Holographic Subregion Complexity from Kinematic Space

TL;DR

This paper establishes a volume formula that expresses bulk volumes on a fixed AdS$_3$ slice as a kinematic-space integral, $\frac{\mathrm{vol}(Q)}{4G_N}=\frac{1}{2\pi}\int_\mathcal{K} \lambda_Q \omega$, thereby connecting geometric quantities to the CFT entanglement structure. It then derives an explicit, entanglement-entropies-based expression for holographic subregion complexity in the vacuum, confirming that $\mathrm{vol}(\Sigma)$ can be computed purely from $S(u,v)$ via a double integral over kinematic space. The method extends to excited states (conical defects and BTZ black holes), where non-minimal geodesics and horizon contributions must be included, and yields a lower bound on complexity given by entanglement data alone. These results illuminate how vacuum subregion complexity is universal and encoded in the spectrum of single-interval entropies, while non-vacuum states reveal additional structure such as entwinement and thermal contributions. Collectively, the work strengthens the holographic complexity=volume proposal by providing a concrete CFT framework to compute subregion complexity and by clarifying the role of entanglement in holographic geometry.

Abstract

We consider the computation of volumes contained in a spatial slice of AdS$_3$ in terms of observables in a dual CFT. Our main tool is kinematic space, defined either from the bulk perspective as the space of oriented bulk geodesics, or from the CFT perspective as the space of entangling intervals. We give an explicit formula for the volume of a general region in the spatial slice as an integral over kinematic space. For the region lying below a geodesic, we show how to write this volume purely in terms of entangling entropies in the dual CFT. This expression is perhaps most interesting in light of the complexity=volume proposal, which posits that complexity of holographic quantum states is computed by bulk volumes. An extension of this idea proposes that the holographic subregion complexity of an interval, defined as the volume under its Ryu-Takayanagi surface, is a measure of the complexity of the corresponding reduced density matrix. If this is true, our results give an explicit relationship between entanglement and subregion complexity in CFT, at least in the vacuum. We further extend many of our results to conical defect and BTZ black hole geometries.

Figures (18)

• Figure 1: In AdS$_3$/CFT$_2$ the RT proposal states that the entanglement entropy of the region $A$ is given by the length of the geodesic $\gamma_A$ in the constant time slice that connects the boundary points of $A$. The volume of the region $\Sigma$ below $\gamma_A$ is proposed to be a measure for the complexity of the reduced density matrix corresponding to $A$.
• Figure 2: We can parametrize geodesics via their endpoints $u$ and $v$ or via the position of their center $\theta$ and their opening angle $\alpha$. The tuples $(\theta,\alpha)$ and $(\theta+\pi, \pi-\alpha)$ correspond to the same geodesic, but with opposite orientation. The geodesic with the orientation of the red arrow is associated with the entangling interval $[u,v]$, the geodesic with the orientation of the blue arrow is associated with the complement $[u,v]^c$.
• Figure 3: A point $p$ that lies in the constant time slice of asymptotic AdS$_3$ is associated with the set of all geodesics that intersect $p$ (LHS). This set is a curve in $\mathcal{K}$, the so-called point curve of $p$. The geodesic distance of two points $p$ and $p'$ is given, up to a proportionality factor, by the volume of the region $\Delta_{pp'}$ in $\mathcal{K}$ that is bounded by the point curves of $p$ and $p'$ (RHS). Since $(\theta, \alpha=0)$ correspond to boundary points of $\text{AdS}_3$, the lower boundary of $\mathcal{K}$ is identified with the constant time slice of the CFT depicted in green (LHS).
• Figure 4: The volume of a region $Q$ on the constant time slice is given by an integral over the chord lengths of all geodesics. The chord length of a geodesic is the length of the segment of the geodesic that lies inside of $Q$ (depicted in red).
• Figure 5: The disc $D_R$ associates an opening angle $\alpha_R$ on its boundary to each geodesic $(\theta,\alpha)$. Geodesics of the form $(\theta,\alpha_*)$ are tangent to $D_R$.