# nLab tensor network

Contents

### Context

#### Monoidal categories

monoidal categories

# Contents

## 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.

## References

### General

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

reviewed in