# 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 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 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