🕳️ ER=EPR: Entanglement Entropy and Wormhole Geometry
The theoretical framework you’re asking about is the ER=EPR conjecture, proposed by Leonard Susskind and Juan Maldacena in 2013. This conjecture provides a radical conceptual unification of quantum entanglement (EPR) and spacetime geometry/wormholes (ER), suggesting that they are two sides of the same fundamental reality. 1. The ER=EPR Identity
* EPR (Einstein-Podolsky-Rosen): Refers to quantum entanglement, a non-local connection between two quantum systems (like an entangled pair of particles).
* ER (Einstein-Rosen): Refers to a non-traversable wormhole (or Einstein-Rosen bridge), a solution in General Relativity that connects two distant regions of spacetime.
The conjecture states that any two maximally entangled black holes are connected by a wormhole. The non-local quantum connection is physically equivalent to a geometric tunnel. 2. Entanglement Entropy as Geometric Length
The connection is made quantitative and rigorous primarily through the AdS/CFT Correspondence (Holographic Duality) and the concept of Entanglement Entropy (S):
* Entanglement Entropy (S): This is the von Neumann entropy of a subsystem (\rho_A = \text{Tr}_B[\rho_{AB}]), which quantifies the degree of quantum correlation (entanglement) between two subsystems, A and B.
* The Ryu-Takayanagi (RT) Formula: In the holographic framework, this formula relates the entanglement entropy (S) of a region of the boundary quantum field theory (CFT) to the minimal area (\mathcal{A}) of a surface in the higher-dimensional bulk spacetime (Anti-de Sitter space, AdS).
Where \mathcal{A}(\Gamma_A) is the minimal surface area in the bulk whose boundary matches the boundary of subsystem A, and G_N is Newton’s constant. 3. The Wormhole-Entropy Link
* Black Hole Entropy: The area of a black hole’s event horizon (\mathcal{A}_{Horizon}) is related to its entropy by the Bekenstein-Hawking formula: S_{BH} = \frac{\mathcal{A}_{Horizon}}{4G_N}. The RT formula generalizes this concept.
* Entangled Black Holes: The ER=EPR conjecture relates the entanglement entropy between the two black holes in a maximally entangled pair (the Thermofield Double state) directly to the area of the event horizons on the two sides of the wormhole connecting them.
* Wormhole Dynamics (Complexity): A particularly powerful extension of this idea relates the internal geometry of the wormhole—specifically its spatial volume or “length”—to the quantum complexity of the entangled state. As the entangled state evolves in time, the non-traversable wormhole grows longer inside, a geometric process that quantifies the computational effort (complexity) required to un-entangle the two sides.
This entire framework suggests that spacetime geometry itself is an emergent phenomenon arising from the fundamental structure of quantum entanglement.
The video below discusses the ER=EPR concept as a way to potentially unify General Relativity and Quantum Mechanics. What Does ER=EPR Really Mean?

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