We define a novel class of tensor networks motivated by the Python's Lunch Conjecture (PLC) in local tensor network models of the black hole interior. We start from the observation that, for extended black brane states with short-range correlations, the PLC predicts a complexity that is smaller than the upper bound for generic short-range correlated states. We argue that the PLC makes implicit assumptions about the fine structure of the relevant tensor networks modeling gravity that render them non-generic. We demonstrate this explicitly in random tensor network models of the python's lunch, where the exponential complexity is not generally controlled by the PLC exponent. We trace the difference with the PLC to a lack of "computational covariance" in random tensor networks: while the PLC is motivated by an ability to arbitrarily decompose space into low-complexity units provided certain basic rules are followed, we show that random tensor networks do not generically have this property. We propose another class of tensor networks built from what we call "twirled perfect tensors" that do satisfy the computational covariance property and have a complexity bounded by the PLC value. We still find a discrete limitation from local postselection that appears to be absent in gravity. Moreover, we show that this class of tensor networks combines desirable holographic features of perfect tensor networks and random tensor networks, for example, it obeys a lattice Ryu-Takayanagi formula for arbitrary boundary subregions. Though motivated by holography, these tensor networks provide a flexible framework with potential applications beyond quantum gravity.
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50 pages + appendices. Watch at 5-minute video abstract at https://youtu.be/AX-cSk-iI6o. v3: corrected an estimate, added clarifications
We define a novel class of tensor networks motivated by the Python's Lunch Conjecture (PLC) in local tensor network models of the black hole interior. We start from the observation that, for extended black brane states with short-range correlations, the PLC predicts a complexity that is smaller than the upper bound for generic short-range correlated states. We argue that the PLC makes implicit assumptions about the fine structure of the relevant tensor networks modeling gravity that render them non-generic. We demonstrate this explicitly in random tensor network models of the python's lunch, where the exponential complexity is not generally controlled by the PLC exponent. We trace the difference with the PLC to a lack of "computational covariance" in random tensor networks: while the PLC is motivated by an ability to arbitrarily decompose space into low-complexity units provided certain basic rules are followed, we show that random tensor networks do not generically have this property. We propose another class of tensor networks built from what we call "twirled perfect tensors" that do satisfy the computational covariance property and have a complexity bounded by the PLC value. We still find a discrete limitation from local postselection that appears to be absent in gravity. Moreover, we show that this class of tensor networks combines desirable holographic features of perfect tensor networks and random tensor networks, for example, it obeys a lattice Ryu-Takayanagi formula for arbitrary boundary subregions. Though motivated by holography, these tensor networks provide a flexible framework with potential applications beyond quantum gravity.
Tensor networks provide a natural framework for exploring holographic duality because they obey entanglement area laws. They have been used to construct explicit toy models realizing many of the interesting structural features of the AdS/CFT correspondence, including the non-uniqueness of bulk operator reconstruction in the boundary theory. In this article, we explore the holographic properties of networks of random tensors. We find that our models naturally incorporate many features that are analogous to those of the AdS/CFT correspondence. When the bond dimension of the tensors is large, we show that the entanglement entropy of all boundary regions, whether connected or not, obey the Ryu-Takayanagi entropy formula, a fact closely related to known properties of the multipartite entanglement of assistance. We also discuss the behavior of Rényi entropies in our models and contrast it with AdS/CFT. Moreover, we find that each boundary region faithfully encodes the physics of the entire bulk entanglement wedge, i.e., the bulk region enclosed by the boundary region and the minimal surface. Our method is to interpret the average over random tensors as the partition function of a classical ferromagnetic Ising model, so that the minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the analog of a bulk field, we find that our model reproduces the expected corrections to the Ryu-Takayanagi formula: the bulk minimal surface is displaced and the entropy is augmented by the entanglement of the bulk field. Increasing the entanglement of the bulk field ultimately changes the minimal surface behavior topologically, in a way similar to the effect of creating a black hole. Extrapolating bulk correlation functions to the boundary permits the calculation of the scaling dimensions of boundary operators, which exhibit a large gap between a small number of low-dimension operators and the rest. While we are primarily motivated by the AdS/CFT duality, the main results of the article define a more general form of bulk-boundary correspondence which could be useful for extending holography to other spacetimes.
