nLab
tensor network

Idea

The term tensor network has become popular in quantum physics for essentially what in monoidal category theory is referred to a 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

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

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)

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)