Tensor chain and constraints in tensor networks

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 /CF T 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

Quantum error correction and entanglement spectrum in tensor networks

“…The number of layers is approximately proportional to the distance between two CP tensor chains beside the geodesic. It is interesting to notice that given a tiling of tensor network, this distance becomes larger with the increase of κ c , which is observed if we compare different tensor networks in this paper with each other and is proved in [34]. Such tendency implies that the correlation between two endpoints carried by the MPS becomes stronger as well.…”

“…As a result, the tensor network with κ c = 1 in this paper is not a candidate of holography. Furthermore, among the examples of tensor networks considered in this paper, the tensor network with κ c = 3/2 has more likelihood to mimic the AdS holography since it exhibits both features of QEC and non-flat ES, which motivates us to propose some strategy to construct tensor networks in more general setup which could capture the desirable holographic aspects of AdS space, which will be explored in [34].…”

“…The detailed analysis is given in [34]. Here we just argue that all of these tensor chains form a set S D with infinite number of elements, and satisfy κ ≤ κ c .…”

“…This phenomenon results from the boundary effect in a network with finite layers, where besides the lower edges of M c , the edges at the end of M c need to be contracted as well. The boundary effect of greedy algorithm is investigated with details in [34].…”

“…We remark that such a relation still holds in general cases. A detailed analysis on tensor networks with general tiling and general constraints is given in [34], where the tensor networks with general constraints in terms of tensor chains are classified based on their properties of QEC and ES, with the power of CP reduced interior angle κ c . The four tensor networks considered in this paper are typical examples of their own class.…”