Relative Entropy and Holography

TL;DR

The paper investigates relative entropy in holography by comparing the vacuum to neighboring states using spherical entangling surfaces. It establishes that linear perturbations saturate the first-law relation ΔS = Δ⟨H⟩, while quadratic and higher-order effects render Δ⟨H⟩ > ΔS and ensure the positivity of relative entropy; this is shown across simple bulk backgrounds and generalized via Fefferman–Graham expansion. The results support the holographic entropy formula, illuminate the structure of the modular Hamiltonian, and hint at vacuum-state tomography, with broader implications for Bekenstein-type bounds and energy localization in quantum field theories. The analysis spans higher-dimensional AdS/CFT setups, two-dimensional CFTs, and special geometries like slabs and annuli, revealing both local and nonlocal features of entanglement in holographic states.

Abstract

Relative entropy between two states in the same Hilbert space is a fundamental statistical measure of the distance between these states. Relative entropy is always positive and increasing with the system size. Interestingly, for two states which are infinitesimally different to each other, vanishing of relative entropy gives a powerful equation $ΔS=ΔH$ for the first order variation of the entanglement entropy $ΔS$ and the expectation value of the \modu Hamiltonian $ΔH$. We evaluate relative entropy between the vacuum and other states for spherical regions in the AdS/CFT framework. We check that the relevant equations and inequalities hold for a large class of states, giving a strong support to the holographic entropy formula. We elaborate on potential uses of the equation $ΔS=ΔH$ for vacuum state tomography and obtain modified versions of the Bekenstein bound.

Figures (3)

• Figure 1: (Colour Online) Extremal surfaces in the high temperature phase. The figures show a cross-section of the AdS$_3$ black hole at constant $t$. (a) For sufficiently small $\Delta\phi$, the holographic entanglement entropy (\ref{['define']}) is evaluated with the red geodesic. The dashed green geodesic passing on the other side of the black hole is not homologous to the interval $V$, however, it would yield the entanglement entropy for the complementary interval $\bar{V}$. (b) For large $\Delta\phi$, the dominant saddle-point (in green) has two disconnected components, i.e., the geodesic homologous to $\bar{V}$ and the geodesic wrapping around the horizon.
• Figure 2: Comparing $\Delta \langle H\rangle$ and $\Delta S$ in the high temperature phase. Panel (a) shows the log of the relative entropy and panel (b), the ratio $\Delta S/\Delta \langle H\rangle$, both as functions of angular size $\Delta\phi\in (0,2\pi)$. The different curves are for $\beta/R=2\pi \frac{i}{10}$ with $i=1,...,10$. Curves corresponding to higher temperature (smaller $\beta$) have greater relative entropy in (a) and lower ratios $\Delta S/\Delta \langle H\rangle$ in (b).
• Figure 3: The annular region on the AdS boundary is shown with the two solid lines. When the radius $R_1$ and $R_2$ of the annulus approach each other the minimal surface has the shape of a half torus connecting the two spheres (left panel). When $R_2/R_1$ is greater than a certain value the minimal surface is formed by the two spherical caps ending at the spheres of radius $R_1$ and $R_2$ at the boundary (right pannel).