We establish a quantitative equivalence between the holographic duality of theAdS/CFT correspondence and the theory of quantum error-correcting codes (QECC).By identifying the code subspace with the low-energy bulk effective field theoryand the physical Hilbert space with the boundary CFT, we show that the RyuTakayanagi formula for entanglement entropy directly determines the code distanced —the minimal weight of an undetectable error. We prove a holographic Hammingbound: d ≤ Area(γmin)/(4Gℏln2), with equality for perfect tensor networks. Thegravitational constant G thus quantifies the error-correcting capacity of spacetime.A testable prediction is derived: the code distance for a black hole horizon saturates the bound, explaining the unitarity of Hawking evaporation as a property of amaximum-distance separable code. We further show that the entanglement wedgereconstruction theorem is equivalent to the Knill–Laflamme conditions for QECC
We establish a quantitative equivalence between the holographic duality of theAdS/CFT correspondence and the theory of quantum error-correcting codes (QECC).By identifying the code subspace with the low-energy bulk effective field theoryand the physical Hilbert space with the boundary CFT, we show that the RyuTakayanagi formula for entanglement entropy directly determines the code distanced —the minimal weight of an undetectable error. We prove a holographic Hammingbound: d ≤ Area(γmin)/(4Gℏln2), with equality for perfect tensor networks. Thegravitational constant G thus quantifies the error-correcting capacity of spacetime.A testable prediction is derived: the code distance for a black hole horizon saturates the bound, explaining the unitarity of Hawking evaporation as a property of amaximum-distance separable code. We further show that the entanglement wedgereconstruction theorem is equivalent to the Knill–Laflamme conditions for QECC
Holographic quantum error-correcting code, the quantum-information structure hypothesized for the AdS/CFT correspondence, has being attracting increasing attention in new directions interrelating the studies of quantum gravity and quantum simulation. In this work, we initiate a novel approach for building holographic code that can be generally applied in potentially broad and interdisciplinary contexts. Our approach takes an "opposite" route to the conventional paradigm that is based on bulk tensor-networks. As illustrated in an exact model, we start from scalable descriptions of boundary qudits which can guide succinct quantum-circuit simulations, and rigorously show how the bulk qudits and the encoding structure emerge from boundary entanglement patterns. By investigating the entanglement patterns, we systematically unfold the hypothetical structure for bulk reconstruction and the details of the Ryu-Takayanagi formula in the formalism of operator-algebra quantum error correction, demonstrating desired properties that are not yet proved in the established models. Our work might offer a fresh perspective for the study of holographic code.
We study the holographic map in AdS/CFT, as modeled by a quantum error correcting code with exact complementary recovery. We show that the map is determined by local conditional expectations acting on the operator algebras of the boundary/physical Hilbert space. Several existing results in the literature follow easily from this perspective. The Black Hole area law, and more generally the Ryu-Takayanagi area operator, arises from a central sum of entropies on the relative commutant. These entropies are determined in a state independent way by the conditional expectation. The conditional expectation can also be found via a minimization procedure, similar to the minimization involved in the RT formula. For a local net of algebras associated to connected boundary regions, we show the complementary recovery condition is equivalent to the existence of a standard net of inclusions -- an abstraction of the mathematical structure governing QFT superselection sectors given by Longo and Rehren. For a code consisting of algebras associated to two disjoint regions of the boundary theory we impose an extra condition, dubbed dual-additivity, that gives rise to phase transitions between different entanglement wedges. Dual-additive codes naturally give rise to a new split code subspace, and an entropy bound controls which subspace and associated algebra is reconstructable. We also discuss known shortcomings of exact complementary recovery as a model of holography. For example, these codes are not able to accommodate holographic violations of additive for overlapping regions. We comment on how approximate codes can fix these issues.
Recent work showed holographic error correcting codes to have simple universal features at O(1/G). In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are associated with flat entanglement spectra indicating maximal entanglement between appropriate subspaces. We extend such results to one-loop order (O(1) corrections) by controlling both higher-derivative corrections to the bulk effective action and dynamical quantum fluctuations below the cutoff. This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in G, ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow. A final corollary requires interesting cancelations to occur in the bulk renormalization-group flow of holographic quantum codes. Intermediate technical results include showing the Lewkowycz-Maldacena computation of RT entropy to take the form of a Hamilton-Jacobi variation of the action with respect to boundary conditions, corresponding results for higher-derivative actions, and generalizations to allow RT surfaces with finite conical angles.
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.
Abstract Recent work showed holographic error correcting codes to have simple universal features at O(1/G). In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are associated with flat entanglement spectra indicating maximal entanglement between appropriate subspaces. We extend such results to one-loop order (O(1) corrections) by controlling both higher-derivative corrections to the bulk effective action and dynamical quantum fluctuations below the cutoff. This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in G, ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow. A final corollary requires interesting cancelations to occur in the bulk renormalization-group flow of holographic quantum codes. Intermediate technical results include showing the Lewkowycz-Maldacena computation of RT entropy to take the form of a Hamilton-Jacobi variation of the action with respect to boundary conditions, corresponding results for higher-derivative actions, and generalizations to allow RT surfaces with finite conical angles.
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.