A bstract We propose an algebraic definition of ER=EPR in the G N → 0 limit, which associates bulk spacetime connectivity/disconnectivity to the operator algebraic structure of a quantum gravity system. The new formulation not only includes information on the amount of entanglement, but also more importantly the structure of entanglement . We give an independent definition of a quantum wormhole as part of the proposal. This algebraic version of ER=EPR sheds light on a recent puzzle regarding spacetime disconnectivity in holographic systems with $$ \mathcal{O} $$ O (1 /G N ) entanglement. We discuss the emergence of quantum connectivity in the context of black hole evaporation and further argue that at the Page time, the black hole-radiation system undergoes a transition involving the transfer of an emergent type III 1 subalgebra of high complexity operators from the black hole to radiation. We argue this is a general phenomenon that occurs whenever there is an exchange of dominance between two competing quantum extremal surfaces.
A bstract We propose an algebraic definition of ER=EPR in the G N → 0 limit, which associates bulk spacetime connectivity/disconnectivity to the operator algebraic structure of a quantum gravity system. The new formulation not only includes information on the amount of entanglement, but also more importantly the structure of entanglement . We give an independent definition of a quantum wormhole as part of the proposal. This algebraic version of ER=EPR sheds light on a recent puzzle regarding spacetime disconnectivity in holographic systems with O O (1 /G N ) entanglement. We discuss the emergence of quantum connectivity in the context of black hole evaporation and further argue that at the Page time, the black hole-radiation system undergoes a transition involving the transfer of an emergent type III 1 subalgebra of high complexity operators from the black hole to radiation. We argue this is a general phenomenon that occurs whenever there is an exchange of dominance between two competing quantum extremal surfaces.
A bstract We present an operator algebraic perspective on the black hole information problem. For a black hole after Page time that is entangled with the early radiation we formulate a version of the information puzzle that is well-posed in the G → 0 limit. We then give a description of the information recovery protocol in terms of von Neumann algebras using elements of the Jones index theory of type II 1 subfactors. The subsequent evaporation and recovery steps are represented by Jones’s basic construction, and an operation called the canonical shift. A central element in our description is the Jones projection, which leads to an entanglement swap and implements an operator algebraic version of a quantum teleportation protocol. These aspects are further elaborated on in a microscopic model based on type I algebras. Finally, we argue that in the emergent type III algebra the canonical shift may be interpreted as a spacetime translation and, hence, that at the microscopic level “translation = teleportation”.
[This is a creative exploration of a concept with AI -it should not be taken as correct mathematics.] We present a detailed examination of the conjectured correspondence between Einstein-Rosen (ER) bridges and Einstein-Podolsky-Rosen (EPR) entanglement, denoted as ER=EPR. This examination concentrates on the quantitative structures and derivations supporting this correspondence, primarily within the setting of the Anti-de Sitter/Conformal Field Theory (AdS/CFT) duality. We analyze the realization of this correspondence in the eternal black hole geometry and its dual Thermofield Double (TFD) state. A complete exposition of the holographic entanglement entropy prescriptions is provided, including the derivation of the Quantum Extremal Surface (QES) prescription via the Lewkowycz-Maldacena argument concerning the gravitational path integral and conical singularities, and the subsequent derivation of the Jafferis-Lewkowycz-Maldacena-Suh (JLMS) relation establishing the equality of bulk and boundary relative entropies. We investigate the relationship between the temporal evolution of the spacetime geometry, characterized by the growth of the Einstein-Rosen bridge, and the evolution of quantum complexity. A complete derivation of the maximal volume growth rate is provided, carefully analyzing the geometry inside the horizon. Furthermore, a complete derivation of the action growth rate, supporting the Complexity=Action conjecture, is provided, clarifying the precise contribution of bulk, boundary, and joint terms in the gravitational action according to the definitions established by Lehner et al. for manifolds with null boundaries. The correspondence is extended to typical entangled states, characterized by Einstein-Rosen caterpillars. We derive the effective Hamiltonian governing the ensemble average of the generating random circuits and establish the quantitative relationship between the geometric length of the wormhole and the measure of quantum randomness (k-designs), utilizing the results of Magán, Sasieta, and Swingle. The role of QES and the derivation of the Page curve via replica wormholes are analyzed, detailing the calculations in Jackiw-Teitelboim gravity theories following Penington et al. Mechanisms for rendering wormholes traversable via boundary interactions (Gao-Jafferis-Wall construction) and operational interpretations based on Local Operationsand Classical Communication protocols are also discussed, emphasizing the quantitative consistency of these interconnected concepts.
