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arXiv:2211.12439

Asymptotically isometric codes for holography

Thomas Faulkner, Min Li

Published Nov 22, 2022• 8 citations
arXivHTMLPDF

Abstract

The holographic principle suggests that the low energy effective field theory of gravity, as used to describe perturbative quantum fields about some background has far too many states. It is then natural that any quantum error correcting code with such a quantum field theory as the code subspace is not isometric. We discuss how this framework can naturally arise in an algebraic QFT treatment of a family of CFT with a large-$N$ limit described by the single trace sector. We show that an isometric code can be recovered in the $N \rightarrow \infty$ limit when acting on fixed states in the code Hilbert space. Asymptotically isometric codes come equipped with the notion of simple operators and nets of causal wedges. While the causal wedges are additive, they need not satisfy Haag duality, thus leading to the possibility of non-trivial entanglement wedge reconstructions. Codes with complementary recovery are defined as having extensions to Haag dual nets, where entanglement wedges are well defined for all causal boundary regions. We prove an asymptotic version of the information disturbance trade-off theorem and use this to show that boundary theory causality is maintained by net extensions. We give a characterization of the existence of an entanglement wedge extension via the asymptotic equality of bulk and boundary relative entropy or modular flow. While these codes are asymptotically exact, at fixed $N$ they can have large errors on states that do not survive the large-$N$ limit. This allows us to fix well known issues that arise when modeling gravity as an exact codes, while maintaining the nice features expected of gravity, including, among other things, the emergence of non-trivial von Neumann algebras of various types.

