nLab
tensor network

Contents

Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

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.

References

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

In holographic entanglement entropy

The use of tensor netwaorks 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

In higher parallel transport

Discussion in relation to higher parallel transport:

Created on February 3, 2019 at 05:36:07. See the history of this page for a list of all contributions to it.