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arXiv:2305.16028 · Journal of High Energy Physics

Information loss, mixing and emergent type III1 factors

Keiichiro Furuya, Nima Lashkari, Mudassir Moosa, Shoy Ouseph

Published Aug 18, 2023• 25 citations
arXivHTMLPDF

Abstract

Abstract A manifestation of the black hole information loss problem is that the two-point function of probe operators in a large Anti-de Sitter black hole decays in time, whereas, on the boundary CFT, it is expected to be an almost periodic function of time. We point out that the decay of the two-point function (clustering in time) holds important clues to the nature of observable algebras, states, and dynamics in quantum gravity. We call operators that cluster in time “mixing” and explore the necessary and sufficient conditions for mixing. The information loss problem is a special case of the statement that in type I algebras, there exists no mixing operators. We prove that, in a thermofield double state (KMS state), if mixing operators form an algebra (close under multiplication), the resulting algebra must be a von Neumann type III1 factor. In other words, the physically intuitive requirement that all nonconserved operators should exponentially mix is so strong that it fixes the observable algebra to be an exotic algebra called a type III1 factor. More generally, for an arbitrary out-of-equilibrium state of a general quantum system (von Neumann algebra), we show that if the set of operators that mix under modular flow forms an algebra, it is a type III1 von Neumann factor. In a theory of Generalized Free Fields (GFF), we show that if the two-point function clusters in time, all operators are mixing, and the algebra is a type III1 factor. For example, in 𝒩 = 4 SYM, above the Hawking-Page phase transition, clustering of the single trace operators implies that the algebra is a type III1 factor, settling a recent conjecture of Leutheusser and Liu. We explicitly construct the C∗-algebra and von Neumann subalgebras of GFF associated with time bands and, more generally, open sets of the bulk spacetime using the HKLL reconstruction map.

DOI
10.1007/jhep08(2023)111

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Abstract A manifestation of the black hole information loss problem is that the two-point function of probe operators in a large Anti-de Sitter black hole decays in time, whereas, on the boundary CFT, it is expected to be an almost periodic function of time. We point out that the decay of the two-point function (clustering in time) holds important clues to the nature of observable algebras, states, and dynamics in quantum gravity. We call operators that cluster in time “mixing” and explore the necessary and sufficient conditions for mixing. The information loss problem is a special case of the statement that in type I algebras, there exists no mixing operators. We prove that, in a thermofield double state (KMS state), if mixing operators form an algebra (close under multiplication), the resulting algebra must be a von Neumann type III1 factor. In other words, the physically intuitive requirement that all nonconserved operators should exponentially mix is so strong that it fixes the observable algebra to be an exotic algebra called a type III1 factor. More generally, for an arbitrary out-of-equilibrium state of a general quantum system (von Neumann algebra), we show that if the set of operators that mix under modular flow forms an algebra, it is a type III1 von Neumann factor. In a theory of Generalized Free Fields (GFF), we show that if the two-point function clusters in time, all operators are mixing, and the algebra is a type III1 factor. For example, in 𝒩 = 4 SYM, above the Hawking-Page phase transition, clustering of the single trace operators implies that the algebra is a type III1 factor, settling a recent conjecture of Leutheusser and Liu. We explicitly construct the C∗-algebra and von Neumann subalgebras of GFF associated with time bands and, more generally, open sets of the bulk spacetime using the HKLL reconstruction map.
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The central question in this article is how information does leak out from black holes. Relying on algebraic arguments and the concept of superselection sectors, we propose the existence of certain operators whose correlations extend across the black hole atmosphere and range into the interior. Contained in the full algebra, these black hole intertwiners will not belong to the subalgebra describing semiclassical bulk physics. We study this proposal in the context of operator reconstructions for code spaces containing a large number of microstates. As long as the atmosphere is excluded from a particular subsystem, the global state seen under the action of the associated algebra is maximally mixed and therefore described by a single classical background. Once the relevant correlations are encoded, i.e. if the algebra is sufficiently enlarged, perfect state distinguishability becomes possible. We arrive at this by computing the von Neumann entropy which may explain the result obtained by applying the quantum extremal surface prescription to the mixed state. We then examine these insights in the context of black hole evaporation and argue that information is transferred to the radiation via black hole intertwiners. We derive the Page curve. The mechanism above suggests that black hole information is topologically protected. An infalling observer would experience no drama. This may resolve the unitarity problem without running into any firewall or state puzzle, the latter being evident in generalized entropy computations. We also examine the question of how certain wormhole topologies may be understood given these findings. We argue that their occurrence in gravity replica computations may be related to the maximal correlation between radiation and atmosphere surrounding the old black hole. This may suggest a connection between topology change and near horizon quantum gravitational effects.
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Mathematical Applications of Science Fiction We present a rigorous derivation of the spectral inversion map for modular Hamiltonians within the framework of Type III von Neumann algebras, particularly those arising in the algebraic formulation of holographic field theories. By introducing a measure of Non-Commutative Kolmogorov Complexity defined on the predual space of the operator algebra, we establish a direct correspondence between the algorithmic information content of a quantum state and the spectral density of its associated modular operator. We prove that the reconstruction of bulk operators from boundary data is formally equivalent to a spectral inversion problem subject to an algorithmic entropy bound. To this end, we construct the Valerius–Kolmogorov Operator, a unitary mapping that retrieves information from the island regions of the entanglement wedge by inverting the modular flow. We provide detailed proofs of the existence, uniqueness, and stability of this inversion, showing that unitary information retrieval is possible precisely when the complexity growth of the modular flow is subexponential. This framework resolves the trans-temporal information leakage paradox in black hole evaporation models by interpreting Hawking radiation as a complexity-encoded modular scramble.