Tensor chain and constraints in tensor networks

TL;DR

This paper introduces a framework for tensor networks with constraints, using tensor chains on hyperbolic space, to model AdS/CFT correspondence and analyze quantum error correction and entanglement spectrum.

Contribution

It generalizes hyperinvariant tensor networks by incorporating tensor constraints and classifies networks based on QEC and entanglement properties.

Findings

Tensor chains can be classified by their reduced interior angle.

The framework models AdS3/CFT2 correspondence.

Tensor network properties relate to quantum error correction and entanglement spectrum.

Abstract

This paper accompanies with our recent work on quantum error correction (QEC) and entanglement spectrum (ES) in tensor networks (arXiv:1806.05007). We propose a general framework for planar tensor network state with tensor constraints as a model for $AdS_3/CFT_2$ correspondence, which could be viewed as a generalization of hyperinvariant tensor networks recently proposed by Evenbly. We elaborate our proposal on tensor chains in a tensor network by tiling $H^2$ space and provide a diagrammatical description for general multi-tensor constraints in terms of tensor chains, which forms a generalized greedy algorithm. The behavior of tensor chains under the action of greedy algorithm is investigated in detail. In particular, for a given set of tensor constraints, a critically protected (CP) tensor chain can be figured out and evaluated by its average reduced interior angle. We classify tensor networks according to their ability of QEC and the flatness of ES. The corresponding geometric description of critical protection over the hyperbolic space is also given.