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Complete Toy Models of Holographic Duality

Tamara Kohler, Toby S. Cubitt

Published Oct 21, 2018• 6 citations
OpenAlex

Abstract

Tensor networks have previously been used to construct holographic quantum error correcting codes (HQECC), which are toy models of the AdS/CFT correspondence. HQECC have been able to exhibit many of the interesting features of the duality, such as complementary recovery and redundant encoding. However, in previous HQECC the boundary Hamiltonian which results from mapping a local bulk Hamiltonian to the boundary is a non-local Hamiltonian, with global terms that act over the entire boundary. In this work, we combine HQECC with recently developed Hamiltonian simulation theory to construct a bulk-boundary mapping in which local bulk Hamiltonians map to local boundary Hamiltonians. This allows us to construct a toy model of a complete holographic duality between models, not just states and observables. This mapping of local Hamiltonians extends the toy models to encompass the relationship between bulk and boundary energy scales, and in particular the mapping between bulk and boundary time dynamics. We discuss what insight this gives into toy models of black hole formation.

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Tensor networks have previously been used to construct holographic quantum error correcting codes (HQECC), which are toy models of the AdS/CFT correspondence. HQECC have been able to exhibit many of the interesting features of the duality, such as complementary recovery and redundant encoding. However, in previous HQECC the boundary Hamiltonian which results from mapping a local bulk Hamiltonian to the boundary is a non-local Hamiltonian, with global terms that act over the entire boundary. In this work, we combine HQECC with recently developed Hamiltonian simulation theory to construct a bulk-boundary mapping in which local bulk Hamiltonians map to local boundary Hamiltonians. This allows us to construct a toy model of a complete holographic duality between models, not just states and observables. This mapping of local Hamiltonians extends the toy models to encompass the relationship between bulk and boundary energy scales, and in particular the mapping between bulk and boundary time dynamics. We discuss what insight this gives into toy models of black hole formation.
Holographic quantum error correcting codes (HQECC) have been proposed as toy models for the AdS/CFT correspondence, and exhibit many of the features of the duality. HQECC give a mapping of states and observables. However, they do not map local bulk Hamiltonians to local Hamiltonians on the boundary. In this work, we combine HQECC with Hamiltonian simulation theory to construct a bulk-boundary mapping between local Hamiltonians, whilst retaining all the features of the HQECC duality. This allows us to construct a duality between models, encompassing the relationship between bulk and boundary energy scales and time dynamics. \n \nIt also allows us to construct a map in the reverse direction: from local boundary Hamiltonians to the corresponding local Hamiltonian in the bulk. Under this boundary-to-bulk mapping, the bulk geometry emerges as an approximate, low-energy, effective theory living in the code-space of an (approximate) HQECC on the boundary. At higher energy scales, this emergent bulk geometry is modified in a way that matches the toy models of black holes proposed previously for HQECC. Moreover, the duality on the level of dynamics shows how these toy-model black holes can form dynamically.
A bstract Holographic quantum error correcting codes (HQECC) have been proposed as toy models for the AdS/CFT correspondence, and exhibit many of the features of the duality. HQECC give a mapping of states and observables. However, they do not map local bulk Hamiltonians to local Hamiltonians on the boundary. In this work, we combine HQECC with Hamiltonian simulation theory to construct a bulk-boundary mapping between local Hamiltonians, whilst retaining all the features of the HQECC duality. This allows us to construct a duality between models , encompassing the relationship between bulk and boundary energy scales and time dynamics. It also allows us to construct a map in the reverse direction: from local boundary Hamiltonians to the corresponding local Hamiltonian in the bulk. Under this boundary-to-bulk mapping, the bulk geometry emerges as an approximate, low-energy, effective theory living in the code-space of an (approximate) HQECC on the boundary. At higher energy scales, this emergent bulk geometry is modified in a way that matches the toy models of black holes proposed previously for HQECC. Moreover, the duality on the level of dynamics shows how these toy-model black holes can form dynamically.
A bstract The AdS/CFT correspondence realises the holographic principle where information in the bulk of a space is encoded at its border. We are yet a long way from a full mathematical construction of AdS/CFT, but toy models in the form of holographic quantum error correcting codes (HQECC) have replicated some interesting features of the correspondence. In this work we construct new HQECCs built from random stabilizer tensors that describe a duality between models encompassing local Hamiltonians whilst exactly obeying the Ryu-Takayanagi entropy formula for all boundary regions. We also obtain complementary recovery of local bulk operators for any boundary bipartition. Existing HQECCs have been shown to exhibit these properties individually, whereas our mathematically rigorous toy models capture these features of AdS/CFT simultaneously, advancing further towards a complete construction of holographic duality.
Tensor networks implementing quantum error correcting codes have recently been used to construct toy models of holographic duality explicitly realizing some of the more puzzling features of the AdS/CFT correspondence. These models reproduce the Ryu-Takayanagi entropy formula for boundary intervals, and allow bulk operators to be mapped to the boundary in a redundant fashion. These exactly solvable, explicit models have provided valuable insight but nonetheless suffer from many deficiencies, some of which we attempt to address in this article. We propose a new class of tensor network models that subsume the earlier advances and, in addition, incorporate additional features of holographic duality, including: (1) a holographic interpretation of all boundary states, not just those in a “code” subspace, (2) a set of bulk states playing the role of “classical geometries” which reproduce the Ryu-Takayanagi formula for boundary intervals, (3) a bulk gauge symmetry analogous to diffeomorphism invariance in gravitational theories, (4) emergent bulk locality for sufficiently sparse excitations, and (5) the ability to describe geometry at sub-AdS resolutions or even flat space.
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].
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.