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arXiv:1906.05501 · Journal of High Energy Physics

Shrinking of operators in quantum error correction and AdS/CFT

Hayato Hirai

Published Dec 1, 2019• 2 citations
arXivHTMLPDF

Abstract

Abstract We first show that a class of operators acting on a given bipartite pure state on ℋ A ⊗ ℋ B can shrink its supports on ℋ A ⊗ ℋ B to only ℋ A or ℋ B while keeping its mappings. Using this result, we show how to systematically construct the decoders of the quantum error-correcting codes against erasure errors. The implications of the results for the operator dictionary in the AdS/CFT correspondence are also discussed. The “sub- algebra code with complementary recovery” introduced in the recent work of Harlow is a quantum error-correcting code that shares many common features with the AdS/CFT correspondence. We consider it under the restriction of the bulk (logical) Hilbert space to a subspace that generally has no tensor factorization into subsystems. In this code, the central operators of the reconstructed algebra on the boundary subregion can emerge as a consequence of the restriction of the bulk Hilbert space. Finally, we show a theorem in this code which implies the validity of not only the entanglement wedge reconstruction but also its converse statement with the central operators.

DOI
10.1007/jhep12(2019)128

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Abstract We first show that a class of operators acting on a given bipartite pure state on ℋ A ⊗ ℋ B can shrink its supports on ℋ A ⊗ ℋ B to only ℋ A or ℋ B while keeping its mappings. Using this result, we show how to systematically construct the decoders of the quantum error-correcting codes against erasure errors. The implications of the results for the operator dictionary in the AdS/CFT correspondence are also discussed. The “sub- algebra code with complementary recovery” introduced in the recent work of Harlow is a quantum error-correcting code that shares many common features with the AdS/CFT correspondence. We consider it under the restriction of the bulk (logical) Hilbert space to a subspace that generally has no tensor factorization into subsystems. In this code, the central operators of the reconstructed algebra on the boundary subregion can emerge as a consequence of the restriction of the bulk Hilbert space. Finally, we show a theorem in this code which implies the validity of not only the entanglement wedge reconstruction but also its converse statement with the central operators.
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 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].
In this note we prove a simple theorem in quantum information theory, which implies that bulk operators in the Anti-de Sitter / Conformal Field Theory (AdS/CFT) correspondence can be reconstructed as CFT operators in a spatial subregion AAA, provided that they lie in its entanglement wedge. This is an improvement on existing reconstruction methods, which have at most succeeded in the smaller causal wedge. The proof is a combination of the recent work of Jafferis, Lewkowycz, Maldacena, and Suh on the quantum relative entropy of a CFT subregion with earlier ideas interpreting the correspondence as a quantum error correcting code.
The AdS/CFT correspondence is an explicit realization of the holographic principle relating a theory of gravity in a volume of space to a lower dimensional quantum field theory on its boundary. By exploiting elements of quantum error correction, qubit toy models of this correspondence have been constructed for which the bulk logical operators are representable by operators acting on the boundary. Given a boundary subregion, wedges in the volume space are used to enclose the bulk qubits for which logical operators are reconstructable on that boundary subregion. In this thesis a number of different wedges, such as the causal wedge, greedy entanglement wedge and minimum entanglement wedge, are examined. More specifically, Monte-Carlo simulations of boundary erasure are performed with various toy models to study the differences between wedges and the effect on these wedge by the type of the model, non-uniform boundaries and stacking of models. It has been found that the minimum entanglement wedge is the best approximate for the true geometric wedge. This is illustrated by an example toy model for which an operator beyond the greedy entanglement wedge was also reconstructed. In addition, by calculating the entropy of these subregions, the viability of a mutual information wedge is rejected. Only for particular connected boundary subregions was the inclusion of the central tensor by the geometric wedge associated to a rise in mutual information.
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.
Quantum error correcting codes with finite-dimensional Hilbert spaces have yielded new insights on bulk reconstruction in AdS/CFT. In this paper, we give an explicit construction of a quantum error correcting code where the code and physical Hilbert spaces are infinite-dimensional. We define a von Neumann algebra of type II1{\mathrm{II}}_{1}II1​ acting on the code Hilbert space and show how it is mapped to a von Neumann algebra of type II1{\mathrm{II}}_{1}II1​ acting on the physical Hilbert space. This toy model demonstrates the equivalence of entanglement wedge reconstruction and the exact equality of bulk and boundary relative entropies in infinite-dimensional Hilbert spaces.