The geometry of spacetime in the bulk (including wormholes) corresponds to quantum states on the boundary (including entangled states like EPR pairs).
This duality suggests that spacetime geometry and quantum entanglement are deeply related, providing a mathematical bridge between GR (which governs the bulk) and QFT (which governs the boundary).
ER = EPR might be understood as a holographic duality in AdS/CFT, where entanglement (EPR pairs) is reflected in the bulk as connected wormholes (ER bridges).
2. Geometrizing Quantum Information: To formalize ER = EPR, we need to geometrize quantum information, meaning we need to describe quantum entanglement (which is an abstract quantum phenomenon) in terms of spacetime geometry. This could be done through:
Quantum Error Correction and Tensor Networks:Â
Quantum error correction codes and tensor network models have been used to describe the entanglement structure of quantum systems and how it might correspond to spacetime geometry.Â
In this view:
Quantum states of entangled particles can be mapped onto networks of quantum gates or tensors, which in turn can be mapped onto a spacetime geometry.
Tensor networks like MERA (Multiscale Entanglement Renormalization Ansatz) suggest a holographic interpretation of quantum states, where the geometry of spacetime (like a wormhole) emerges from entanglement patterns in quantum systems.
If we can rigorously formalize this idea, we could connect the entanglement entropy of quantum states to geometric quantities like the area of wormhole mouths or the volume of ER bridges. This would provide a mathematical tool to bridge GR and QFT.
3. Entanglement Entropy and Spacetime Emergence: Entanglement entropy measures how entangled a system is, and in some cases (such as in black hole physics), it is related to the area of the event horizon. This suggests a connection between quantum information and the geometry of spacetime.
