### Abstract

We examine the use of string diagrams and the mathematics of category theory in the description of quantum states by tensor networks. This approach lead to a unification of several ideas, as well as several results and methods that have not previously appeared in either side of the literature. Our approach enabled the development of a tensor network framework allowing a solution to the quantum decomposition problem which has several appealing features. Specifically, given an n-body quantum state S, we present a new and general method to factor S into a tensor network of clearly defined building blocks. We use the solution to expose a previously unknown and large class of quantum states which we prove can be sampled efficiently and exactly. This general framework of categorical tensor network states, where a combination of generic and algebraically defined tensors appear, enhances the theory of tensor network states.

Original language | English |
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Article number | 042172 |

Number of pages | 39 |

Journal | AIP Advances |

Volume | 1 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2 Dec 2010 |

### Keywords

- quant-ph
- cond-mat.other
- cs.CC
- cs.LO
- math-ph
- math.MP

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## Cite this

Biamonte, J. D., Clark, S. R., & Jaksch, D. (2010). Categorical Tensor Network States.

*AIP Advances*,*1*(4), [042172]. https://doi.org/10.1063/1.3672009