DOI
10.48550/arxiv.2211.12439

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The holographic principle suggests that the low energy effective field theory of gravity, as used to describe perturbative quantum fields about some background has far too many states. It is then natural that any quantum error correcting code with such a quantum field theory as the code subspace is not isometric. We discuss how this framework can naturally arise in an algebraic QFT treatment of a family of CFT with a large-NNN limit described by the single trace sector. We show that an isometric code can be recovered in the N→∞N \rightarrow \inftyN→∞ limit when acting on fixed states in the code Hilbert space. Asymptotically isometric codes come equipped with the notion of simple operators and nets of causal wedges. While the causal wedges are additive, they need not satisfy Haag duality, thus leading to the possibility of non-trivial entanglement wedge reconstructions. Codes with complementary recovery are defined as having extensions to Haag dual nets, where entanglement wedges are well defined for all causal boundary regions. We prove an asymptotic version of the information disturbance trade-off theorem and use this to show that boundary theory causality is maintained by net extensions. We give a characterization of the existence of an entanglement wedge extension via the asymptotic equality of bulk and boundary relative entropy or modular flow. While these codes are asymptotically exact, at fixed NNN they can have large errors on states that do not survive the large-NNN limit. This allows us to fix well known issues that arise when modeling gravity as an exact codes, while maintaining the nice features expected of gravity, including, among other things, the emergence of non-trivial von Neumann algebras of various types.
In the context of quantum theories of spacetime, one overarching question is how quantum information in the bulk spacetime is encoded holographically in boundary degrees of freedom.It is particularly interesting to understand the correspondence between bulk subregions and boundary subregions in order to address the emergence of locality in the bulk quantum spacetime.For the AdS/CFT correspondence, it is known that this bulk information is encoded redundantly on the boundary in the form of an error-correcting code.Having access only to a subregion of the boundary is as if part of the holographic code has been damaged by noise and rendered inaccessible.In quantum-information science, the problem of recovering information from a damaged code is addressed by the theory of universal recovery channels.We apply and extend this theory to address the problem of relating bulk and boundary subregions in AdS/CFT, focusing on a conjecture known as entanglement wedge reconstruction.Existing work relies on the exact equivalence between bulk and boundary relative entropies, but these are only approximately equal in bulk effective field theory, and in similar situations it is known that predictions from exact entropic equalities can be qualitatively incorrect.We show that the framework of universal recovery channels provides a robust demonstration of the entanglement wedge reconstruction conjecture as well as new physical insights.Most notably, we find that a bulk operator acting in a given boundary region's entanglement wedge can be expressed as the response of the boundary region's modular Hamiltonian to a perturbation of the bulk state in the direction of the bulk operator.This formula can be interpreted as a noncommutative version of Bayes's rule that attempts to undo the noise induced by restricting to only a portion of the boundary.To reach these conclusions, we extend the theory of universal recovery channels to finite-dimensional operator algebras and demonstrate that recovery channels approximately preserve the multiplicative structure of the operator algebra.
Bulk reconstruction is a central problem in AdS/CFT, and entanglement wedge reconstruction is its subregion version. We argue that this subregion statement should be separated from the stronger holographic quantum error correction interpretation, in which one region-independent logical bulk operator has code-preserving representatives in several boundary regions. A simple locality argument shows that such a common reconstruction must commute with the code-preserving local algebras in the complementary regions. This is the mechanism realized in HaPPY-type codes: the erased regions are blind to a protected logical algebra. An ordinary finite NNN holographic CFT does not have such a protected invisible sector for supergravity fields. Its low-energy local observables, in particular, suitably smeared stress tensors, detect the physical support and gravitational dressing of ordinary bulk operators, up to possible center or superselection data. Thus, there is no such holographic quantum error correction and the N=∞N=\inftyN=∞ agreement of global and subregion HKLL formulae is a free-theory statement. What remains is entanglement wedge reconstruction without holographic quantum error correction, or subregion complementarity: each boundary region has its own code-preserving low-energy algebra and its own region-adapted bulk interpretation, rather than a shared logical operator.
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.
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.
We explicitly construct a class of holographic quantum error correction codes with non-trivial centers in the code subalgebra. Specifically, we use the Bacon-Shor codes and perfect tensors to construct a gauge code (or a stabilizer code with gauge-fixing), which we call the holographic hybrid code. This code admits a local log-depth encoding/decoding circuit, and can be represented as a holographic tensor network which satisfies an analog of the Ryu-Takayanagi formula and reproduces features of the sub-region duality. We then construct approximate versions of the holographic hybrid codes by “skewing” the code subspace, where the size of skewing is analogous to the size of the gravitational constant in holography. These approximate hybrid codes are not necessarily stabilizer codes, but they can be expressed as the superposition of holographic tensor networks that are stabilizer codes. For such constructions, different logical states, representing different bulk matter content, can “back-react” on the emergent geometry, resembling a key feature of gravity. The locality of the bulk degrees of freedom becomes subspace-dependent and approximate. Such subspace-dependence is manifest from the point of view of the “entanglement wedge” and bulk operator reconstruction from the boundary. Exact complementary error correction breaks down for certain bipartition of the boundary degrees of freedom; however, a limited, state-dependent form is preserved for particular subspaces. We also construct an example where the connected two-point correlation functions can have a power-law decay. Coupled with known constraints from holography, a weakly back-reacting bulk also forces these skewed tensor network models to the “large N limit” where they are built by concatenating a large N number of copies.
Gauge/gravity duality is often described as a quantum error correcting code. However, as seen in the Jafferis-Lewkowycz-Maldacena-Suh (JLMS) formula, exact quantum error correction with complementary recovery (and thus entanglement wedge reconstruction) emerges only in the limit G→0G \to 0G→0. As a result, precise arguments controlling error terms have focused on what we call `small' codes which, as G→0G \to 0G→0, describe only perturbative excitations near a given classical solution. Such settings are quite restrictive and, in particular, they prohibit discussion of any modular flow that would change the classical background. As a result, they forbid consideration of modular flows generated by semiclassical bulk states at order-one modular parameters. In contrast, we present a single `large' code for the bulk theory that can accommodate such flows and, in particular, in the G→0G \to 0G→0 limit includes superpositions of states associated with distinct classical backgrounds. This large code is assembled from small codes that each satisfy an approximate Faulkner-Lewkowycz-Maldacena formula. In this extended setting we clarify the meaning of the (approximate) JLMS relation between bulk and boundary modular Hamiltonians and quantify its validity in an appropriate class of states.