nLab
tensor network

Idea

The term tensor network has become popular in quantum physics for essentially what in mathematical physics is known as Penrose notation and in monoidal category theory is referred to as string diagrams.

The term rose to prominence in quantum physics partly with discussion of finite quantum mechanics in terms of dagger-compact categories but then mainly via its use in holographic entanglement entropy

For finite quantum mechanics in $\dagger$-compact categories

Application to finite quantum mechanics in terms of dagger-compact categories… (see there).

For holographic entanglement entropy

Application to holographic entanglement entropy (…)

graphics grabbed from Harlow 18

graphics grabbed from Harlow 18

In this context the Ryu-Takayanagi formula for holographic entanglement entropy has an exact proof PYHP 15, Theorem 2.

Related concepts

• tensor, tensor product, tensor category

• string diagram

• neural network

References

General

• Jacob Biamonte, Ville Bergholm, Tensor Networks in a Nutshell, Contemporary Physics (arxiv:1708.00006)

• Jacob Biamonte, Lectures on Quantum Tensor Networks (arXiv:1912.10049)

Relation to spin chains:

• Mari Carmen Banuls, Michal P. Heller, Karl Jansen, Johannes Knaute, Viktor Svensson, From spin chains to real-time thermal field theory using tensor networks (arXiv:1912.08836)

In holographic entanglement entropy

The use of tensor networks as a tool in holographic entanglement entropy goes back to

• Brian Swingle, Entanglement Renormalization and Holography (arXiv:0905.1317)

• Brian Swingle, Constructing holographic spacetimes using entanglement renormalization (arXiv:1209.3304)

Further interpretation in terms of quantum error correcting codes is due to

• Ahmed Almheiri, Xi Dong, Daniel Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP 1504:163,2015 (arXiv:1411.7041)

• Fernando Pastawski, Beni Yoshida, Daniel Harlow, John Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, JHEP 06 (2015) 149 (arXiv:1503.06237)

reviewed in

• Daniel Harlow, TASI Lectures on the Emergence of Bulk Physics in AdS/CFT (arXiv:1802.01040)

Discussion in the context of the p-adic AdS/CFT correspondence:

• Matthew Heydeman, Matilde Marcolli, Ingmar Saberi, Bogdan Stoica, Tensor networks, $p$-adic fields, and algebraic curves: arithmetic and the $AdS_3/CFT_2$ correspondence (arXiv:1605.07639)

• Arpan Bhattacharyya, Ling-Yan Hung, Yang Lei, Wei Li, Tensor network and (p-adic) AdS/CFT, (arXiv:1703.05445)

See also

• Ning Bao, Geoffrey Penington, Jonathan Sorce, Aron C. Wall, Beyond Toy Models: Distilling Tensor Networks in Full AdS/CFT (arXiv:1812.01171)

• Ning Bao, Geoffrey Penington, Jonathan Sorce, Aron C. Wall, Holographic Tensor Networks in Full AdS/CFT (arXiv:1902.10157)

• Alexander Jahn, Marek Gluza, Fernando Pastawski, Jens Eisert, Majorana dimers and holographic quantum error-correcting codes (arXiv:1905.03268)

In higher parallel transport

Discussion in relation to higher parallel transport:

• Arthur Parzygnat, Two-dimensional algebra in lattice gauge theory (arXiv:1802.01139)