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Environment and Classical Channels in Categorical Quantum Mechanics

  • Bob Coecke
  • Simon Perdrix
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6247)

Abstract

We present a both simple and comprehensive graphical calculus for quantum computing. We axiomatize the notion of an environment, which together with the axiomatic notion of classical structure enables us to define classical channels, quantum measurements and classical control. If we moreover adjoin the axiomatic notion of complementarity, we obtain sufficient structural power for constructive representation and correctness derivation of typical quantum informatic protocols.

Keywords

Quantum Channel Classical Structure Physical Review Letter Classical Data Dense Code 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. IEEE-LiCS 2004 (2004), Revision: arXiv:quant-ph/0808.1023Google Scholar
  2. 2.
    Abramsky, S., Coecke, B.: Abstract physical traces. Theory and Applications of Categories 14, 111–124 (2005), arXiv:0910.3144 Google Scholar
  3. 3.
    Barnum, H., Caves, C.M., Fuchs, C.A., Jozsa, R., Schumacher, B.: Noncommuting mixed states cannot be broadcast. Physical Review Letters 76, 2818–2821 (1996), arXiv:quant-ph/9511010Google Scholar
  4. 4.
    Bennett, C.H., Brassard, G.: Quantum cryptography: Public key distribution and coin tossing. IEEE-CSSP (1984)Google Scholar
  5. 5.
    Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wooters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Physical Review Letters 70, 1895–1899 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bennet, C.H., Wiesner, S.: Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Physical Review Letters 69, 2881–2884 (1992)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Carboni, A., Walters, R.F.C.: Cartesian bicategories I. Journal of Pure and Applied Algebra 49, 11–32 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chiribella, G., D’Ariano, G.M., Perinotti, P.: Probabilistic theories with purification (2009), arXiv:0908.1583Google Scholar
  9. 9.
    Coecke, B.: Axiomatic description of mixed states from Selinger’s CPM-construction. Electronic Notes in Theoretical Computer Science 210, 3–13 (2008)CrossRefMathSciNetGoogle Scholar
  10. 10.
    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); Extended version: arXiv:0906.4725Google Scholar
  11. 11.
    Coecke, B., Edwards, B., Spekkens, R.W.: Phase groups and the origin of non-locality for qubits. In: ENTCS-QPL 2009 (to appear, 2010), arXiv:1003.5005Google Scholar
  12. 12.
    Coecke, B., Kissinger, A.: The compositional structure of multipartite quantum entanglement. In: ICALP 2010 (2010), arXiv:1002.2540Google Scholar
  13. 13.
    Coecke, B., Paquette, E.O.: Categories for the practicing physicist. In: Coecke, B. (ed.) New Structures for Physics. Lecture Notes in Physics, pp. 167–271. Springer, Heidelberg (2009), arXiv:0905.3010Google Scholar
  14. 14.
    Coecke, B., Paquette, E.O., Perdrix, S.: Bases in diagrammatic quantum protocols. Electronic Notes in Theoretical Computer Science 218, 131–152 (2008)CrossRefGoogle Scholar
  15. 15.
    Coecke, B., Paquette, E.O., Pavlovic, D.: Classical and quantum structuralism. In: Mackie, I., Gay, S. (eds.) Semantic Techniques for Quantum Computation, pp. 29–69. Cambridge University Press, Cambridge (2009), arXiv:0904.1997Google Scholar
  16. 16.
    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), arXiv:quant-ph/0608035Google Scholar
  17. 17.
    Coecke, B., Pavlovic, D., Vicary, J.: A new description of orthogonal bases (2008), arXiv:0810.0812Google Scholar
  18. 18.
    Coecke, B., Wang, B.-S., Wang, Q.-L., Wang, Y.-J., Zhang, Q.-Y.: Graphical calculus for quantum key distribution. In: ENTCS-QPL 2009 (to appear, 2010)Google Scholar
  19. 19.
    Dixon, L., Duncan, R., Kissinger, A.: quantomatic, http://dream.inf.ed.ac.uk/projects/quantomatic/
  20. 20.
    Duncan, R., Perdrix, S.: Graph states and the necessity of Euler decomposition. In: CiE 2009. LNCS, vol. 5635. Springer, Heidelberg (2009), arXiv:0902.0500Google Scholar
  21. 21.
    Duncan, R., Perdrix, S.: Rewriting measurement-based quantum computations with generalised flow. In: ICALP 2010 (2010)Google Scholar
  22. 22.
    Ekert, A.: Quantum cryptography based on Bell’s theorem. Physical Review Letters 67, 661–663 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Joyal, A., Street, R.: The Geometry of tensor calculus I. Advances in Mathematics 88, 55–112 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Kelly, G.M., Laplaza, M.L.: Coherence for compact closed categories. Journal of Pure and Applied Algebra 19, 193–213 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Lack, S.: Composing PROPs. Theory and Applications of Categories 13, 147–163 (2004)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, Heidelberg (2000)Google Scholar
  27. 27.
    Penrose, R.: Applications of negative dimensional tensors. In: Welsh, D. (ed.) Combinatorial Mathematics and its Applications, pp. 221–244. Academic Press, London (1971)Google Scholar
  28. 28.
    Selinger, P.: Dagger compact closed categories and completely positive maps. Electronic Notes in Theoretical Computer Science 170, 139–163 (2007)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Selinger, P.: A survey of graphical languages for monoidal categories. In: Coecke, B. (ed.) New Structures for Physics, pp. 275–337. Springer, Heidelberg (2009), arXiv:0908.3347Google Scholar
  30. 30.
    Zurek, W.H.: Decoherence and the Transition from Quantum to Classical. Physics Today 44, 36–44 (1991), arXiv:quant-ph/0306072 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Bob Coecke
    • 1
  • Simon Perdrix
    • 1
  1. 1.Laboratoire d’Informatique de GrenobleOxford University Computing Laboratory CNRS 

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