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arXiv:1804.00441 · Chinese Physics C

Quantum bit threads of MERA tensor network in large c limit *

Chong-Bin Chen, Fu-Wen Shu, Meng-He Wu

Published Jun 29, 2020• 16 citations
arXivHTMLPDF

Abstract

Abstract The Ryu-Takayanagi (RT) formula plays a large role in the current theory of gauge-gravity duality and emergent geometry phenomena. The recent reinterpretation of this formula in terms of a set of “bit threads” is an interesting effort in understanding holography. In this study, we investigate a quantum generalization of the “bit threads” based on a tensor network, with particular focus on the multi-scale entanglement renormalization ansatz (MERA). We demonstrate that, in the large c limit, isometries of the MERA can be regarded as “sources” (or “sinks”) of the information flow, which extensively modifies the original picture of bit threads by introducing a new variable ρ: density of the isometries. In this modified picture of information flow, the isometries can be viewed as generators of the flow. The strong subadditivity and related properties of the entanglement entropy are also obtained in this new picture. The large c limit implies that classical gravity can emerge from the information flow.

DOI
10.1088/1674-1137/44/7/075102

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Abstract The Ryu-Takayanagi (RT) formula plays a large role in the current theory of gauge-gravity duality and emergent geometry phenomena. The recent reinterpretation of this formula in terms of a set of “bit threads” is an interesting effort in understanding holography. In this study, we investigate a quantum generalization of the “bit threads” based on a tensor network, with particular focus on the multi-scale entanglement renormalization ansatz (MERA). We demonstrate that, in the large c limit, isometries of the MERA can be regarded as “sources” (or “sinks”) of the information flow, which extensively modifies the original picture of bit threads by introducing a new variable ρ: density of the isometries. In this modified picture of information flow, the isometries can be viewed as generators of the flow. The strong subadditivity and related properties of the entanglement entropy are also obtained in this new picture. The large c limit implies that classical gravity can emerge from the information flow.
We consider the special case of random tensor networks (RTNs) endowed with gauge symmetry constraints on each tensor. We compute the R\'enyi entropy for such states and recover the Ryu-Takayanagi (RT) formula in the large-bond regime. The result provides first of all an interesting new extension of the existing derivations of the RT formula for RTNs. Moreover, this extension of the RTN formalism brings it in direct relation with (tensorial) group field theories (and spin networks), and thus provides new tools for realizing the tensor network/geometry duality in the context of background-independent quantum gravity, and for importing quantum gravity tools into tensor network research.
Extending the holographic principle to de Sitter (dS) spacetimes remains one of the most vital open frontiers in quantum gravity, where a microscopic, bottom-up tensor-network framework that relates boundary quantum data to emergent de Sitter spacetime is still lacking. In this work, we first show the emergence of de Sitter spacetime from boundary entanglement by formulating a non-unitary continuous multi-scale entanglement renormalization ansatz (cMERA) for a concrete non-Hermitian critical fermion chain. Within this emergent spacetime, we analyze the associated geodesics and show that they act as extremal Ryu-Takayanagi (RT) surfaces undergoing a smooth timelike-to-null transition. Remarkably, we demonstrate that this continuum trajectory suggests a distinct tensor-network architecture in which the bond-counting contribution naturally truncates at the discrete timelike-to-null transition toward the deep infrared. In the resulting architecture, the null ray along the horizon is represented by zero-cost links, since the associated cut severs no tensor legs. This network structure successfully reproduces the logarithmic scaling of non-unitary critical entanglement entropy, offering a bond-counting picture for the de Sitter RT formula. Our results provide the long-sought dS/(c)MERA correspondence at the level of both emergent spacetime and discrete holographic entanglement.
We show that the gradient of entanglement entropy across a holographic boundary, as described by the Ryu–Takayanagi formula, generates an effective force thatobeys Einstein’s field equations. By representing the entanglement structure of aquantum many-body system as a MERA tensor network, we demonstrate that thevariation of the entanglement entropy with respect to the boundary metric yieldsthe Einstein tensor. The result provides a constructive link between quantum information theory and classical gravity, in which spacetime curvature emerges asa thermodynamic consequence of entanglement renormalization. A testable prediction is derived: the entanglement-driven acceleration of a test particle near aholographic screen deviates from the Newtonian prediction by a factor proportionalto the mutual information density.
Abstract In this work, we compute the entanglement entropy in continuous icMERA tensor networks for large N models at strong coupling. Our results show that the 1/N quantum corrections to the Fisher information metric (interpreted as a local bond dimension of the tensor network) in an icMERA circuit, are related to quantum corrections to the minimal area surface in the Ryu-Takayanagi formula. Upon picking two different non-Gaussian entanglers to build the icMERA circuit, the results for the entanglement entropy only differ at subleading orders in 1/GN, i.e., at the structure of the quantum corrections in the bulk. The fact that the large N part of the entropy can be always related to the leading area term of the holographic calculation is very suggestive. These results, constitute the first tensor network calculations at large N and strong coupling simultaneously, pushing the field of tensor network descriptions of the emergence of dual spacetime geometries from the structure of entanglement in quantum field theory.
A bstract We provide a derivation of the Ryu-Takayanagi (RT) formula in 3D gravity for generic boundary subregions — including RT surface phase transitions — directly from the dual two-dimensional conformal field theory (CFT). Our approach relies on the universal statistics of the algebraic conformal data and the large- c behavior of conformal blocks with Cardy boundaries involved. We observe the emergence of 3D multi-boundary black holes with Karch-Randall branes from entangled states of any number of CFTs with and without Cardy boundaries. The RT formula is obtained directly from the CFT in the high-temperature regime. Two direct applications are: 1) A simple derivation of the multi-interval entanglement entropy for the vacuum state of a single CFT; 2) A CFT-based detection of the emergence of replica wormholes in the context of entanglement islands and black hole microstate counting. Our framework yields the first holographic random tensor network that faithfully captures the entanglement structure of holographic CFTs. These results imply that bulk spacetime geometries indeed emerge from the eigenstate thermalization hypothesis (ETH) in the dual field theory in the large- c limit — a paradigm we refer to as It from ETH .
Bit threads, arising as the convex dual of the minimal-surface formula for holographic entanglement entropy, are line-like structures with nontrivial bulk trajectories. Their physical picture has long been associated with Bell-pair interpretations, yet the meaning of their detailed trajectories, particularly their nonuniqueness, remains unclear. We propose to apply Coecke's notion of quantum information flow (QIF) in protocol-network geometry, developed within categorical quantum mechanics, to a holographic setup, and show that its trajectories within the discrete holographic bulk---the tensor network---obey the divergence-free and density-bound conditions of bit threads. From this process-centered perspective, bit-thread nonuniqueness becomes natural: for a given resource state, different protocols realizing the same distillation task can give rise to different QIF trajectories. We further analyze stabilizer-type holographic tensor networks using ZX string diagrams and show that QIF trajectories can probe finer entanglement structure beyond the level captured by entanglement entropy.