Can Bell-CHSH and relative entropy techniques probe holographic spacetime structure?

Statement

The subregion-subalgebra-duality in AdS/CFT identifies bulk spacetime regions with emergent type III₁ boundary subalgebras --- the same algebraic setting in which Marcelo’s Weyl operator Bell-CHSH tests and Araki-Uhlmann relative entropy computations operate. Can these tools, developed for flat-space QFT, be applied to holographic boundary theories to extract information about bulk causal structure, spacetime connectivity, or gravitational entropy?

Why It Matters

The Bell inequality program and relative entropy program have developed a sophisticated and computationally tractable framework for probing entanglement in type III${}_1$ algebras using Weyl operators and Tomita-Takesaki modular theory. If this framework can be adapted to holographic CFTs, it would provide new computable probes of bulk geometry. In particular:

• Bell-CHSH violations between boundary subregions could detect whether the corresponding bulk regions are causally connected or disconnected.
• Relative entropy between boundary states could provide a direct algebraic route to the gravitational entropy, bypassing the Ryu-Takayanagi area formula.
• The crossed product construction connecting type III${}_1$ (boundary) to type II${}_{\infty }$ (gravitational) could be tested by comparing relative entropy computations at the two levels.

What We Know

• Liu’s review (2025-liu-lectures-entanglement-vna) establishes that boundary subalgebras in the large $N$ limit are type III${}_1$ and that modular flows generate bulk geometric flows.
• The algebraic ER=EPR proposal identifies spacetime connectivity with algebraic commutant structure, suggesting that entanglement measures (Bell violations, relative entropy) between boundary regions encode bulk topology.
• For free-field QFT in flat space, the Bell-CHSH and relative entropy techniques are well-developed and yield explicit results. In holographic CFTs, the boundary theory is strongly coupled, making direct computation far more difficult.
• The crossed product construction provides a bridge between relative entropy (type III${}_1$) and gravitational entropy (type II${}_{\infty }$), but explicit computations have been done only in limited settings (JT gravity, static observers).

Possible Approaches

1. Large $N$ free-field models. Consider holographic CFTs in limits where the boundary theory simplifies (e.g., generalized free fields at large $N$), and compute Bell-CHSH violations and relative entropy using the Weyl operator formalism adapted to the boundary. Compare with known bulk geometric data.

2. Thermofield double state. The thermofield double state in a holographic CFT is dual to the two-sided eternal black hole. The left and right boundary algebras are type III${}_1$ factors. Apply the Weyl operator Bell test to these complementary algebras and study how the Bell violation depends on the black hole temperature (which controls the entanglement).

3. Relative entropy and RT formula. Compute Araki-Uhlmann relative entropy between states on a boundary subregion algebra and compare with the bulk generalized entropy. The modular Hamiltonian for ball-shaped regions in CFTs is known explicitly (Casini-Huerta-Myers), which could make the computation tractable.

4. Toy models. Use 2d holographic CFTs or tensor network models (where the type III${}_1$ structure is approximated by large but finite systems) to test whether Bell-CHSH violations between boundary subregions detect bulk connectivity changes (e.g., phase transitions in the RT surface).

Related Questions

• embezzlement-cost-relative-entropy --- the crossed product provides a framework for computing embezzlement cost, which is also related to gravitational entropy
• relative-entropy-interacting-theories --- applying relative entropy to holographic (strongly coupled) theories requires going beyond free fields
• type III structure in gauge theories --- the type III${}_1$ classification in gauge theories with Gribov restriction is an analogous structural question

This question is tracked under the exploratory gauge-gravity-duality area and is the originating question for the holographic-bell-program connection page.