Abstract
A sort of planar tensor networks with tensor constraints is investigated as a model for holography. We study the greedy algorithm generated by tensor constraints and propose the notion of critical protection (CP) against the action of greedy algorithm. For given tensor constraints, a CP tensor chain can be defined. We further find that the ability of quantum error correction (QEC), the non-flatness of entanglement spectrum (ES) and the correlation function can be quantitatively evaluated by the geometric structure of CP tensor chain. Four classes of tensor networks with different properties of entanglement is discussed. Thanks to tensor constraints and CP, the correlation function is reduced into a bracket of Matrix Production State and the result agrees with the one in conformal field theory.
Highlights
Quantum entanglement plays a key role in understanding the structure of spacetime from the emergent point of view [1,2]
We further find that the ability of quantum error correction (QEC), the non-flatness of entanglement spectrum (ES) and the correlation function can be quantitatively evaluated by the geometric structure of critical protection (CP) tensor chain
In this paper the notion of critical protection based on tensor chain has been proposed to describe the behavior of tensor networks under the action of greedy algorithm
Summary
Quantum entanglement plays a key role in understanding the structure of spacetime from the emergent point of view [1,2]. For the vacuum in AdS3=CFT2 correspondence, Renyi entropy satisfies Cardy-Calabrese formula and a nonflat ES is inherent [5,6] Another remarkable feature of AdS space is the subsystem duality, which states that a local operator in the bulk can be reconstructed in a subsystem A on the boundary if it is located within the entanglement wedge of A [7,8,9,10,11,12,13]. The key ingredient of hyperinvariant tensor networks is to impose multitensor constraints, which demand certain product of multiple tensors to form an isometric mapping This sort of tensor networks can accomplish QEC as perfect tensors, and generate non-flat ES as MERA, qualitatively capturing both holographic features of AdS spacetime. A geometric quantity κc, named as the average reduced interior angle of a tensor chain, will be proposed to measure the ability of QEC and justify the flatness of ES
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