Constructing holographic spacetimes using entanglement renormalization (1209.3304v1)

Abstract: We elaborate on our earlier proposal connecting entanglement renormalization and holographic duality in which we argued that a tensor network can be reinterpreted as a kind of skeleton for an emergent holographic space. Here we address the question of the large $N$ limit where on the holographic side the gravity theory becomes classical and a non-fluctuating smooth spacetime description emerges. We show how a number of features of holographic duality in the large $N$ limit emerge naturally from entanglement renormalization, including a classical spacetime generated by entanglement, a sparse spectrum of operator dimensions, and phase transitions in mutual information. We also address questions related to bulk locality below the AdS radius, holographic duals of weakly coupled large $N$ theories, Fermi surfaces in holography, and the holographic interpretation of branching MERA. Some of our considerations are inspired by the idea of quantum expanders which are generalized quantum transformations that add a definite amount of entropy to most states. Since we identify entanglement with geometry, we thus argue that classical spacetime may be built from quantum expanders (or something like them).

• The paper establishes that tensor networks in entanglement renormalization offer a theoretical scaffold for constructing holographic spacetimes in the large-N limit.
• It shows that classical geometry emerges from quantum states via quasi-random tensor networks that mimic quantum expanders.
• The work reveals a sparse spectrum of operator dimensions correlating with bulk locality and disentanglement in strongly interacting quantum theories.

Exploring the Connections between Entanglement Renormalization and Holographic Duality

The paper "Constructing Holographic Spacetimes Using Entanglement Renormalization," authored by Brian Swingle, investigates the potential intrinsic relationship between entanglement renormalization (ER) and holographic duality. This paper provides a compelling theoretical framework suggesting that the tensor networks used in entanglement renormalization may serve as scaffolds for constructing holographic spacetimes, particularly under the large $N$ limit—a fundamental context in which the gravity theory becomes classical and spacetime achieves smoothness without fluctuations.

Theoretical Foundations and Implications

Swingle begins by grounding the discussion in the conceptual overlap between ER and holography. Entanglement renormalization is inherently concerned with organizing quantum information across different scales, thus introducing a natural geometrical interpretation of quantum states. This approach mirrors aspects of the AdS/CFT correspondence, established within string theory, where a bulk gravitational theory in a higher-dimensional space is dual to a conformal field theory (CFT) on the boundary. In particular, when moving towards the large $N$ limit, these theories predict a classical geometric space emergent from quantum entanglement.

One of the pivotal discussions in this paper is the emergence of classical geometry from quantum states. Swingle posits that ER can depict a continuum picture by employing generic — in a sense, quasi-random — tensors, which mimic quantum expanders: constructs in quantum information theory that introduce controlled entanglement into the system. The paper draws on the notion that classical spacetime can be seen as emerging from entanglement structure, providing support to the conjecture that spacetime in theories like AdS/CFT is fundamentally quantum in origin.

Spectrum of Operator Dimensions and Scaling Mechanisms

A notable highlight is the emergence of a sparse spectrum of operator dimensions in the large $N$ regime. Swingle demonstrates that strongly interacting theories, described by ER with quasi-random tensors, lead naturally to a preference for operators with large scaling dimensions throughout the tensor network. This intrinsic mechanism offers an explanation for the behavior observed in holographic duality, where a sparse spectrum correlates with bulk locality and the disentanglement of degrees of freedom in field theory. Consequently, this perspective provides a bridge between the microscopic entanglement properties encapsulated in ER and macroscopic emergent geometries in holography.

Mutual Information, Causality, and Additional Geometrical Constructs

Mutual information plays a significant role in linking the two theories. Swingle contends that ER naturally accounts for mutual information in a way that aligns with holographic predictions, including phase transitions in mutual information that reflect changes in the underlying entanglement structure. He explores `RG causality' within ER, suggesting that causal cones formed during renormalization could map onto identifiable bulk regions in a holographic dual, potentially aligning with ideas related to bulk reconstruction and quantum error correction.

The motivation extends to branching ER networks, which can model the growth of additional dimensions and consequently systems like Fermi surfaces exhibiting a logarithmic violation of the area law in their entanglement properties. These branching MERA constructions parallel scenarios in holography where geometry and entropic measures of boundary systems indicate hidden dimensional structures and strongly correlated phases.

Broader Implications and Future Directions

Conclusively, Swingle substantializes the hypothesis that ER is not merely a computational tool for many-body systems but a potential parallel realization of holography when adapted to the large $N$ context. The theoretical implications are profound, suggesting that tools from quantum information could further explore quantum gravity's microstructures and provide a new lens through which spacetime and quantum matter relationships can be discerned.

The paper encourages further investigation into the detailed matching of ER and holographic predictions, especially concerning whereabouts the Ryu-Takayanagi formula holds or requires modification beyond leading orders in $N$. Additionally, potential applications in more computational settings or more physically realized systems (e.g., in finite temperature states involving black holes) remain significant questions driving the field forward.

This work is a linchpin in fostering dialogue between condensed matter physics, quantum information, and high-energy theoretical physics, urging a reexamination of the foundational assumptions behind spacetime emergence from quantum entanglement. As the connection between these domains continues to unravel, the triangulation of insights from various branches of physics promises to illuminate the intricate tapestry of our universe's quantum underpinnings.