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arXiv:2610.02168

A baby universe from a large family: booklet cosmology states and quantum error correction

Jingshu Dai, Binye Dong, Cheng Peng

Published Oct 1, 2026• 0 citations
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Abstract

We propose an extension of the cosmological-state construction of Antonini, Sasieta and Swingle (AS$^2$) to three or more holographic CFTs. Each CFT resides on the asymptotic boundary of an AdS page, and all pages are glued along a common interface by imposing the multiway junction conditions. The associated states are prepared by Euclidean evolution with a multilinear insertion. In appropriate heavy insertion limit the bulk geometry develop a closed universe in the center and we refer the state in this limit as a booklet cosmological state. Extending the effective Gaussian assumption used in AS$^2$, we model the three-page insertion by a circular complex Gaussian random tensor, yielding a tripartite Haar state in flat energy windows. In this Gaussian model of the cosmology-to-boundary map, each arm is a network branch associated with one boundary CFT. In the simplest example of three pages with three equal output Hilbert-space dimension $b$, we show that a prescribed code of dimension $K$ is approximately recoverable from any two arms, with vanishing error and high probability as $K/b\to0$.

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We propose an extension of the cosmological-state construction of Antonini, Sasieta and Swingle (AS2^22) to three or more holographic CFTs. Each CFT resides on the asymptotic boundary of an AdS page, and all pages are glued along a common interface by imposing the multiway junction conditions. The associated states are prepared by Euclidean evolution with a multilinear insertion. In appropriate heavy insertion limit the bulk geometry develop a closed universe in the center and we refer the state in this limit as a booklet cosmological state. Extending the effective Gaussian assumption used in AS2^22, we model the three-page insertion by a circular complex Gaussian random tensor, yielding a tripartite Haar state in flat energy windows. In this Gaussian model of the cosmology-to-boundary map, each arm is a network branch associated with one boundary CFT. In the simplest example of three pages with three equal output Hilbert-space dimension bbb, we show that a prescribed code of dimension KKK is approximately recoverable from any two arms, with vanishing error and high probability as K/b→0K/b\to0K/b→0.
We propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindlerwedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed in [1].
We revisit whether a semiclassical closed baby universe in AdS/CFT necessarily possess a trivial one-dimensional Hilbert space or may instead carry a large entropy. Recent results on Haar random encoding suggest a breakdown of complementary recovery, in which no logical operators can be reconstructed from individual bipartite subsystems. Motivated by this, we propose an interpretation where a baby universe emerges as logical degrees of freedom that cannot be accessed from either boundary alone, assuming pseudorandom dynamics in holographic CFT correlators. We then analyze two conceptual puzzles: an apparent cloning of baby-universe microstates and its eventual fate at the singularity. Both puzzles are avoided because no single boundary observer can access the baby-universe degrees of freedom, be it classical or quantum, reflecting an emergent form of complementarity due to the structure of random encoding. In this interpretation, observers arise naturally: the same heavy operator that prepares the baby-universe geometry also serves as observer-like degrees of freedom that define an observer-dependent baby-universe microstate.
We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large-N CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For “tree” tensor networks (i.e., those that are constructed by discretizing spacetime with nonintersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous using a version of one-shot entanglement distillation in the CFT. Using the known quantum error correcting properties of AdS/CFT, we show that bulk legs can be added to the tensor networks to create holographic quantum error correcting codes. These codes behave similarly to previous holographic tensor network toy models, but describe actual bulk excitations in continuum AdS/CFT. By assuming some natural generalizations of the “holographic entanglement of purification” conjecture, we are able to construct tensor networks for more general bulk discretizations, leading to finer-grained networks that partition the information content of a Ryu-Takayanagi surface into tensor-factorized subregions. While the granularity of such a tensor network must be set larger than the string/Planck scales, we expect that it can be chosen to lie well below the AdS scale. However, we also prove a no-go theorem which shows that the bulk-to-boundary maps cannot all be isometries in a tensor network with intersecting Ryu-Takayanagi surfaces.
Holographic quantum-error correcting codes are models of bulk/boundary dualities such as the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, where a higher-dimensional bulk geometry is associated with the code's logical degrees of freedom. Previous discrete holographic codes based on tensor networks have reproduced the general code properties expected from continuum AdS/CFT, such as complementary recovery. However, the boundary states of such tensor networks typically do not exhibit the expected correlation functions of CFT boundary states. In this work, we show that a new class of exact holographic codes, extending the previously proposed hyperinvariant tensor networks into quantum codes, produce the correct boundary correlation functions. This approach yields a dictionary between logical states in the bulk and the critical renormalization group flow of boundary states. Furthermore, these codes exhibit a state-dependent breakdown of complementary recovery as expected from AdS/CFT under small quantum gravity corrections.
Holographic quantum-error correcting codes are models of bulk/boundary dualities such as the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, where a higher-dimensional bulk geometry is associated with the code's logical degrees of freedom. Previous discrete holographic codes based on tensor networks have reproduced the general code properties expected from continuum AdS/CFT, such as complementary recovery. However, the boundary states of such tensor networks typically do not exhibit the expected correlation functions of CFT boundary states. In this work, we show that a new class of exact holographic codes, extending the previously proposed hyperinvariant tensor networks into quantum codes, produce the correct boundary correlation functions. This approach yields a dictionary between logical states in the bulk and the critical renormalization group flow of boundary states. Furthermore, these codes exhibit a state-dependent breakdown of complementary recovery as expected from AdS/CFT under small quantum gravity corrections.
A bstract We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large- N CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For “tree” tensor networks (i.e., those that are constructed by discretizing spacetime with non­ intersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous using a version of one-shot entanglement distillation in the CFT. Using the known quantum error correcting properties of AdS/CFT, we show that bulk legs can be added to the tensor networks to create holographic quantum error correcting codes. These codes behave similarly to previous holographic tensor network toy models, but describe actual bulk excitations in continuum AdS/CFT. By assuming some natural generalizations of the “holographic entanglement of purification” conjecture, we are able to construct tensor networks for more general bulk discretizations, leading to finer-grained networks that partition the information content of a Ryu-Takayanagi surface into tensor-factorized subregions. While the granularity of such a tensor network must be set larger than the string/Planck scales, we expect that it can be chosen to lie well below the AdS scale. However, we also prove a no-go theorem which shows that the bulk-to-boundary maps cannot all be isometries in a tensor network with intersecting Ryu-Takayanagi surfaces.