If spacetime is built out of quantum bits, does the shape of space depend on how the bits are entangled? The ER=EPR conjecture relates the entanglement entropy of a collection of black holes to the cross sectional area of Einstein-Rosen (ER) bridges (or wormholes) connecting them. We show that the geometrical entropy of classical ER bridges satisfies the subadditivity, triangle, strong subadditivity, and Cadney-Linden-Winter inequalities. These are nontrivial properties of entanglement entropy, so this is evidence for ER=EPR. We further show that the entanglement entropy associated with classical ER bridges has nonpositive tripartite information. This is not a property of entanglement entropy, in general. For example, the entangled four qubit pure state ∣GHZ4⟩=(∣0000⟩+∣1111⟩)/2 has positive tripartite information, so this state cannot be described by a classical ER bridge. Large black holes with massive amounts of entanglement between them can fail to have a classical ER bridge if they are built out of ∣GHZ4⟩ states. States with nonpositive tripartite information are called monogamous. We conclude that classical ER bridges require monogamous EPR correlations.
We construct a class of wormhole geometries supported by the non-local gravitational self-energy that regularizes the particle and black-hole sectors of spacetime. Using this framework, inspired by T-duality, we show that two entangled particles (or particle-black-hole pairs) naturally source an Einstein-Rosen-type geometry in which the required violation of the strong energy condition arises from intrinsic quantum-gravity effects rather than from ad hoc exotic matter, which is matter that violates the null energy condition. We classify the resulting wormholes, analyze their horizons, throat structure and embedding properties, and we identify the exotic energy needed at the minimal surface. Imposing the ER=EPR requirement of non-traversability and the absence of a macroscopic throat, we find that only the zero-throat geometry is compatible with an entanglement-induced Einstein-Rosen bridge, providing a concrete realization of ER=EPR within a fully regular spacetime. Finally, we briefly discuss possible implications for microscopic ER networks from vacuum fluctuations, replica-wormhole interpretations of Hawking radiation, and possible links to entanglement-driven dark-energy scenarios.
We present a collection of recent results concerning quantum information theory applied to quantum gravity. We first study the entanglement structure of Euclidean path integral states in SU(2) Chern-Simons theory, where we elucidate a connection between topological entanglement and quantum mechanical entanglement. We prove that the topology of certain three-manifolds controls the entanglement structure of the resulting quantum state, and conjecture a more general relationship for arbitrary three-manifolds. We then analyze the quantum circuit complexity of the time evolution operator in the Sachdev-Ye-Kitaev model, a theory of near-extremal black hole microstates. We find that this complexity grows linearly for a time exponential in the entropy, modulo a caveat concerning global obstructions to the growth of the distance function along geodesics, which we do not rule out. This constitutes a partial proof of Susskind’s conjecture about the complexity growth of black holes. Finally, we address the black hole information paradox in three dimensions by considering a toy model of black hole microstates in the form of an end-of-the-world brane. This brane carries a quantum theory which is itself holographic, and we compute entanglement entropies in the glued dual geometry by using a Ryu-Takayanagi formula where the minimal entropy surface can pass through the gluing surface.
With an aim towards understanding the time-dependence of entanglement entropy in generic quantum field theories, we propose a covariant generalization of the holographic entanglement entropy proposal of hep-th/0603001. Apart from providing several examples of possible covariant generalizations, we study a particular construction based on light-sheets, motivated in similar spirit to the covariant entropy bound underlying the holographic principle. In particular, we argue that the entanglement entropy associated with a specified region on the boundary in the context of the AdS/CFT correspondence is given by the area of a co-dimension two bulk surface with vanishing expansions of null geodesics. We demonstrate our construction with several examples to illustrate its reduction to the holographic entanglement entropy proposal in static spacetimes. We further show how this proposal may be used to understand the time evolution of entanglement entropy in a time varying QFT state dual to a collapsing black hole background. Finally, we use our proposal to argue that the Euclidean wormhole geometries with multiple boundaries should be regarded as states in a non-interacting but entangled set of QFTs, one associated to each boundary.