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Foundations of Network Diagrams: Dynamical Systems, Bayesian Networks and Quantum Processes Filippo Bonchi University of Pisa Quantum Teleportation Quantum Teleportation 1932: von Neumanns original formulation of quantum theory based on

  1. Foundations of Network Diagrams: Dynamical Systems, Bayesian Networks and Quantum Processes Filippo Bonchi University of Pisa

  2. Quantum Teleportation

  3. Quantum Teleportation 1932: von Neumann’s original formulation of quantum theory based on Hilbert spaces

  4. Quantum Teleportation 1932: von Neumann’s original formulation of quantum theory based on Hilbert spaces New York Times headline of May 4, 1935.

  5. Quantum Teleportation 1932: von Neumann’s original formulation of quantum theory based on Hilbert spaces 1935: EPR weirdness of non-locality: "spooky action at distance" New York Times headline of May 4, 1935.

  6. Quantum Teleportation 1932: von Neumann’s original formulation of quantum theory based on Hilbert spaces 1935: EPR weirdness of non-locality: "spooky action at distance" 1993: Bennet et al. conceived the feature of quantum teleportation. New York Times headline of May 4, 1935.

  7. Quantum Teleportation 1932: von Neumann’s original formulation of quantum theory based on Hilbert spaces 1935: EPR weirdness of non-locality: "spooky action at distance" 1993: Bennet et al. conceived the feature of quantum teleportation. New York Times headline of May 4, 1935. Why did it take so long?

  8. Quantum Pictorialism

  9. Quantum Pictorialism Reasoning about quantum systems via Hilbert spaces is rather incovenient, pretty much like programming a distributed application in Assembly

  10. Quantum Pictorialism Reasoning about quantum systems via Hilbert spaces is rather incovenient, pretty much like programming a distributed application in Assembly ¨ ˛ 1 ` i 1 ` i 1 ` i 1 ` i 1 ` i 1 ´ i 1 ´ i 1 ` i ´ ´ 1 ` i 1 ´ i 1 ´ i 1 ` i 1 ` i 1 ` i 1 ` i 1 ` i ´ ´ ˚ ‹ π 1 ` i 1 ´ i 1 ´ i 1 ` i 1 ´ i 1 ´ i 1 ´ i 1 ´ i ´ ´ 2 ˚ ‹ 1 ´ i 1 ´ i 1 ´ i 1 ´ i 1 ` i 1 ´ i 1 ´ i 1 ` i ´ ´ 1 ˚ ‹ vs. π ˚ ‹ 2 4 1 ` i 1 ´ i 1 ´ i 1 ` i 1 ´ i 1 ´ i 1 ´ i 1 ´ i ´ ´ ˚ ‹ π 1 ´ i 1 ´ i 1 ´ i 1 ´ i 1 ` i 1 ´ i 1 ´ i 1 ` i ´ ´ ˚ ‹ 2 ˝ ‚ 1 ` i 1 ` i 1 ` i 1 ` i 1 ` i 1 ´ i 1 ´ i 1 ` i ´ ´ 1 ` i 1 ´ i 1 ´ i 1 ` i 1 ` i 1 ` i 1 ` i 1 ` i ´ ´

  11. Quantum Pictorialism Reasoning about quantum systems via Hilbert spaces is rather incovenient, pretty much like programming a distributed application in Assembly ¨ ˛ 1 ` i 1 ` i 1 ` i 1 ` i 1 ` i 1 ´ i 1 ´ i 1 ` i ´ ´ 1 ` i 1 ´ i 1 ´ i 1 ` i 1 ` i 1 ` i 1 ` i 1 ` i ´ ´ ˚ ‹ π 1 ` i 1 ´ i 1 ´ i 1 ` i 1 ´ i 1 ´ i 1 ´ i 1 ´ i ´ ´ 2 ˚ ‹ 1 ´ i 1 ´ i 1 ´ i 1 ´ i 1 ` i 1 ´ i 1 ´ i 1 ` i ´ ´ 1 ˚ ‹ vs. π ˚ ‹ 2 4 1 ` i 1 ´ i 1 ´ i 1 ` i 1 ´ i 1 ´ i 1 ´ i 1 ´ i ´ ´ ˚ ‹ π 1 ´ i 1 ´ i 1 ´ i 1 ´ i 1 ` i 1 ´ i 1 ´ i 1 ` i ´ ´ ˚ ‹ 2 ˝ ‚ 1 ` i 1 ` i 1 ` i 1 ` i 1 ` i 1 ´ i 1 ´ i 1 ` i ´ ´ 1 ` i 1 ´ i 1 ´ i 1 ` i 1 ` i 1 ` i 1 ` i 1 ` i ´ ´ Developing an high level language for quantum system would boost the discovery of quantum features and the development of quantum technologies

