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Introduction
Maldacena & Susskind’s “ER=EPR” hypothesis claims that two physical systems in a quantum entangled (“EPR”, or Einstein-Podolsky-Rosen) state are connected by an Einstein-Rosen (“ER”) wormhole. More generally, it claims that spacetime topology is the “dual” of quantum entanglement: physical systems in a quantum entangled state can be described as systems connected by a wormhole, and vice-versa. Our project’s goals were to assess the motivations for “ER=EPR”, including AdS/CFT theory and the Black Hole Firewall Problem, and to construct a conceptual map that relates characteristics of wormholes to characteristics of entanglement.
Quantum Entanglement | Wormhole |
Entanglement entropy For a bipartite system in a pure state, the degree to which is entangled with $$B$$ is given by the entanglement entropy of , defined by | Minimal cross-sectional area of throat For two regions $A$ and $B$ connected by a wormhole, the size of the wormhole at an instant is the minimal cross-sectional area of its throat. |
No superluminal signaling For a bipartite system , entanglement correlations between and cannot be used to send superluminal signals. | Non-Traversability Any future-directed curve that passes through a wormhole must become spacelike along some of its extent. |
No-Cloning If a multipartite entangled state is the result of cloning one of the substates of a bipartite entangled state , and: (a) all -clones remain entangled with ; (b) if the entanglement between and any -clone is broken, the entanglement between and the others remains intact;then it facilitates superluminal signaling. | Topology change If a wormhole has formed as a result of topology change, and if it violates the null energy condition, then it is traversable. |
Entanglement non-detectability There is no projection operator onto the set of entangled vectors. So there is no observable that represents entanglement in terms of a linear operator. | Wormhole non-detectability There is no local curvature measurement that can distinguish between a one-sided black hole spacetime and a two-sided black hole (i.e., wormhole) spacetime. |
Entanglement conservation For a bipartite system , the entanglement entropy of subsystem is invariant under local operations. | Area conservation The maximal cross-sectional area of a wormhole is invariant under local operations performed near its mouths. |
Entanglement monogamy If and are maximally entangled, then neither nor is entangled with any other system . (, for any entanglement measure .) | Wormhole monogamy If a "maximal" wormhole connects regions A and B, then neither nor can be connected by a wormhole to any other region . (, for mutual information measure .) |
We have now created a conceptual map between entanglement and wormholes. While this map exposes intriguing relations that support ER=EPR, additional questions remain. First, we need to further specify the meaning of non-detectability, since it risks making ER=EPR unverifiable. Second, entanglement monogamy is potentially contradictory to ER=EPR's resolution to the Firewall Paradox: cannot be wormhole-connected with since it is maximally entangled with . Finally, more conceptual work could be done on the relation between ER=EPR and recent derivations of the Einstein equations from entanglement first principles.