Causal Holographic Information (1204.1698v2)

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.

• The paper introduces causal holographic information as a new measure quantifying information in boundary field theories through a causal wedge construction.
• It contrasts this measure with holographic entanglement entropy by highlighting differences such as the failure to satisfy strong subadditivity.
• The study shows that in symmetric, maximally entangled cases, causal holographic information coincides with entanglement entropy, guiding future quantum gravity research.

Causal Holographic Information: A Summary

The paper "Causal Holographic Information" by Veronika E. Hubeny and Mukund Rangamani proposes a novel approach to understanding the holographic reconstruction of spacetime, with a focus on quantifying the "amount of information" contained in specific spatial regions of boundary field theories. The work introduces the concept of causal holographic information, denoted as $\chi_{\cal A}$, which seeks to measure the information content of a spatial region ${\cal A}$ within boundary field theories using a causal wedge construction. This measure is based on the area of a specific bulk surface, the causal information surface $\csf{\cal A}$, which is defined as a co-dimension two extremal surface within a bulk causal wedge.

The causal wedge construction is motivated by the desire to identify how much of the bulk geometry can be reconstructed from a given part of the boundary field theory when the entire field theory path integral maps to the overall bulk geometry. Causal structure is considered a fundamental aspect of spacetime and is used to delineate the boundaries of the bulk regions associated with boundary regions.

Key Components

1. Causal Wedge and Causal Holographic Information:
   • The causal wedge, denoted as ${\cal C}[{\cal A}]$, is the region in the asymptotically anti-de Sitter (AdS) space that is both causally influenced by and influences a boundary region ${\cal A}$.
   • $\chi_{\cal A}$ is defined as the area of the causal information surface in Planck units, providing a measure of information that is pertinent to the bulk causal wedge.
2. Comparison with Holographic Entanglement Entropy:
   • The authors compare $\chi_{\cal A}$ with the notion of holographic entanglement entropy, $S_{\cal A}$, which uses extremal surfaces in the bulk to compute the entanglement entropy of a boundary region.
   • While $S_{\cal A}$ is traditionally tied to entanglement entropy, the causal holographic information $\chi_{\cal A}$ is proposed not to capture entropy in the same way and is shown not to satisfy strong subadditivity, a key property of entanglement entropy.
3. Agreement in Specific Cases:
   • In some scenarios, such as certain symmetric configurations in 1+1 dimensional conformal field theories (CFTs), $\chi_{\cal A}$ agrees with $S_{\cal A}$. These instances tend to be cases where the regions share maximal entanglement with their complements.
   • Such agreements are ascribed to situations where the information is maximally entangled, suggesting equivalence between these quantities under specific symmetrical and maximally entangled conditions.
4. Implications and Speculations:
   • $\chi_{\cal A}$ is proposed to serve as a potential upper bound for entanglement entropy, although it does not exhibit the properties of a traditional entropy measure.
   • The concept encourages revisiting the reconstruction of bulk geometry, proposing that the causal wedge represents the minimal bulk area recoverable from boundary data.
   • The authors speculate on the potential of $\chi_{\cal A}$ to inspire new insights into the encoding of bulk geometry by boundary theories and the further understanding required to unify these descriptions.

Conclusion and Future Directions

While the causal holographic information offers a new perspective and measure regarding boundary theory data's reflection of bulk geometry, further research is needed to understand how this concept can be systematically applied and possibly reconciled with other measures such as entanglement entropy. This paper suggests a meaningful direction for future exploration in theories of quantum gravity and AdS/CFT correspondence, potentially revealing deeper insights into the entanglement structure and data encoding of quantum gravity theories.