[Paper Review] Holographic entanglement entropy beyond coherent states

Curtis T. Asplund|arXiv (Cornell University)|May 22, 2012

Black Holes and Theoretical Physics31 references3 citations

TL;DR

This paper investigates holographic entanglement entropy in strongly-coupled conformal field theories (CFTs) for non-coherent states—specifically superpositions of spatially separated coherent states analogous to Bell states. Using the Ryu-Takayanagi formula, it identifies additional O(N²) contributions beyond the area law, implying that such states lack a classical bulk geometry dual and are incompatible with standard holographic descriptions, particularly relevant for local quenches.

ABSTRACT

We study entanglement entropy for a class of states in quantum field theory that are entangled superpositions of coherent states with well-separated supports, analogous to Einstein-Podolsky-Rosen or Bell states. We calculate the contributions beyond the area law in a simple model. In the case of strongly coupled conformal field theories, we argue that these states are holographically dual to superpositions of bulk geometries. We note that for these states one can use the Ryu-Takayanagi holographic entanglement entropy formula to calculate some terms in the entanglement entropy, but that there can be additional O(N^2) contributions. We argue that this class of states includes those generated by local quenches and thus that these cannot be described by a classical dual geometry. These considerations may be important for more fine grained treatments of holographic thermalization.

Motivation & Objective

• To investigate entanglement entropy in non-coherent CFT states that are superpositions of coherent states with well-separated supports.
• To determine whether such states can be described by a classical bulk geometry in the context of AdS/CFT.
• To identify additional O(N²) contributions to entanglement entropy beyond the Ryu-Takayanagi area law.
• To assess the implications for holographic thermalization, particularly in the context of local quenches.
• To generalize the Ryu-Takayanagi formula to non-coherent states and explore its limitations.

Proposed method

• Constructs a model of entangled CFT states as superpositions of coherent field states localized in spatially separated regions A and B, analogous to Bell states.
• Assumes a CFT with a holographic dual (e.g., N=4 SYM) and applies the Ryu-Takayanagi formula to compute entanglement entropy for a region A.
• Identifies that while the area law term is captured by the formula, additional O(N²) contributions arise due to quantum superposition of distinct bulk geometries.
• Analyzes the state after a local quench, showing it resembles the constructed non-coherent superposition and thus cannot be described by a single classical geometry.
• Uses the entanglement structure of such states to argue for non-classical duals, especially in strongly-coupled CFTs.
• Compares the entanglement entropy of entangled vs. unentangled superpositions to isolate non-area-law contributions.

Experimental results

Research questions

• RQ1 Can the Ryu-Takayanagi formula capture entanglement entropy for CFT states that are superpositions of coherent states with separated supports?
• RQ2 What is the nature and magnitude of entanglement entropy contributions beyond the area law in such non-coherent states?
• RQ3 Do local quench states in strongly-coupled CFTs admit a classical bulk geometry dual?
• RQ4 How do O(N²) contributions to entanglement entropy arise in holographic descriptions of non-coherent states?
• RQ5 Can the holographic entanglement entropy formula be generalized to describe superpositions of distinct bulk geometries?

Key findings

• For superpositions of coherent states with well-separated supports, the Ryu-Takayanagi formula captures the area law term but misses additional O(N²) contributions to entanglement entropy.
• These O(N²) contributions arise due to quantum superposition of distinct bulk geometries, implying no single classical geometry can describe the full state.
• Local quench states in CFTs are shown to be of this non-coherent type and thus cannot be described by a classical bulk geometry.
• The entanglement entropy from a local quench in 2D CFTs scales with the central charge, consistent with O(N²) contributions in the holographic limit.
• The results suggest that standard holographic thermalization models based on classical geometries (e.g., Vaidya) may be insufficient for non-coherent states, even to leading order in N.
• The study implies that more general, non-geometric dual descriptions are required for fine-grained treatments of holographic thermalization.