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.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science: LICS 2004, pp. 415–425. IEEE Computer Society, Los Alamitos (2004)
Coecke, B., Duncan, R.: Interacting quantum observables. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 298–310. Springer, Heidelberg (2008)
Hein, M., Dür, W., Eisert, J., Raussendorf, R., Van den Nest, M., Briegel, H.J.: Entanglement in graph states and its applications. In: Proceedings of the International School of Physics “Enrico Fermi” on Quantum Computers, Algorithms and Chaos (July 2005)
Raussendorf, R., Briegel, H.J.: A one-way quantum computer. Phys. Rev. Lett. 86, 5188–5191 (2001)
Van den Nest, M., Dehaene, J., De Moor, B.: Graphical description of the action of local clifford transformations on graph states. Physical Review A 69, 022316 (2004)
Kelly, G., Laplaza, M.: Coherence for compact closed categories. Journal of Pure and Applied Algebra 19, 193–213 (1980)
Joyal, A., Street, R.: The geometry of tensor categories i. Advances in Mathematics 88, 55–113 (1991)
Selinger, P.: Dagger compact closed categories and completely positive maps. In: Proceedings of the 3rd International Workshop on Quantum Programming Languages (2005)
Duncan, R.: Types for Quantum Computation. PhD thesis. Oxford University (2006)
Coecke, B., Pavlovic, D., Vicary, J.: A new description of orthogonal bases. Math. Structures in Comp. Sci., 13 (2008), http://arxiv.org/abs/0810.0812 (to appear)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Duncan, R., Perdrix, S. (2009). Graph States and the Necessity of Euler Decomposition. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds) Mathematical Theory and Computational Practice. CiE 2009. Lecture Notes in Computer Science, vol 5635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03073-4_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-03073-4_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03072-7
Online ISBN: 978-3-642-03073-4
eBook Packages: Computer ScienceComputer Science (R0)