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Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information

Donald M. Marolf, Henry Maxfield

Published Aug 1, 2020• 277 citations
OpenAlex

Abstract

In the 1980's, work by Coleman and by Giddings and Strominger linked the\nphysics of spacetime wormholes to `baby universes' and an ensemble of theories.\nWe revisit such ideas, using features associated with a negative cosmological\nconstant and asymptotically AdS boundaries to strengthen the results, introduce\na change in perspective, and connect with recent replica wormhole discussions\nof the Page curve. A key new feature is an emphasis on the role of null states.\nWe explore this structure in detail in simple topological models of the bulk\nthat allow us to compute the full spectrum of associated boundary theories. The\ndimension of the asymptotically AdS Hilbert space turns out to become a random\nvariable $Z$, whose value can be less than the naive number $k$ of independent\nstates in the theory. For $k>Z$, consistency arises from an exact degeneracy in\nthe inner product defined by the gravitational path integral, so that many a\npriori independent states differ only by a null state. We argue that a similar\nproperty must hold in any consistent gravitational path integral. We also\ncomment on other aspects of extrapolations to more complicated models, and on\npossible implications for the black hole information problem in the individual\nmembers of the above ensemble.

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In the 1980's, work by Coleman and by Giddings and Strominger linked the\nphysics of spacetime wormholes to `baby universes' and an ensemble of theories.\nWe revisit such ideas, using features associated with a negative cosmological\nconstant and asymptotically AdS boundaries to strengthen the results, introduce\na change in perspective, and connect with recent replica wormhole discussions\nof the Page curve. A key new feature is an emphasis on the role of null states.\nWe explore this structure in detail in simple topological models of the bulk\nthat allow us to compute the full spectrum of associated boundary theories. The\ndimension of the asymptotically AdS Hilbert space turns out to become a random\nvariable ZZZ, whose value can be less than the naive number kkk of independent\nstates in the theory. For k>Zk>Zk>Z, consistency arises from an exact degeneracy in\nthe inner product defined by the gravitational path integral, so that many a\npriori independent states differ only by a null state. We argue that a similar\nproperty must hold in any consistent gravitational path integral. We also\ncomment on other aspects of extrapolations to more complicated models, and on\npossible implications for the black hole information problem in the individual\nmembers of the above ensemble.
Naively, resolving the black hole information paradox requires microscopic details about quantum gravity. Recent work suggests that, instead, a unitary Page curve can be recovered by adding disorder-averaged replica instantons to the path integral, though their origin is unclear. In this Letter, we show how replica instantons and disorder averaging emerge naturally in an effective theory built from typical microscopic states. We relate replica instantons to a moment expansion of simple operators, and find a microcanonical description in terms of wormholes and Euclidean black holes.
A bstract Recent work has shown how to obtain the Page curve of an evaporating black hole from holographic computations of entanglement entropy. We show how these computations can be justified using the replica trick, from geometries with a spacetime wormhole connecting the different replicas. In a simple model, we study the Page transition in detail by summing replica geometries with different topologies. We compute related quantities in less detail in more complicated models, including JT gravity coupled to conformal matter and the SYK model. Separately, we give a direct gravitational argument for entanglement wedge reconstruction using an explicit formula known as the Petz map; again, a spacetime wormhole plays an important role. We discuss an interpretation of the wormhole geometries as part of some ensemble average implicit in the gravity description.
Abstract In this paper I argue that the Hilbert space of states of a holographic, traversable wormhole does not factorize into the tensor product of the boundary Hilbert spaces. After presenting the general argument I analyze two examples: the scalar sectors of the BTZ geon and the AdS2 eternal wormhole. Utilizing real-time holography I derive the Hilbert spaces, identify the dual states and evaluate correlation functions. I show that the number of peculiarities associated with the wormhole and black hole physics emerges once the factorization is a priori assumed. This includes null states and null operators, highly entangled vacuum states and the cross-boundary interactions all emerging as avatars of non-factorization.
The AdS/CFT correspondence relates quantum entanglement between boundary conformal field theories and geometric connections in the dual asymptotically anti-de Sitter spacetime. We consider entangled states in the n-fold tensor product of a 1 + 1 dimensional CFT Hilbert space defined by the Euclidean path integral over a Riemann surface with n holes. In one region of moduli space, the dual bulk state is a black hole with n asymptotically AdS3 regions connected by a common wormhole, while in other regions the bulk fragments into disconnected components. We study the entanglement structure and compute the wave function explicitly in the puncture limit of the Riemann surface in terms of CFT n-point functions. We also use AdS minimal surfaces to measure entanglement more generally. In some regions of the moduli space the entanglement is entirely multipartite, though not of the GHZ type. However, even when the bulk is completely connected, there are regions of the moduli space in which the entanglement is instead almost entirely bipartite: significant entanglement occurs only between pairs of CFTs. We develop new tools to analyze intrinsically n-partite entanglement, and use these to show that for some wormholes with n similar sized horizons there is intrinsic entanglement between all n parties, and that the distillable entanglement between the asymptotic regions is at least (n + 1)/2 partite.
[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.
According to the holographic principle, the information content assigned to a gravitational region is processed by its lower dimensional boundary. As an example setup compatible with this principle, the AdS/CFT correspondence relies on the existence of D-branes in superstring theory. Black hole complementarity is inevitably linked to holography and states that information associated with the collapsed pure state is reflected in the near horizon region. Yet, if this is so, it is indispensable to understand the mechanism that makes black holes viewed from the outside evolve unitarily. We here argue that the information preserving quantum atmosphere of the black hole emerges from hidden variables on its horizon which would necessitate going beyond a probabilistic description within standard quantum theory. In AdS/CFT, this would mean that the completion of the semiclassical subalgebra to the complete boundary algebra has to be traced back to the emergent near horizon Hilbert space structure. The present investigations suggest that spacetime horizons, in general, may play a crucial role in restoring a long speculated ontology in quantum mechanics.