Causal Holographic Information

TL;DR

The authors introduce causal holographic information, $\chi_{\cal A}$, derived from the area of the causal information surface $\Xi_{\cal A}$ within the bulk causal wedge $\blacklozenge_{\cal A}$ anchored to a boundary region ${\cal A}$. They show that $\chi_{\cal A}$ is not a von Neumann entropy and generally differs from the entanglement entropy $S_{\cal A}$, though it provides an upper bound on $S_{\cal A}$ and coincides with $S_{\cal A}$ in several symmetric or maximally entangled scenarios. The paper analyzes precise matches in 1+1D CFTs on ${\bf S}^1$, and in cases with spherical entangling surfaces or vacuum states in $d$-dimensional CFTs, where conformal mappings relate $S_{\cal A}$ to thermal entropies and the surfaces align. The discussion emphasizes the potential of $\chi_{\cal A}$ as a lower bound on recoverable bulk information from boundary data and considers extensions via a family of $\chi_{\cal Q}$ to aid bulk reconstruction and metric determination, outlining future research directions in holographic information and bulk emergence.

Abstract

We propose a measure of holographic information based on a causal wedge construction. The motivation behind this comes from an attempt to understand how boundary field theories can holographically reconstruct spacetime. We argue that given the knowledge of the reduced density matrix in a spatial region of the boundary, one should be able to reconstruct at least the corresponding bulk causal wedge. In attempt to quantify the `amount of information' contained in a given spatial region in field theory, we consider a particular bulk surface (specifically a co-dimension two surface in the bulk spacetime which is an extremal surface on the boundary of the bulk causal wedge), and propose that the area of this surface, measured in Planck units, naturally quantifies the information content. We therefore call this area the causal holographic information. We also contrast our ideas with earlier studies of holographic entanglement entropy. In particular, we establish that the causal holographic information, whilst not being a von Neumann entropy, curiously enough agrees with the entanglement entropy in all cases where one has a microscopic understanding of entanglement entropy.

Figures (6)

• Figure 1: Illustration of the causal sets $D$ and $J$ associated with a 1-dimensional spacelike region ${\cal A}$. The future (past) domain of dependence$D^{\pm}[{\cal A}]$ is the set of points which are fully determined by future (past) evolution of the 'initial data' on ${\cal A}$. The future (past) domain of influence$J^{\pm}[{\cal A}]$ is the set of points which can be causally influenced by (or can influence) ${\cal A}$.
• Figure 2: Illustration of various causal sets associated with the boundary region ${\cal A}$ (color online). AdS boundary is the plane at $z=0$ on the right; the bulk extends to the left of this plane. The region ${\cal A}$ is the red segment at $z=0, t=0$. The future and past bulk domains of influence of ${\cal A}$ are bounded by yellow and green surfaces respectively, and future and past boundaries of the bulk causal wedge $\blacklozenge_{{\cal A}}$ are indicated by the red and blue surfaces respectively. Their intersection with the AdS boundary encloses the boundary domain of dependence $\Diamond_{\cal A}$, and their intersection with each other (light-blue curve) corresponds to the causal information surface ${\Xi}_{{\cal A}}$. For simplicity we illustrate these constructs in Poincare AdS; the causal wedge in global AdS$_3$ appears in Fig. \ref{['f:cwedge3d']}(a) (which shows ${\cal A}$ corresponding to half the circle; causal wedge of any other interval would be obtained simply by translating one of the null planes with respect to the other).
• Figure 3: Sketch to illustrate the fact the causal information surfaces ${\Xi}_{\cal A}$ and ${\Xi}_{{\cal A}^c}$ for a region ${\cal A}$ and its complement ${\cal A}^c$ have to lie closer to the respective boundary regions than the common extremal surface ${\mathfrak E}_{\cal A} = {\mathfrak E}_{{\cal A}^c}$.
• Figure 4: Illustration of the causal wedges $\blacklozenge_{{\cal A}}$ in three dimensional asymptotically globally AdS$_{3}$ spacetimes. The three figures correspond to the three geometries described in Table \ref{['t:adsthree']}. For convenience we have chosen the region ${\cal A}$ to be a half of the boundary ${\bf S}^1$, i.e., $\varphi_0 = \pi$. At the intersection of the $\partial_+(\blacklozenge_{{\cal A}})$ and $\partial_-(\blacklozenge_{{\cal A}})$ lies the causal information surface ${\Xi}_{\cal A}$ which as we discuss in the text is the same as the extremal surface ${\mathfrak E}_{\cal A}$ in these examples. Note that for the static spacetimes (a) and (b) which correspond to AdS3 and the static BTZ geometry, the surfaces at a fixed time slice $t=0$ as shown, while for the stationary rotating BTZ geometry (c), this surface dips above and below the $t=0$ slice in the bulk. [Note that for ease of visualization, we have changed the viewpoint between the three plots. Also, note that the 'seams' are just numerical glitches.]
• Figure 5: Sketch accompanying the argument in main text for why the extremal surface ${\mathfrak E}_{\cal A}$ cannot lie closer to the boundary than the causal information surface ${\Xi}_{\cal A}$. Left: we argue that at $p$, $\Theta_{\alpha} < \Theta_{\beta}$. Right: impossible situation, since it contradicts the physical requirement that $\Theta_{{\Xi}_{\tilde{{\cal A}}}} \ge 0$ and $\Theta_{{\mathfrak E}_{\cal A}} = 0$ everywhere.