Holographic duality from random tensor networks (1601.01694v3)

Abstract: Tensor networks provide a natural framework for exploring holographic duality because they obey entanglement area laws. They have been used to construct explicit toy models realizing many interesting structural features of the AdS/CFT correspondence, including the non-uniqueness of bulk operator reconstruction in the boundary theory. In this article, we explore the holographic properties of networks of random tensors. We find that our models naturally incorporate many features that are analogous to those of the AdS/CFT correspondence. When the bond dimension of the tensors is large, we show that the entanglement entropy of boundary regions, whether connected or not, obey the Ryu-Takayanagi entropy formula, a fact closely related to known properties of the multipartite entanglement of assistance. Moreover, we find that each boundary region faithfully encodes the physics of the entire bulk entanglement wedge. Our method is to interpret the average over random tensors as the partition function of a classical ferromagnetic Ising model, so that the minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the analog of a bulk field, we find that our model reproduces the expected corrections to the Ryu-Takayanagi formula: the minimal surface is displaced and the entropy is augmented by the entanglement of the bulk field. Increasing the entanglement of the bulk field ultimately changes the minimal surface topologically in a way similar to creation of a black hole. Extrapolating bulk correlation functions to the boundary permits the calculation of the scaling dimensions of boundary operators, which exhibit a large gap between a small number of low-dimension operators and the rest. While we are primarily motivated by AdS/CFT duality, our main results define a more general form of bulk-boundary correspondence which could be useful for extending holography to other spacetimes.

• The paper demonstrates that random tensor networks naturally reproduce the Ryu-Takayanagi entropy formula for boundary regions.
• The authors map the entanglement structure to a classical Ising model, showcasing effective entanglement wedge reconstruction.
• The study confirms bidirectional holographic codes and converging Renyi entropies, underscoring emergent quantum error correction.

Insightful Overview of "Holographic Duality from Random Tensor Networks"

The paper "Holographic duality from random tensor networks" by Hayden et al. provides a novel approach to understanding holographic duality through the use of random tensor networks. This approach is motivated by the entanglement area laws obeyed by tensor networks, aligning them naturally with the Ryu-Takayanagi (RT) entropy formula central to the AdS/CFT correspondence. The work presented in the paper explores how networks of random tensors can model key features of holographic duality, including entanglement behavior and bulk-boundary correspondence.

Key Contributions

1. Emergence of RT Entropy Formula: The authors demonstrate that tensor networks constructed from random tensors reproduce the RT entropy formula for boundary regions, encapsulating both connected and disconnected regions. This result is significant as it supports the notion that holographic duality might naturally emerge from random tensor structures.
2. Entanglement Analysis: By mapping the problem to classical statistical models, specifically a ferromagnetic Ising model, the authors explore the entanglement properties within the tensor network. Unique entanglement wedge reconstruction is achieved, indicating that each boundary strongly encodes the bulk's physical information faithfully.
3. Behavior of Renyi Entropies: The paper contrasts Renyi entropies in random tensor models with those in AdS/CFT, noting that all Renyi entropies in the model converge to the same value in the large bond dimension limit. This convergence highlights a simplification that arises from using random tensors but diverges in complexity from gravity duals in AdS/CFT.
4. Bidirectional Holographic Codes: An important result is the validation of bidirectional holographic codes (BHCs). The random tensors provide a bulk-to-boundary isometry with error correction properties, showcasing emergent quantum error correction behaviors and strengthening the argument for viewing holographic duality through quantum information lenses.

Implications and Future Directions

Theoretical Insights: The use of random tensor networks provides a rich theoretical framework that aligns with quantum gravity requirements, notably demonstrating intrinsic nonlocality and emergent sub-AdS locality within the bulk theory. The findings suggest that gravitation and holography might inherently involve random structures, offering a new path for exploring quantum gravity models.

Practical Applications: From the perspective of quantum error correction and information storage, the results imply that random tensor networks could provide robust templates for encoding information in a manner reminiscent of holographic duality. This could be particularly pertinent for developing quantum systems that rely on error correction in highly entangled states.

Speculation on Future Developments: The paper paves the way for expanding the scope of holographic theories to non-AdS spacetimes using generalized forms of bulk-boundary correspondences. As computation and entanglement properties get integrated into holographic considerations, future work could explore time-dependent and dynamic settings, potentially informing better our understanding of holography in non-static or evolving spacetimes.

In conclusion, "Holographic duality from random tensor networks" contributes a sophisticated perspective to the holographic principle. It reaffirms entanglement's pivotal role and posits that randomness, through tensor networks, might be a fundamental characteristic of holography. This work positions itself as a bridge toward more generalized and perhaps more practical implementations of holographic principles in quantum gravity and quantum information science.