>[!warning] >This content has not been peer reviewed. # Geometry can be computed from entanglement ## What it is (established result) In the AdS/CFT correspondence, the **Ryu–Takayanagi (RT) formula** relates the entanglement entropy \(S_A\) of a boundary region \(A\) in the CFT to the area of a minimal surface \(\gamma_A\) in the bulk whose boundary matches \(\partial A\): \[ S_A = \frac{\mathrm{Area}(\gamma_A)}{4 G_N \hbar} \quad \text{(in appropriate units)}. \] This provides a precise instance of the idea that **spacetime geometry encodes quantum entanglement**: changes in entanglement structure correspond to changes in geometric data (areas / distances) in the dual description \([1,2]\). --- ## How RRT / RST uses it RST does **not** assume AdS/CFT, but uses the RT idea as a structural mapping: - Treat **distance** as an emergent quantity derived from a substrate’s “entanglement / coherence” relations. - In a simulator, maintain a symmetric entanglement proxy \(E_{ij}\) and define a dynamic distance \(d_{ij}\) as a monotone decreasing function of \(E_{ij}\) (e.g. \(d_{ij} \propto -\log E_{ij}\)). - As nodes share coherent workload, \(E_{ij}\) rises and \(d_{ij}\) shrinks; if coherence decays, distances grow — providing an explicit knob for “de-synchronisation” style expansion. The `Reality Engine` demo implements this as an inspectable `RyuTakayanagiMetric` plug-in. --- ## Links | Role | Link | |:---|:---| | Simulator application | [[further applications/Reality Engine/Reality Engine (RST)]] | | Simulator code | [[further applications/Reality Engine/Reality Engine - Code]] | --- ## References [1] S. Ryu and T. Takayanagi, "Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence," *Physical Review Letters* **96**, 181602 (2006), `https://doi.org/10.1103/PhysRevLett.96.181602`. [2] V. E. Hubeny, M. Rangamani, and T. Takayanagi, "A covariant holographic entanglement entropy proposal," *Journal of High Energy Physics* **2007**, 062 (2007), `https://doi.org/10.1088/1126-6708/2007/07/062`.