Holographic Entanglement Entropy (1609.01287v2)

Abstract: We review the developments in the past decade on holographic entanglement entropy, a subject that has garnered much attention owing to its potential to teach us about the emergence of spacetime in holography. We provide an introduction to the concept of entanglement entropy in quantum field theories, review the holographic proposals for computing the same, providing some justification for where these proposals arise from in the first two parts. The final part addresses recent developments linking entanglement and geometry. We provide an overview of the various arguments and technical developments that teach us how to use field theory entanglement to detect geometry. Our discussion is by design eclectic; we have chosen to focus on developments that appear to us most promising for further insights into the holographic map. This is a draft of a few chapters of a book which will appear sometime in the near future, to be published by Springer. The book in addition contains a discussion of application of holographic ideas to computation of entanglement entropy in strongly coupled field theories, and discussion of tensor networks and holography, which we have chosen to exclude from the current manuscript.

• The paper establishes that entanglement entropy in quantum field theories is computed via classical gravitational dynamics using the RT and HRT proposals.
• It reveals that the area formula for extremal surfaces mirrors black hole entropy and drives phase transitions in strongly coupled systems.
• The work implies that spacetime geometry may emergently arise from quantum entanglement, providing new insights into gravity and quantum cosmology.

Holographic Entanglement Entropy: Quantum and Geometric Interfaces

The paper "Holographic Entanglement Entropy" by Mukund Rangamani and Tadashi Takayanagi provides a comprehensive exploration of the intricate relationship between quantum entanglement and spacetime geometry in the context of gauge/gravity duality. This essay offers a structured overview of their work, highlighting key results, implications, and theoretical developments that stem from the holographic perspective on entanglement, especially through the lens of the AdS/CFT correspondence.

The primary focus is on how entanglement entropy in quantum field theories (QFTs) can be computed using classical gravitational dynamics, inherently suggesting a duality between quantum and geometric descriptions. This endeavor leverages the RT and HRT (Ryu-Takayanagi and Hubeny-Rangamani-Takayanagi) proposals, which relate the entanglement entropy of a boundary region in a QFT to the area of an extremal surface in a higher-dimensional anti-de Sitter (AdS) spacetime. The RT proposal initially addressed static configurations, while the HRT extension accommodated dynamic settings, thereby broadening the scope and applicability of the holographic entanglement hypothesis.

Key Numerical Results and Claims

One of the pivotal results in holographic studies is the area formula for entanglement entropy, reminiscent of the Bekenstein-Hawking entropy for black holes, yet applicable here to extremal surfaces anchored to the boundary of an AdS space. This approach effectively captures the leading order contribution in the large central charge limit. For example, the entanglement entropy of spherical regions shows a surprisingly universal behavior linked to central charges of the underlying CFT, thereby providing a bridge to classical notions of geometry.

Interestingly, the paper reviews applications and calculations, elucidating scenarios where entanglement entropy facilitates understanding phase transitions between geometrical states, evidenced in CFTs at large central charge where certain holographic duals induce phase transitions analogous to those in statistical mechanics. The phase space involved delineates between connected and disconnected extremal surfaces, contingent on the region's size relative to boundary conditions, thereby making bold assertions on the geometrical realization of linked quantum regions.

Theoretical and Practical Implications

The implications of holographic entanglement transcend the boundaries of traditional quantum theory, proposing a framework where spacetime itself emerges from quantum entanglement. This fosters a paradigm shift towards understanding gravity as an emergent phenomenon from the entanglement structure within a quantum system. Practically, this could revolutionize areas such as quantum information theory, black hole thermodynamics, and even quantum cosmology, by subsuming gravity under quantum principles through purely entanglement-driven geometric constructs.

Another pivotal claim is the universality of certain entanglement behaviors at large central charge, further implying that holography captures an essential aspect of strongly-coupled systems. This linkage is not merely coincidental; it uncovers potential pathways for exploring non-perturbative regimes in field theories through classical geometric duals.

Speculation on Future Developments

Looking ahead, the rich interplay between quantum entanglement and geometry poses intriguing questions especially about the nature of spacetime locality and the privacy of bulk information segments (entanglement wedges) within boundary theories. Exploring the limits and breakdowns of entanglement heralds new inquiries into quantum error correction, as the bulk-boundary mapping suggests redundancy and fault tolerance inherent in spacetime fabric.

There is an optimistic horizon where further granularity in holographic dualities might provide deeper insights into the quantum structure of spacetime, potentially reconciling the apparent discrepancies with quantum gravity formalisms. The speculative yet promising nature of holographic entanglement entropy offers a substantial bedrock for future explorations into marrying information theory with gravitational dynamics, further guided by this groundbreaking discourse that Rangamani and Takayanagi so intricately illustrate.

In conclusion, "Holographic Entanglement Entropy" serves as a cornerstone for understanding the quantum-geometric interface, bringing us a step closer to unraveling the profound mysteries of the universe via the elegant dance of mathematics and physics in the AdS/CFT framework.