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A tensor network formalism of thermofield dynamics is introduced. The formalism relates the original Hilbert space with its tilde space by a product of two copies of a tensor network. Then, their interface becomes an event horizon, and the logarithm of the tensor rank corresponds to the black hole entropy. Eventually, multiscale entanglement renormalization anzats (MERA) reproduces an AdS black hole at finite temperature. Our finding shows rich functionalities of MERA as efficient graphical representation of AdS/CFT correspondence.
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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.
This work concerns the quantum Lorentzian and Euclidean black hole non-linear sigma models. For the Euclidean black hole sigma model an equilibrium density matrix is proposed, which reproduces the modular invariant partition function from the 2001 paper of Maldacena, Ooguri and Son. For the Lorentzian black hole sigma model, using its formulation as a gauged ${rm SL}(2,mathbb{R})$ WZW model, we describe the linear and Hermitian structure of its space of states and also propose an expression for the equilibrium density matrix. Our analysis is guided by the results of the study of a certain critical, integrable spin chain. In the scaling limit, the latter exhibits the key features of the Lorentzian black hole sigma model including the same global symmetries, the same algebra of extended conformal symmetry and a continuous spectrum of conformal dimensions.
Techniques for approximately contracting tensor networks are limited in how efficiently they can make use of parallel computing resources. In this work we demonstrate and characterize a Monte Carlo approach to the tensor network renormalization group method which can be used straightforwardly on modern computing architectures. We demonstrate the efficiency of the technique and show that Monte Carlo tensor network renormalization provides an attractive path to improving the accuracy of a wide class of challenging computations while also providing useful estimates of uncertainty and a statistical guarantee of unbiased results.
We argue that the late time behavior of horizon fluctuations in large anti-de Sitter (AdS) black holes is governed by the random matrix dynamics characteristic of quantum chaotic systems. Our main tool is the Sachdev-Ye-Kitaev (SYK) model, which we use as a simple model of a black hole. We use an analytically continued partition function $|Z(beta +it)|^2$ as well as correlation functions as diagnostics. Using numerical techniques we establish random matrix behavior at late times. We determine the early time behavior exactly in a double scaling limit, giving us a plausible estimate for the crossover time to random matrix behavior. We use these ideas to formulate a conjecture about general large AdS black holes, like those dual to 4D super-Yang-Mills theory, giving a provisional estimate of the crossover time. We make some preliminary comments about challenges to understanding the late time dynamics from a bulk point of view.
In this article, we investigate the quantum circuit complexity and entanglement entropy in the recently studied black hole gas framework using the two-mode squeezed states formalism written in arbitrary dimensional spatially flat cosmological Friedmann-Lema$hat{i}$tre-Robertson-Walker (FLRW) background space-time. We compute the various complexity measures and study the evolution of these complexities by following two different prescriptions viz. Covariant matrix method and Nielsens method. Independently, using the two-mode squeezed states formalism we also compute the Renyi and Von-Neumann entanglement entropy, which show an inherent connection between the entanglement entropy and quantum circuit complexity. We study the behaviour of the complexity measures and entanglement entropy separately for three different spatial dimensions and observe various significant different features in three spatial dimensions on the evolution of these quantities with respect to the scale factor. Furthermore, we also study the underlying behaviour of the equilibrium temperature with two of the most essential quantities i.e. rate of change of complexity with scale factor and the entanglement entropy. We observe that irrespective of the spatial dimension, the equilibrium temperature depends quartically on entanglement entropy.
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