# Graph States and the necessity of Euler Decomposition

*Ross Duncan and Simon Perdrix*

### Abstract

Coecke and Duncan recently introduced a categorical formalisation of the interaction of complementary quantum observables. In this paper we use their diagrammatic language to study graph states, a computationally interesting class of quantum states. We give a graphical proof of the fixpoint property of graph states. We then introduce a new equation, for the Euler decomposition of the Hadamard gate, and demonstrate that Van den Nest's theorem–locally equivalent graphs represent the same entanglement–is equivalent to this new axiom. Finally we prove that the Euler decomposition equation is not derivable from the existing axioms.

Book Title

Computability in Europe: Mathematical Theory and Computational Practice (CiE'09)

Editor

Ambos−Spies‚ K. and Löwe‚ B. and Merkle‚ W.

Keywords

quantum computing; categorical quantum mechanics; graphical calculi; measurement−based quantum computing; entanglement

Note

Preprint available at http://arxiv.org/abs/0902.0500

Pages

167–177

Publisher

Springer

Series

Lecture Notes in Computer Science

Volume

5635

Year

2009