Motivated by the ability to consistently apply the Ryu-Takayanagi prescription for general convex surfaces and the relationship between entanglement and geometry in tensor networks, we introduce a novel, covariant bulk object---the holographic slice. The holographic slice is found by considering the continual removal of short-range information in a boundary state. It thus provides a natural interpretation as the bulk dual of a series of coarse-grained holographic states. The slice possesses many desirable properties that provide consistency checks for its boundary interpretation. These include the monotonicity of both area and entanglement entropy, uniqueness, and the inability to probe beyond late-time black hole horizons. Additionally, the holographic slice illuminates physics behind entanglement shadows, as minimal-area extremal surfaces anchored to a coarse-grained boundary may probe entanglement shadows. This lets the slice flow through shadows. To aid in developing intuition for these slices, many explicit examples of holographic slices are investigated. Finally, the relationship to tensor networks and renormalization (particularly in AdS/CFT) is discussed.
Holographic tensor networks serve as toy models for the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, capturing many of its essential features in a concrete manner. However, existing holographic tensor network models remain far from a complete theory of quantum gravity. A key obstacle is their discrete structure, which only approximates the semi-classical geometry of gravity in a qualitative sense. In \cite{Lin:2024dho}, it was shown that a network of partial-entanglement-entropy (PEE) threads, which are bulk geodesics with a specific density distribution, generates a perfect tessellation of AdS space. Moreover, such PEE-network tessellations can be constructed for more highly symmetric geometries using the Crofton formula. In this paper, we assign a quantum state to each vertex in the PEE network and develop several holographic tensor network models: (1) the factorized PEE tensor network, which takes the form of a tensor product of EPR pairs; (2) the HaPPY-like PEE tensor network constructed from perfect tensors; and (3) the random PEE tensor network. In all these models, we reproduce the exact Ryu-Takayanagi formula by showing that the minimal number of cuts along a homologous surface in the network exactly equals the area of that surface.
This paper constructs an exactly solvable holographic model of de Sitter quantum gravity from the {4,3,3} tessellation of S³. Eight cubes tile the three-sphere; placing a perfect tensor at each cell, contracting across shared faces, and designating one cell as the observer produces a holographic model in 3+1 dimensions. Every result is exact: the Ryu-Takayanagi entropy formula holds with zero bulk entanglement, the de Sitter entropy is S = 6 log χ, the Hilbert space dimension is χ⁶ (confirming Banks's finite-dimensionality conjecture as a theorem), the reduced state is maximally mixed (reproducing the Gibbons-Hawking temperature), the Page curve is reproduced exactly, and the observer algebra is the full matrix algebra Mat(χ⁶) with a unique trace state. The proofs use only the perfect tensor property and the graph structure of the tessellation. No approximation, no large-N limit, and no holographic dictionary beyond the tensor network itself is required.
In this thesis, we study a couple of different but related questions: holography and quantum error correction, the gravitationally dressed operators and locality in holographic theory, introducing generalized versions of the class of continuous matrix product state as the most famous class of tensor network, and the notion of Krylov complexity of matrix quantum mechanics. In the first part, we use the notion of the Petz map and apply it to reconstruct operators in the entanglement wedge. Moreover, for geometries that contain black holes, we generalize the notion of the Petz map by itself and reconstruct black hole interior modes. In particular, we show that the Petz map reconstruction of the black hole interior is equivalent to the Papadodimas-Raju Proposal. In the second part, within the AdS/CFT correspondence, we identify a class of CFT operators which represent diff-invariant and approximately local observables in the gravitational dual. The interpretation of these observables is that they are not gravitationally dressed with respect to the boundary, but instead to features of the state. We also provide evidence that there are bulk observables whose commutator vanishes to all orders in 1/N with the entire algebra of single-trace operators defined in a space-like separated time-band. In the third part, we defined two new classes of continuous tensor networks by generalizing the class of continuous matrix product states. One is appropriate to describe the ground state of the relativistic field theory at strong coupling and the other is proper for the theories on compact spacetime. We conjecture that the possible bulk dual of it in the AdS/CFT can be an evaporating black hole microstates with the end of the word branes. In the last project, we study the Krylov complexity for 1-matrix quantum mechanics. In the ground state, the Lancsoz coefficients have linear behavior. The bn coefficients have a positive slope while an coefficients have a negative slope. In the thermal states, the an coefficients are zero while bn have two branches of linear growth. Although matrix quantum mechanics is a solvable theory rather than chaotic, we find the linear behavior of the Lancsoz coefficients of this theory.
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.