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.
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.
The black hole information puzzle arises from a discrepancy between conclusions drawn from general relativity and quantum theory about the nature of the radiation emitted by a black hole. According to Hawking's original argument, the radiation is thermal and its entropy thus increases monotonically as the black hole evaporates. Conversely, due to the reversibility of time evolution according to quantum theory, the radiation entropy should start to decrease after a certain time, as predicted by the Page curve. This decrease has been confirmed by new calculations based on the replica trick, which also exhibit its geometrical origin: spacetime wormholes that form between the replicas. Here we analyse the discrepancy between these and Hawking's original conclusions from a quantum information theory viewpoint, using in particular the quantum de Finetti theorem. The theorem implies the existence of extra information,
W, which is neither part of the black hole nor the radiation, but plays the role of a reference. The entropy obtained via the replica trick can then be identified to be the entropy
S(R∣W) of the radiation conditioned on the reference
W, whereas Hawking's original result corresponds to the non-conditional entropy
S(R). The entropy
S(R∣W), which mathematically is an ensemble average, gains an operational meaning in an experiment with
N independently prepared black holes: For large
N, it equals the normalised entropy of their joint radiation,
S(R1⋯RN)/N. The discrepancy between this entropy and
S(R) implies that the black holes are correlated. The replica wormholes may thus be interpreted as the geometrical representation of this correlation. Our results also suggest a many-black-hole extension of the widely used random unitary model, which we support with non-trivial checks.
A bstract We show how to capture both the non-unitary Page curve and replica wormhole-like contributions that restore unitarity in a toy quantum system with random dynamics. The motivation is to find the simplest dynamical model that captures this aspect of gravitational physics. In our model, we evolve with an ensemble of Hamiltonians with GUE statistics within microcanonical windows. The entropy of the averaged state gives the non-unitary curve, the averaged entropy gives the unitary curve, and the difference comes from matrix index contractions in the Haar averaging that connect the density matrices in a replica wormhole-like manner.
A bstract We construct and study an ensemble of non-isometric error correcting codes in a toy model of an evaporating black hole in two-dimensional dilaton gravity. In the preferred bases of Euclidean path integral states in the bulk and Hamiltonian eigenstates in the boundary, the encoding map is proportional to a linear transformation with independent complex Gaussian random entries of zero mean and unit variance. Using measure concentration, we show that the typical such code is very likely to preserve pairwise inner products in a set S of states that can be subexponentially large in the microcanonical Hilbert space dimension of the black hole. The size of this set also serves as an upper limit on the bulk effective field theory Hilbert space dimension. Similar techniques are used to demonstrate the existence of state-specific reconstructions of S -preserving code space unitary operators. State-specific reconstructions on subspaces exist when they are expected to by entanglement wedge reconstruction. We comment on relations to complexity theory and the breakdown of bulk effective field theory.
Black holes are windows into the deepest structures of reality, where quantum mechanics, thermodynamics, and geometry converge. In this pedagogical preprint we explore how quantum entanglement entropy provides a microscopic foundation for black hole thermodynamics. Starting from rst principles of quantum information theory, we introduce entanglement entropy, the area law, and Hawking radiation, then proceed to modern results: the Ryu-Takayanagi formula, quantum extremal surfaces, the Page curve, replica wormholes, and the island formula. Throughout, the emphasis is on physical intuition. No prior knowledge of string theory or loop quantum gravity is assumed.