Interacting Quantum Observables

  • Bob Coecke
  • Ross Duncan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5126)


We formalise the constructive content of an essential feature of quantum mechanics: the interaction of complementary quantum observables, and information flow mediated by them. Using a general categorical formulation, we show that pairs of mutually unbiased quantum observables form bialgebra-like structures. We also provide an abstract account on the quantum data encoded in complex phases, and prove a normal form theorem for it. Together these enable us to describe all observables of finite dimensional Hilbert space quantum mechanics. The resulting equations suffice to perform computations with elementary quantum gates, translate between distinct quantum computational models, establish the equivalence of entangled quantum states, and simulate quantum algorithms such as the quantum Fourier transform. All these computations moreover happen within an intuitive diagrammatic calculus.


Classical Structure Classical Point Bloch Sphere Monoidal Structure Quantum Observable 
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  1. 1.
    Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Abramsky, S., Coecke, B. (eds.) Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS). IEEE Computer Science Press, Los Alamitos (2004); Abstract physical traces. Theory and Applications of Categories 14, 111–124 (2005)Google Scholar
  2. 2.
    Carboni, A., Walters, R.F.C.: Cartesian bicategories I. Journal of Pure and Applied Algebra 49, 11–32 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Coecke, B.: De-linearizing linearity: projective quantum axiomatics from strong compact closure. ENTCS 170, 49–72 (2007)Google Scholar
  4. 4.
    Coecke, B., Paquette, E.O.: POVMs and Naimark’s theorem without sums (2006) (to appear in ENTCS), arXiv:quant-ph/0608072Google Scholar
  5. 5.
    Coecke, B., Pavlovic, D.: Quantum measurements without sums. In: Chen, G., Kauffman, L., Lamonaco, S. (eds.) Mathematics of Quantum Computing and Technology, pp. 567–604. Taylor and Francis, Abington (2007)Google Scholar
  6. 6.
    Coecke, B., Pavlovic, D., Vicary, J.: Dagger Frobenius algebras in FdHilb are bases. Oxford University Computing Laboratory Research Report RR-08-03 (2008)Google Scholar
  7. 7.
    Danos, V., Kashefi, E., Panangaden, P.: The measurement calculus. Journal of the ACM 54(2) (2007), arXiv:quant-ph/0412135Google Scholar
  8. 8.
    Joyal, A., Street, R.: The Geometry of tensor calculus I. Advances in Mathematics 88, 55–112 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kelly, G.M., Laplaza, M.L.: Coherence for compact closed categories. Journal of Pure and Applied Algebra 19, 193–213 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Klappenecker, A., Rötteler, M.: Constructions of mutually unbiased bases. LNCS, vol. 2948, pp. 137–144. Springer, Heidelberg (2004)Google Scholar
  11. 11.
    Kock, J.: Frobenius Algebras and 2D Topological Quantum Field Theories. In: Composing PROPs. Theory and Applications of Categories, vol. 13, pp. 147–163. Cambridge University Press, Cambridge (2003)Google Scholar
  12. 12.
    Selinger, P.: Dagger compact closed categories and completely positive maps. ENTCS, 170, 139–163 (2005),
  13. 13.
    Street, R.: Quantum Groups: A Path to Current Algebra, Cambridge UP (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Bob Coecke
    • 1
  • Ross Duncan
    • 1
  1. 1.Oxford University Computing Laboratory 

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