)\end{aligned}

Here the endpoints lie at z equal to the radial cutoff, and width is their horizontal separation there. Moving the cutoff changes the exact length. Subtracting a reference interval at the same cutoff removes the logarithmic divergence in the continuum limit.

The leading continuum entropy chooses a pairing

\begin{aligned}S_A&=\frac{\mathcal L_A}{4G_N}\\c&=\frac{3L}{2G_N}\\\frac{I_{\mathrm{RT}}}{c}&=\frac13\max\!\left(0,\log\frac{\ell^2}{d(2\ell+d)}\right)\end{aligned}

The final expression is for equal disjoint intervals in a vacuum with a leading classical Einstein-gravity description. The two noncrossing homology-compatible pairings compete. The continuum switch occurs when the gap divided by width equals the square root of two minus one. Finite-cutoff geometry and quantum corrections need separate treatment.

Common difficulties

A finite difference needs a limit

Typical misconceptionSubtracting two lengths makes the answer independent of every finite cutoff.

Better mental modelTheir logarithmic divergences cancel, but finite-cutoff corrections remain. The left plot shows those corrections approaching the continuum value.

The shaded region lies in a spatial slice

Typical misconceptionThe shaded region is a visible wormhole or a path for a signal.

Better mental modelIt is the spatial region bounded by the intervals and the selected curves. Causal propagation requires the spacetime metric and a separate time evolution.

Run the experiment

01
Predict how deep the curve goes
Choose One interval reaches deeper. Before widening it, predict whether the deepest point moves toward or away from the boundary. Drag either endpoint outward.
What to observe: The curve reaches deeper as the interval grows. The metric rewards that detour because coordinate motion costs less proper length at larger depth.
02
Make the two pairings exchange places
Choose A shared interior, then increase Gap between intervals. Predict the pairing before the status changes. Compare the shaded region before and after the switch.
What to observe: At close separation the outer and inner curves win together. Farther apart, each interval has its own curve. The right-hand plot follows the continuum leading mutual information; the scene compares exact finite-cutoff lengths.
03
Remove the common divergence
Choose Move the cutoff and open Cutoff, conventions and keyboard. Decrease Radial cutoff. Compare the total length with the length difference in the left plot.
What to observe: The total length grows as endpoints approach the boundary. The difference approaches a finite logarithm because the same divergent endpoint contribution appears in both intervals.

AdS Interval Entanglement

What survives when the cutoff moves?

AdS Interval Entanglement

Core mathematical model

Lengths on a fixed-time AdS slice

Fix the cutoff before comparing lengths

The leading continuum entropy chooses a pairing

Common difficulties

A finite difference needs a limit

The shaded region lies in a spatial slice

Run the experiment

Predict how deep the curve goes

Make the two pairings exchange places

Remove the common divergence

Related experiments and articles

From AdS signals to entanglement geometry and quantum recovery