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

## TL;DR

This paper analytically computes the changes in holographic entanglement entropy and subregion complexity in thermal perturbations of pure-AdS spacetime, revealing sign patterns and proposing a refined first law of entanglement thermodynamics.

## Contribution

It provides higher-order analytic formulas for entanglement entropy and complexity changes, and introduces a refined first law incorporating complexity effects.

## Key findings

- Change in entanglement entropy has definite sign at each order.
- Subregion complexity is negative relative to entanglement entropy at each order (except first order or in three dimensions).
- Proposes a second-order extension of the first law of entanglement thermodynamics including complexity.

## 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.