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A Diagrammatic Approach to Quantum Dynamics

Author Stefano Gogioso

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Stefano Gogioso
  • University of Oxford, UK

Acknowledgements

This publication was made possible through the support of a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.

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Stefano Gogioso. A Diagrammatic Approach to Quantum Dynamics. In 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 139, pp. 19:1-19:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.CALCO.2019.19

Abstract

We present a diagrammatic approach to quantum dynamics based on the categorical algebraic structure of strongly complementary observables. We provide physical semantics to our approach in terms of quantum clocks and quantisation of time. We show that quantum dynamical systems arise naturally as the algebras of a certain dagger Frobenius monad, with the morphisms and tensor product of the category of algebras playing the role, respectively, of equivariant transformations and synchronised parallel composition of dynamical systems. We show that the Weyl Canonical Commutation Relations between time and energy are an incarnation of the bialgebra law and we derive Schrödinger’s equation from a process-theoretic perspective. Finally, we use diagrammatic symmetry-observable duality to prove Stone’s proposition and von Neumann’s Mean Ergodic proposition, recasting the results as two faces of the very same coin.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Categorical semantics
Keywords
  • Quantum dynamics
  • String diagrams
  • Categorical algebra

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References

  1. Samson Abramsky and Bob Coecke. A categorical semantics of quantum protocols. In Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004., pages 415-425. IEEE, 2004. URL: https://doi.org/10.1109/LICS.2004.1319636.
  2. Samson Abramsky and Bob Coecke. Categorical Quantum Mechanics. In K. Engesser, Gabbay D. M., and Lehmann D., editors, Handbook of Quantum Logic and Quantum Structures, pages 261-323. Elsevier, 2009. URL: https://doi.org/10.1016/B978-0-444-52869-8.50010-4.
  3. Samson Abramsky and Chris Heunen. H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics. Clifford Lectures, AMS Proceedings of Symposia in Applied Mathematics, 71:1-24, 2012. URL: http://arxiv.org/abs/1011.6123.
  4. Jeremy Butterfield. On time in quantum physics. In Heather Dyke and Adrian Bardon, editors, A Companion to the Philosophy of Time. John Wiley & Sons Ltd, 2014. Google Scholar
  5. Bob Coecke and Ross Duncan. Interacting quantum observables: categorical algebra and diagrammatics. New Journal of Physics, 13(4):043016, 2011. URL: https://doi.org/10.1088/1367-2630/13/4/043016.
  6. Bob Coecke, Ross Duncan, Aleks Kissinger, and Quanlong Wang. Strong complementarity and non-locality in categorical quantum mechanics. In Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science, pages 245-254. IEEE Computer Society, 2012. URL: https://doi.org/10.1109/LICS.2012.35.
  7. Bob Coecke and Aleks Kissinger. Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, 2017. URL: https://doi.org/10.1017/9781316219317.
  8. Bob Coecke and Dusko Pavlovic. Quantum measurements without sums. In G. Chen, L. Kauffman, and S. Lamonaco, editors, Mathematics of Quantum Computing and Technology. Taylor and Francis, 2007. URL: https://arxiv.org/abs/quant-ph/0608035.
  9. Bob Coecke, Dusko Pavlovic, and Jamie Vicary. A new description of orthogonal bases. Mathematical Structures in Computer Science, 23(3):555-567, 2013. URL: https://doi.org/10.1017/S0960129512000047.
  10. Ross Duncan and Kevin Dunne. Interacting Frobenius Algebras are Hopf. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, pages 535-544. ACM, 2016. URL: https://doi.org/10.1145/2933575.2934550.
