Entanglement Entropy and Subregion Complexity in Thermal Perturbations around Pure-AdS Spacetime

TL;DR

This work computes the changes in holographic entanglement entropy (HEE) and holographic subregion complexity (HSC) for spherical boundary subregions in both uncharged and charged AdS black hole backgrounds, using a perturbative expansion in the subregion size. It provides dimensionally dependent analytic expressions up to third order for HEE (and higher orders numerically) and for HSC up to comparable orders, revealing a consistent anti‑correlation of signs between ΔS and ΔC at each order. The authors connect these results to entanglement thermodynamics, proposing a refined first law that includes a complexity‑driven work term, and show this refinement holds to second order but fails at third, hinting at additional information‑theoretic quantities at play. They also carefully analyze embedding and boundary terms, showing how information encoded in the RT surface partitions between entropy and complexity, and they discuss broader implications for complexity definitions and phase transitions in holographic contexts.

Abstract

We compute the holographic entanglement entropy and subregion complexity of spherical boundary subregions in the uncharged and charged AdS black hole backgrounds, with the \textbf{change} in these quantities being defined with respect to the pure AdS result. This calculation is done perturbatively in the parameter $\frac{R}{z_{\rm h}}$, where $z_{\rm h}$ is the black hole horizon and $R$ is the radius of the entangling region. We provide analytic formulae for these quantities as functions of the boundary spacetime dimension $d$ including several orders higher than previously computed. We observe that the change in entanglement entropy has definite sign at each order and subregion complexity has a negative sign relative to entanglement entropy at each of those orders (except at first order or in three spacetime dimensions, where it vanishes identically). We combine pre-existing work on the "complexity equals volume" conjecture and the conjectured relationship between Fisher information and bulk entanglement to suggest a refinement of the so-called first law of entanglement thermodynamics by introducing a work term associated with complexity. This extends the previously proposed first law, which held to first order, to one which holds to second order. We note that the proposed relation does not hold to third order and speculate on the existence of additional information-theoretic quantities that may also play a role.

Figures (3)

• Figure 1: Plots of $y_1$, $y_2$ and $y_3$ in various dimensions. Note that we do not have $y_3$ for AdS${}_{12}$.
• Figure 2: Plots of $\frac{1}{p^2} y_{(0,1)}$ and $\frac{1}{p^2 (1 + p^2 )} y_{(1,1)}$ in various dimensions.
• Figure 3: Plots of $\Delta s$ and $\Delta c$ to second- and third-order in the uncharged black hole background. The points are explicitly calculated values. The curves are plots of the general formulae.