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arXiv:2407.01666 · Journal of High Energy Physics

Tensor networks for black hole interiors: non-isometries, quantum extremal surfaces, and wormholes

Gracemarie Bueller, Oliver Dewolfe A, Kenneth Higginbotham

Published Oct 1, 2024• 5 citations
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Abstract

A bstract We use hyperbolic tensor networks to construct a holographic map for black hole interiors that adds a notion of locality to the non-isometric codes proposed by Akers, Engelhardt, Harlow, Penington, and Vardhan. We use tools provided by these networks to study the relationship between non-isometries and quantum extremal surfaces behind the horizon. Furthermore, we introduce a limited notion of dynamics for these interior tensor networks based on the qudit models introduced by Akers et al., and study the evolution of quantum extremal surfaces in an evaporating black hole. We also find a tensor network description of a wormhole connecting the black hole interior to the radiation, providing a mechanism for interior states and operators to be encoded in the radiation after the Page time. As a particular case, we construct a tensor network realization of the backwards-forwards maps recently proposed to incorporate non-trivial effective dynamics in dynamical constructions of these non-isometric black hole codes.

DOI
10.1007/jhep10(2024)012

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A bstract We use hyperbolic tensor networks to construct a holographic map for black hole interiors that adds a notion of locality to the non-isometric codes proposed by Akers, Engelhardt, Harlow, Penington, and Vardhan. We use tools provided by these networks to study the relationship between non-isometries and quantum extremal surfaces behind the horizon. Furthermore, we introduce a limited notion of dynamics for these interior tensor networks based on the qudit models introduced by Akers et al., and study the evolution of quantum extremal surfaces in an evaporating black hole. We also find a tensor network description of a wormhole connecting the black hole interior to the radiation, providing a mechanism for interior states and operators to be encoded in the radiation after the Page time. As a particular case, we construct a tensor network realization of the backwards-forwards maps recently proposed to incorporate non-trivial effective dynamics in dynamical constructions of these non-isometric black hole codes.
Abstract We introduce a new holographic map for encoding black hole interiors by including both fundamental and effective dynamics. This holographic map is constructed by evolving a state in the effective, semiclassical gravity description of the interior backwards in time to pull the degrees of freedom outside the black hole, before evolving forwards in time in the fundamental description. We show this “backwards-forwards” map is equivalent to a post-selection map of the type introduced by Akers, Engelhardt, Harlow, Penington, and Vardhan, and in the case of trivial effective interactions reduces to their model, while providing a suitable generalization when those interactions are nontrivial. We show the map is equivariant with respect to time evolution, and independent of any interactions outside the black hole. This construction includes interactions with an infaller in a way that preserves the unitarity of black hole evolution exactly and does not allow for superpolynomial computational complexity.
We introduce a new holographic map for encoding black hole interiors by including both fundamental and effective dynamics. This holographic map is constructed by evolving a state in the effective, semiclassical gravity description of the interior backwards in time to pull the degrees of freedom outside the black hole, before evolving forwards in time in the fundamental description. We show this ``backwards-forwards'' map is equivalent to a post-selection map of the type introduced by Akers, Engelhardt, Harlow, Penington, and Vardhan, and in the case of trivial effective interactions reduces to their model, while providing a suitable generalization when those interactions are nontrivial. We show the map is equivariant with respect to time evolution, and independent of any interactions outside the black hole. This construction includes interactions with an infaller in a way that preserves the unitarity of black hole evolution exactly and does not allow for superpolynomial computational complexity.
A bstract Quantum error correction has given us a natural language for the emergence of spacetime, but the black hole interior poses a challenge for this framework: at late times the apparent number of interior degrees of freedom in effective field theory can vastly exceed the true number of fundamental degrees of freedom, so there can be no isometric (i.e. inner-product preserving) encoding of the former into the latter. In this paper we explain how quantum error correction nonetheless can be used to explain the emergence of the black hole interior, via the idea of “non-isometric codes protected by computational complexity”. We show that many previous ideas, such as the existence of a large number of “null states”, a breakdown of effective field theory for operations of exponential complexity, the quantum extremal surface calculation of the Page curve, post-selection, “state-dependent/state-specific” operator reconstruction, and the “simple entropy” approach to complexity coarse-graining, all fit naturally into this framework, and we illustrate all of these phenomena simultaneously in a soluble model.
A bstract The backwards-forwards map, introduced as a generalization of the non-isometric holographic maps of the black hole interior of Akers, Engelhardt, Harlow, Penington, and Vardhan to include non-trivial dynamics in the effective description, has two possible formulations differing in when the post-selection is performed. While these two forms are equivalent on the set of dynamically generated states — states formed from unitary time evolution acting on well-defined initial configurations of infalling matter — they differ on the generic set of states necessary to describe the apparent world of the infalling observer. We show that while both versions successfully reproduce the Page curve, the version involving post-selection as the final step, dubbed the backwards-forwards-post-selection (BFP) map, has the desirable properties of being non-isometric but isometric on average and providing state-dependent reconstruction of bulk operators, while the other version does not. Thus the BFP map is a suitable non-isometric code describing the black hole interior including interior interactions.
We introduce a general class of toy models to study the quantum information-theoretic properties of black hole radiation. The models are governed by a set of isometries that specify how microstates of the black hole at a given energy evolve to entangled states of a tensor product black-hole/radiation Hilbert space. The final state of the black hole radiation is conveniently summarized by a tensor network built from these isometries. We introduce a set of quantities generalizing the Renyi entropies that provide a complete set of bipartite/multipartite entanglement measures, and give a general formula for the average of these over initial black hole states in terms of the isometries defining the model. For models where the dimension of the final tensor product radiation Hilbert space is the same as that of the space of initial black hole microstates, the entanglement structure is universal, independent of the choice of isometries. In the more general case, we find that models which best capture the “information-free” property of black hole horizons are those whose isometries are tensors corresponding to states of tripartite systems with maximally mixed subsystems.
The non-isometric holographic model of the black hole interior stands out as a potential resolution of the long-standing black hole information puzzle since it remedies the friction between the effective calculation and the microscopic description. In this study, combining the final-state projection model, the non-isometric model of black hole interior and Hayden-Preskill thought experiment, we investigate the information recovery from decoding Hawking radiation and demonstrate the emergence of the Page time in this setup. We incorporate the effective modes into the scrambling inside the horizon, which are usually disregarded in Hayden-Preskill protocols, and show that the Page time can be identified as the transition of information transmission channels from the EPR projection to the local projections. This offers a new perspective on the Page time. We compute the decoupling condition under which retrieving information is feasible and show that this model computes the black hole entropy consistent with the quantum extremal surface calculation. Assuming the full knowledge of the dynamics of the black hole interior, we show how Yoshida-Kitaev decoding strategy can be employed in the modified Hayden-Preskill protocol. Furthermore, we perform experimental tests of both probabilistic and Grover's search decoding strategies on the 7-qubit IBM quantum processors to validate our analytical findings and confirm the feasibility of retrieving information in the non-isometric model. This study would stimulate more interests to explore black hole information problem on the quantum processors.