  11. Richard P Feynman. Simulating physics with computers. International journal of theoretical physics, 21(6):467-488, 1982. URL: https://doi.org/10.1007/BF02650179.
  12. Richard P Feynman. Quantum mechanical computers. Foundations of physics, 16(6):507-531, 1986. URL: https://doi.org/10.1007/BF01886518.
  13. Stefano Gogioso. Categorical Quantum Dynamics. PhD thesis, University of Oxford, 2017. URL: https://arxiv.org/abs/1709.09772.
  14. Stefano Gogioso and Fabrizio Genovese. Infinite-dimensional Categorical Quantum Mechanics. Electronic Proceedings in Theoretical Computer Science, 236:51-69, 2017. URL: https://doi.org/10.4204/EPTCS.236.4.
  15. Stefano Gogioso and Fabrizio Genovese. Towards Quantum Field Theory in Categorical Quantum Mechanics. Electronic Proceedings in Theoretical Computer Science, 266:349-366, 2018. URL: https://doi.org/10.4204/EPTCS.266.22.
  16. Stefano Gogioso and William Zeng. Generalised Mermin-type non-locality arguments. Logical Methods in Computer Science, 15(2), 2019. URL: https://lmcs.episciences.org/5402.
  17. Robert Goldblatt. Lectures on the hyperreals. An introduction to nonstandard analysis. Springer-Verlag, 1998. Google Scholar
  18. Chris Heunen and Martti Karvonen. Monads on dagger categories. Theory and Applications of Categories, 31(35):1016-1043, 2016. URL: https://arxiv.org/abs/1602.04324.
  19. Jan Hilgevoord. Time in quantum mechanics: a story of confusion. Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics, 36(1):29-60, 2005. URL: https://doi.org/10.1016/j.shpsb.2004.10.002.
  20. André Joyal and Ross Street. The geometry of tensor calculus, I. Advances in mathematics, 88(1):55-112, 1991. URL: https://doi.org/10.1016/0001-8708(91)90003-P.
  21. Aleks Kissinger. Pictures of Processes: Automated Graph Rewriting for Monoidal Categories and Applications to Quantum Computing. PhD thesis, University of Oxford, 2012. URL: https://arxiv.org/abs/1203.0202.
  22. Jarrod R. McClean, John A. Parkhill, and Alán Aspuru-Guzik. Feynmanquoterights clock, a new variational principle, and parallel-in-time quantum dynamics. Proceedings of the National Academy of Sciences, 110(41):E3901-E3909, 2013. URL: https://doi.org/10.1073/pnas.1308069110.
  23. Thomas Pashby. Time and quantum theory: A history and a prospectus. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 52:24-38, 2015. URL: https://doi.org/10.1016/j.shpsb.2015.03.002.
  24. Bryan W Roberts. Time, symmetry and structure: A study in the foundations of quantum theory. PhD thesis, University of Pittsburgh, 2012. URL: http://d-scholarship.pitt.edu/12533.
  25. Abraham Robinson. Non-standard analysis. Princeton University Press, 1974. Google Scholar
  26. Peter Selinger. A Survey of Graphical Languages for Monoidal Categories. In Bob Coecke, editor, New Structures for Physics, volume 813 of Lecture Notes in Physics, pages 289-355. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-12821-9_4.
  27. Jamie Vicary. Categorical formulation of finite-dimensional quantum algebras. Communications in Mathematical Physics, 304(3):765-796, 2011. URL: https://doi.org/10.1007/s00220-010-1138-0.
  28. Hermann Weyl. Quantenmechanik und gruppentheorie. Zeitschrift für Physik, 46(1-2):1-46, 1927. URL: https://doi.org/10.1007/BF02055756.
  29. Hermann Weyl. The theory of groups and quantum mechanics. Dover Publications Inc., New York, 1950. Google Scholar
  30. Stanisław L. Woronowicz. Compact matrix pseudogroups. Communications in Mathematical Physics, 111(4):613-665, 1987. URL: https://doi.org/10.1007/BF01219077.
  31. Stanisław L. Woronowicz. Compact quantum groups. Symétries quantiques (Les Houches, 1995), 845(884):98, 1998. Google Scholar
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