### Abstract

We describe a graphical calculus for completely positive maps and in doing so review the theory of open quantum systems and other fundamental primitives of quantum information theory using the language of tensor networks. In particular we demonstrate the construction of tensor networks to pictographically represent the Liouville-superoperator, Choi-matrix, process-matrix, Kraus, and system-environment representations for the evolution of quantum states, review how these representations interrelate, and illustrate how graphical manipulations of the tensor networks may be used to concisely transform between them. To further demonstrate the utility of the presented graphical calculus we include several examples where we provide arguably simpler graphical proofs of several useful quantities in quantum information theory including the composition and contraction of multipartite channels, a condition for whether an arbitrary bipartite state may be used for ancilla assisted process tomography, and the derivation of expressions for the average gate delity and entanglement delity of a channel in terms of each of the dierent representations of the channel.

Original language | English (US) |
---|---|

Pages (from-to) | 759-811 |

Number of pages | 53 |

Journal | Quantum Information and Computation |

Volume | 15 |

Issue number | 9-10 |

State | Published - Jan 1 2015 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Statistical and Nonlinear Physics
- Nuclear and High Energy Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Computational Theory and Mathematics

### Cite this

*Quantum Information and Computation*,

*15*(9-10), 759-811.

}

*Quantum Information and Computation*, vol. 15, no. 9-10, pp. 759-811.

**Tensor networks and graphical calculus for open quantum systems.** / Wood, Christopher J.; Biamonte, Jacob D.; Cory, David G.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Tensor networks and graphical calculus for open quantum systems

AU - Wood, Christopher J.

AU - Biamonte, Jacob D.

AU - Cory, David G.

PY - 2015/1/1

Y1 - 2015/1/1

N2 - We describe a graphical calculus for completely positive maps and in doing so review the theory of open quantum systems and other fundamental primitives of quantum information theory using the language of tensor networks. In particular we demonstrate the construction of tensor networks to pictographically represent the Liouville-superoperator, Choi-matrix, process-matrix, Kraus, and system-environment representations for the evolution of quantum states, review how these representations interrelate, and illustrate how graphical manipulations of the tensor networks may be used to concisely transform between them. To further demonstrate the utility of the presented graphical calculus we include several examples where we provide arguably simpler graphical proofs of several useful quantities in quantum information theory including the composition and contraction of multipartite channels, a condition for whether an arbitrary bipartite state may be used for ancilla assisted process tomography, and the derivation of expressions for the average gate delity and entanglement delity of a channel in terms of each of the dierent representations of the channel.

AB - We describe a graphical calculus for completely positive maps and in doing so review the theory of open quantum systems and other fundamental primitives of quantum information theory using the language of tensor networks. In particular we demonstrate the construction of tensor networks to pictographically represent the Liouville-superoperator, Choi-matrix, process-matrix, Kraus, and system-environment representations for the evolution of quantum states, review how these representations interrelate, and illustrate how graphical manipulations of the tensor networks may be used to concisely transform between them. To further demonstrate the utility of the presented graphical calculus we include several examples where we provide arguably simpler graphical proofs of several useful quantities in quantum information theory including the composition and contraction of multipartite channels, a condition for whether an arbitrary bipartite state may be used for ancilla assisted process tomography, and the derivation of expressions for the average gate delity and entanglement delity of a channel in terms of each of the dierent representations of the channel.

UR - http://www.scopus.com/inward/record.url?scp=84928613570&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84928613570&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84928613570

VL - 15

SP - 759

EP - 811

JO - Quantum Information and Computation

JF - Quantum Information and Computation

SN - 1533-7146

IS - 9-10

ER -