### Abstract

Language | English |
---|---|

Article number | 043016 |

Number of pages | 85 |

Journal | New Journal of Physics |

Volume | 13 |

Issue number | April |

DOIs | |

Publication status | Published - Apr 2011 |

### Fingerprint

### Keywords

- computational physics
- quantum observables
- categorical algebra
- diagrammatics

### Cite this

*New Journal of Physics*,

*13*(April), [043016 ]. https://doi.org/10.1088/1367-2630/13/4/043016

}

*New Journal of Physics*, vol. 13, no. April, 043016 . https://doi.org/10.1088/1367-2630/13/4/043016

**Interacting quantum observables : categorical algebra and diagrammatics.** / Coecke, Bob; Duncan, Ross.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Interacting quantum observables

T2 - New Journal of Physics

AU - Coecke, Bob

AU - Duncan, Ross

PY - 2011/4

Y1 - 2011/4

N2 - This paper has two tightly intertwined aims: (i) to introduce an intuitive and universal graphical calculus for multi-qubit systems, the ZX-calculus, which greatly simplifies derivations in the area of quantum computation and information. (ii) To axiomatize complementarity of quantum observables within a general framework for physical theories in terms of dagger symmetric monoidal categories. We also axiomatize phase shifts within this framework. Using the well-studied canonical correspondence between graphical calculi and dagger symmetric monoidal categories, our results provide a purely graphical formalisation of complementarity for quantum observables. Each individual observable, represented by a commutative special dagger Frobenius algebra, gives rise to an Abelian group of phase shifts, which we call the phase group. We also identify a strong form of complementarity, satisfied by the Z- and X-spin observables, which yields a scaled variant of a bialgebra.

AB - This paper has two tightly intertwined aims: (i) to introduce an intuitive and universal graphical calculus for multi-qubit systems, the ZX-calculus, which greatly simplifies derivations in the area of quantum computation and information. (ii) To axiomatize complementarity of quantum observables within a general framework for physical theories in terms of dagger symmetric monoidal categories. We also axiomatize phase shifts within this framework. Using the well-studied canonical correspondence between graphical calculi and dagger symmetric monoidal categories, our results provide a purely graphical formalisation of complementarity for quantum observables. Each individual observable, represented by a commutative special dagger Frobenius algebra, gives rise to an Abelian group of phase shifts, which we call the phase group. We also identify a strong form of complementarity, satisfied by the Z- and X-spin observables, which yields a scaled variant of a bialgebra.

KW - computational physics

KW - quantum observables

KW - categorical algebra

KW - diagrammatics

U2 - 10.1088/1367-2630/13/4/043016

DO - 10.1088/1367-2630/13/4/043016

M3 - Article

VL - 13

JO - New Journal of Physics

JF - New Journal of Physics

SN - 1367-2630

IS - April

M1 - 043016

ER -