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.
Highlights
The anti-de Sitter (AdS)=conformal field theory (CFT) correspondence [1,2] has proven to be a powerful tool to study quantum field theories by passing to dual gravitational theories (e.g., Refs. [3,4,5])
We compute the holographic entanglement entropy and subregion complexity of spherical boundary subregions in the uncharged and charged anti-de Sitter (AdS) black hole backgrounds, with the change in these quantities being defined with respect to the pure AdS result
This extends the previously proposed first law, which held to first order, to one that holds to second order
Summary
The AdS=CFT correspondence [1,2] has proven to be a powerful tool to study quantum field theories by passing to dual gravitational theories (e.g., Refs. [3,4,5]). There has been some work in finding higher-order corrections in the RT term itself (these are higher order in a small parameter measuring the perturbation away from pure AdS, e.g., the AdS black hole mass) [25,26,27] These corrections are expected to be related to the change of energy density and pressure density of the gravity theory in the same way as the. Combining previous work on holographic complexity [32] and Fisher information [39], we propose a refinement of the first law of entanglement thermodynamics to include a general work term done on the system: ΔE 1⁄4 TΔS þ W.1 These previous works naturally suggest that this work term is related to the change in HSC.
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