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arXiv:2408.00071 · Journal of High Energy Physics

An operator algebraic approach to black hole information

Jeremy van der Heijden, Erik Verlinde

Published Feb 28, 2025• 7 citations
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Abstract

A bstract We present an operator algebraic perspective on the black hole information problem. For a black hole after Page time that is entangled with the early radiation we formulate a version of the information puzzle that is well-posed in the G → 0 limit. We then give a description of the information recovery protocol in terms of von Neumann algebras using elements of the Jones index theory of type II 1 subfactors. The subsequent evaporation and recovery steps are represented by Jones’s basic construction, and an operation called the canonical shift. A central element in our description is the Jones projection, which leads to an entanglement swap and implements an operator algebraic version of a quantum teleportation protocol. These aspects are further elaborated on in a microscopic model based on type I algebras. Finally, we argue that in the emergent type III algebra the canonical shift may be interpreted as a spacetime translation and, hence, that at the microscopic level “translation = teleportation”.

DOI
10.1007/jhep02(2025)207

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A bstract We present an operator algebraic perspective on the black hole information problem. For a black hole after Page time that is entangled with the early radiation we formulate a version of the information puzzle that is well-posed in the G → 0 limit. We then give a description of the information recovery protocol in terms of von Neumann algebras using elements of the Jones index theory of type II 1 subfactors. The subsequent evaporation and recovery steps are represented by Jones’s basic construction, and an operation called the canonical shift. A central element in our description is the Jones projection, which leads to an entanglement swap and implements an operator algebraic version of a quantum teleportation protocol. These aspects are further elaborated on in a microscopic model based on type I algebras. Finally, we argue that in the emergent type III algebra the canonical shift may be interpreted as a spacetime translation and, hence, that at the microscopic level “translation = teleportation”.
A bstract We propose an algebraic definition of ER=EPR in the G N → 0 limit, which associates bulk spacetime connectivity/disconnectivity to the operator algebraic structure of a quantum gravity system. The new formulation not only includes information on the amount of entanglement, but also more importantly the structure of entanglement . We give an independent definition of a quantum wormhole as part of the proposal. This algebraic version of ER=EPR sheds light on a recent puzzle regarding spacetime disconnectivity in holographic systems with O \mathcal{O} O O (1 /G N ) entanglement. We discuss the emergence of quantum connectivity in the context of black hole evaporation and further argue that at the Page time, the black hole-radiation system undergoes a transition involving the transfer of an emergent type III 1 subalgebra of high complexity operators from the black hole to radiation. We argue this is a general phenomenon that occurs whenever there is an exchange of dominance between two competing quantum extremal surfaces.
A bstract In this work, we develop an algebraic description of the Page transition, a key feature in black hole evaporation where the entropy of Hawking radiation follows a unitary Page curve instead of monotonically increasing. By applying concepts from approximate quantum error correction with complementary recovery, we characterize the Page transition as a phase transition in channel recovery. We then generalize the description to infinite-dimensional settings using algebraic relative entropy, which remains valid even in type III factors. For type I/II factors, explicit probes based on relative entropy differences are derived, serving as indicators for the transition at the Page time.
A generalization for the quantum information retrieval protocol recently illustrated by Verlinde and van der Heijden for evaporating black holes is provided to inclusions of type III von Neumann factors. The physical interest of such scenario arises in Quantum Field Theory, where local algebras are type III von Neumann algebras. The formula obtained can be easily interpreted in terms of the statistical dimension of superselection sectors in the case of black holes undergoing charge evaporation, thanks to the index-statistics theorem, leading to a thermodynamic interpretation. A constraint on the values of the index leads to a final remark about the quantization of the charge emitted by the black hole during the evaporation process.
We show that quantum teleportation is intimately connected to Jones' basic construction. This connection leads to generalized, or hybrid teleportation protocols to simultaneously transmit classical and quantum information through finite dimensional von Neumann algebras. We present two operationally concrete teleportation protocols, Hybrid teleportation and our so-called Scaffolding teleportation. Under some mild symmetry assumptions, we establish that any teleportation scheme in the scaffolding framework is equivalent to a teleportation scheme of a special form.
We examine the possibility of reconstructing quantum information emitted by an evaporating black hole using modular operator theory within an algebraic framework. This approach is based on the entanglement structure described by Tomita-Takesaki theory, without relying on a specific model of quantum gravity. In comparison with other solutions to the black hole information paradox (notably the "islands" approach and quantum errorcorrecting codes), our proposal aims to characterize the approximate localizability of interior black hole operators within the Hawking radiation. We also discuss conceptual limitations of this method and potential implementations via tensor networks.
In this work, we develop an algebraic description of the Page transition, a key feature in black hole evaporation where the entropy of Hawking radiation follows a unitary Page curve instead of monotonically increasing. By applying concepts from approximate quantum error correction with complementary recovery, we characterize the Page transition as a phase transition in channel recovery. We then generalize the description to infinite-dimensional settings using algebraic relative entropy, which remains valid even in type III factors. For type I/II factors, explicit probes based on relative entropy differences are derived, serving as indicators for the transition at the Page time.