The Information Theoretic Interpretation of the Length of a Curve

TL;DR

<p>We provide an information-theoretic interpretation of the length of convex curves in AdS$_3$ by mapping bulk geometry to a boundary quantum information task. The key is differential entropy, which expresses curve length as a sum of conditional entropies $S(A_j|B_j)$ arising from a constrained, scale- and location-restricted state merging protocol between boundary regions; the resulting cost equals the curve length (up to $O(1/\,\sqrt{c})$ corrections at large central charge). We prove optimality: no constrained merging strategy can beat this cost, thus tying geometric length to an entanglement-cost minimization problem, and we show that single-shot entropies converge to von Neumann entropies in the large-$c$ limit. The results extend to closed curves, discuss Markov-chain reconstructions, and illuminate how holographic RG and boundary entanglement structure encode bulk geometry, while also outlining limitations and directions for generalization to non-convex surfaces and higher dimensions.

Abstract

In the context of holographic duality with AdS3 asymptotics, the Ryu-Takayanagi formula states that the entanglement entropy of a subregion is given by the length of a certain bulk geodesic. The entanglement entropy can be operationalized as the entanglement cost necessary to transmit the state of the subregion from one party to another while preserving all correlations with a reference party. The question then arises as to whether the lengths of other bulk curves can be interpreted as entanglement costs for some other information theoretic tasks. Building on recent results showing that the length of more general bulk curves is computed by the differential entropy, we introduce a new task called constrained state merging, whereby the state of the boundary subregion must be transmitted using operations restricted in location and scale in a way determined by the geometry of the bulk curve. Our main result is that the cost to transmit the state of a subregion under the conditions of constrained state merging is given by the differential entropy and hence the signed length of the corresponding bulk curve. When the cost is negative, constrained state merging distills entanglement rather than consuming it. This demonstration has two parts: first, we exhibit a protocol whose cost is the length of the curve and second, we prove that this protocol is optimal in that it uses the minimum amount of entanglement. In order to complete the proof, we additionally demonstrate that single-shot smooth conditional entropies for intervals in 1+1-dimensional conformal field theories with large central charge are well approximated by their von Neumann counterparts. We also revisit the relationship between the differential entropy and the maximum entropy among locally consistent density operators, demonstrating large quantitative discrepancy between the two quantities in conformal field theories.

Figures (7)

• Figure 1: a) Geodesic $g_I$, which subtends a boundary interval $I$ (black) and geodesics, which are tangent to $g_I$ on the boundary along with the corresponding boundary intervals $I(x)$ (color). The dashed geodesics contribute zero to integral (\ref{['main']}). b) $a_I(x)$, the linear size of the interval $I(x)$ centered at $x$.
• Figure 2: a) A curve, which asymptotes to the geodesic $g_I$. We have marked in color the geodesics tangent to the curve and the boundary intervals $J(x)$, which they subtend. b) a plot of $a_J(x)$ -- the length of the intervals $J(x)$ as a function of the centerpoint $x$. If the curve asymptotes to the geodesic $g_I$ then $a_J(x)$ must agree with $a_I(x)$ (shown for comparison in dashed gray) outside some interval $(x_L, x_R)$lampros.
• Figure 3: The way to read off $\bar{a}_{I,J}(r)$ from the plot of $a_{I,J}(x)/2$.
• Figure 4: a) The intervals $J(\theta)$ defined by a closed, convex curve in global AdS$_3$. b) The intervals (\ref{['alice0']}-\ref{['bob0']}) initially held by Alice and Bob. We have indicated $J(\theta_N)$ and $J(\theta_{2N})$ and the bulk axis, which joins their centers.
• Figure 5: Intermediate stage in a constrained merging protocol. The top row depicts the interval $I = [L,R]$ with endpoints $L$ and $R$. Step $x$ of the protocol acts on interval $I(x) = [\ell(x),r(x)]$, drawn in the second row. Because $r(x)$ is non-decreasing with $x$, the sites marked by the orange bar have not yet been acted upon. The third row depicts the interval $I(x+1) = [\ell(x+1),r(x+1)]$. Once step $x$ is complete, none of the sites indicated by the green bar will ever be acted upon again since $\ell(x)$ is non-decreasing with $x$. Therefore, after step $x$ the reduced density operators corresponding to the green marked sites on $B$ and the orange marked sites on $A$ should approximate the reduced density operator of the target state ${|\psi\rangle}$.

Theorems & Definitions (3)

• Theorem 3.1
• Theorem A.1
• Theorem D.1