  12. Quantum Pictorialism Reasoning about quantum systems via Hilbert spaces is rather incovenient, pretty much like programming a distributed application in Assembly ¨ ˛ 1 ` i 1 ` i 1 ` i 1 ` i 1 ` i 1 ´ i 1 ´ i 1 ` i ´ ´ 1 ` i 1 ´ i 1 ´ i 1 ` i 1 ` i 1 ` i 1 ` i 1 ` i ´ ´ ˚ ‹ π 1 ` i 1 ´ i 1 ´ i 1 ` i 1 ´ i 1 ´ i 1 ´ i 1 ´ i ´ ´ 2 ˚ ‹ 1 ´ i 1 ´ i 1 ´ i 1 ´ i 1 ` i 1 ´ i 1 ´ i 1 ` i ´ ´ 1 ˚ ‹ vs. π ˚ ‹ 2 4 1 ` i 1 ´ i 1 ´ i 1 ` i 1 ´ i 1 ´ i 1 ´ i 1 ´ i ´ ´ ˚ ‹ π 1 ´ i 1 ´ i 1 ´ i 1 ´ i 1 ` i 1 ´ i 1 ´ i 1 ` i ´ ´ ˚ ‹ 2 ˝ ‚ 1 ` i 1 ` i 1 ` i 1 ` i 1 ` i 1 ´ i 1 ´ i 1 ` i ´ ´ 1 ` i 1 ´ i 1 ´ i 1 ` i 1 ` i 1 ` i 1 ` i 1 ` i ´ ´ Developing an high level language for quantum system would boost the discovery of quantum features and the development of quantum technologies

  13. Network diagrams Electrical Circuits Bayesian Networks Quantum Processes Petri Nets Signal Flow Graphs x x x

  14. Network diagrams Signal Flow Graphs x x x

  15. Network diagrams Diagrammatic languages are not really made of syntax. Signal Flow Graphs x x x

  16. Network diagrams Diagrammatic languages are not really made of syntax. Signal Flow Graphs x x x We are able to describe the behaviour of the whole systems

  17. Network diagrams Diagrammatic languages are not really made of syntax. Signal Flow Graphs x x x We are able to describe the behaviour of the whole systems But not the behaviour of the single components

  18. Network diagrams Diagrammatic languages are not really made of syntax. Signal Flow Graphs x x x ? ? We are able to describe the behaviour of the whole systems But not the behaviour of the single components

  19. Network diagrams Diagrammatic languages are not really made of syntax. Signal Flow Graphs x x x ? ? We are able to describe the behaviour of the whole systems But not the behaviour of the single components The behaviour of the whole system should be "reducible" to the behaviour of its components

  20. Network diagrams Diagrammatic languages are not really made of syntax. Signal Flow Graphs x x x ? ? We are able to describe the behaviour of the whole systems But not the behaviour of the single components The behaviour of the whole system should be "reducible" to the behaviour of its components https://www.azimuthproject.org/azimuth/show/Network+theory

  21. Compositional Modelling There is an emerging, multi-disciplinary field aiming at studying different sorts of networks compositionally , inspired by the algebraic methods of programming language semantics. Diagrams are first-class citizens of the theory. The appropriate algebraic setting is monoidal (and not cartesian ) categories.

  22. https://www.forbes.com/sites/cognitiveworld/2019/07/29/the-future- Compositional Modelling will-be-formulated-using-category-theory/#71a09469625e There is an emerging, multi-disciplinary field aiming at studying different sorts of networks compositionally , inspired by the algebraic methods of programming language semantics. Diagrams are first-class citizens of the theory. The appropriate algebraic setting is monoidal (and not cartesian ) categories.

  23. Signal Flow Graphs Signal Flow Graphs are stream processing circuits widely adopted in Control Theory and Signal Processing k x Claude Shannon. The theory and design of linear differential equation machines (1942).

  24. Signal Flow Graphs Signal Flow Graphs are stream processing circuits widely adopted in Control Theory and Signal Processing c, d ::= k x k x c d c d Claude Shannon. The theory and design of linear differential equation machines (1942).

  25. Sound and Complete Axiomatisation for Signal Flow Graphs = = = = Bialgebra = = = = id0 ≤ ≤ ≤ ≤ ≤ ≤ id0 ≤ ≤

  26. Sound and Complete Axiomatisation for Signal Flow Graphs = = = = Bialgebra = = = = id0 ≤ ≤ ≤ ≤ ≤ ≤ id0 ≤ ≤ https://graphicallinearalgebra.net

  27. Sound and Complete Axiomatisation for Signal Flow Graphs These axioms = = are almost = = the same as Bialgebra those for = Quantum mechanics = = = id0 ≤ ≤ ≤ ≤ ≤ ≤ id0 ≤ ≤ https://graphicallinearalgebra.net

  28. Sound and Complete Axiomatisation for Signal Flow Graphs These axioms = = What is are almost = = the same as going on Bialgebra those for = ? Quantum mechanics = = = id0 ≤ ≤ ≤ ≤ ≤ ≤ id0 ≤ ≤ https://graphicallinearalgebra.net

  29. References Bonchi, Sobocinski, Zanasi - Full Abstraction for Signal Flow Graphs , • POPL, 2015. [see also Fabio Zanasi ph.D thesis - Interacting Hopf Algebras (ENS-Lyon, 2015)] Bonchi, Gadducci, Kissinger, Sobocinski, Zanasi - Rewriting modulo • symmetric monoidal structure - LICS 2016. Bonchi, Soboci ń ski, Zanasi - Interacting Hopf algebras. Journal of Pure • and Applied Algebra (2017). Bonchi, Holland, Piedeleu, Sobocinski, Zanasi - Diagrammatic Algebra: • From Linear to Concurrent Systems , POPL, 2019. [see also Robin Piedeleu Ph.D thesis - Picturing resources in concurrency (Oxford, 2019) ] Bonchi, Piedeleu, Sobocinski, Zanasi - Graphical Affine Algebra , LICS • 2019.

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