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8 - The logic of quantum mechanics – Take II

Published online by Cambridge University Press:  05 June 2016

By
Bob Coecke
Edited by
Jennifer Chubb ,
Ali Eskandarian  and
Valentina Harizanov
Bob Coecke
Affiliation:
University of Oxford
Jennifer Chubb
Affiliation:
University of San Francisco
Ali Eskandarian
Affiliation:
George Washington University, Washington DC
Valentina Harizanov
Affiliation:
George Washington University, Washington DC
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Summary

Abstract. We put forward a new take on the logic of quantum mechanics, following Schrödinger's point of view that it is composition which makes quantum theory what it is, rather than its particular propositional structure due to the existence of superpositions, as proposed by Birkhoff and von Neumann. This gives rise to an intrinsically quantitative kind of logic, which truly deserves the name ‘logic’ in that it also models meaning in natural language, the latter being the origin of logic, that it supports automation, the most prominent practical use of logic, and that it supports probabilistic inference.

The physics and the logic of quantum-ish logic. In 1932 John von Neumann formalized Quantum Mechanics in his book “Mathematische Grundlagen der Quantenmechanik”. This was effectively the official birth of the quantum mechanical formalism which until now, some 75 years later, has remained the same. Quantum theory underpins so many things in our daily lives including chemical industry, energy production and information technology, which arguably makes it the most technologically successful theory of physics ever.

However, in 1935, merely three years after the birth of his brainchild, von Neumann wrote in a letter to American mathematician Garrett Birkhoff: “I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space no more.” (sic)—for more details see [73].

Soon thereafter they published a paper entitled “The Logic of Quantum Mechanics” [13]. Their ‘quantum logic’ was cast in order-theoretic terms, very much in the spirit of the then reigning algebraic view of logic, with the distributive law being replaced with a weaker (ortho)modular law.

This resulted in a research community of quantum logicians [68, 71, 47, 30]. However, despite von Neumann's reputation, and the large body of research that has been produced in the area, one does not find a trace of this activity neither in the mainstream physics, mathematics, nor logic literature. Hence, 75 years later one may want to conclude that this activity was a failure.

What went wrong?

The mathematics of it. Let us consider the raison d'être for the Hilbert space formalism.

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Logic and Algebraic Structures in Quantum Computing , pp. 174 - 198
Publisher: Cambridge University Press
Print publication year: 2016

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References

[1] S., Abramsky and B., Coecke, A categorical semantics of quantum protocols, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LICS), IEEE Computer Society, 2004, Extended version: arXiv:quant-ph/0402130, pp. 415–425.
[2] S., Abramsky and B., Coecke, Abstract physical traces, Theory and Applications of Categories, vol. 14(6) (2005), pp. 111–124.Google Scholar
[3] S., Abramsky and N., Tzevelekos, Introduction to categories and categorical logic, New Structures for Physics (B., Coecke, editor), Lecture Notes in Physics, Springer, 2011, pp. 3–94.
[4] K., Ajdukiewicz, Die syntaktische Konnexität, Studia Philosophica, vol. 1 (1937), pp. 1–27.Google Scholar
[5] J., Aron, Quantum links let computers read, New Scientist, vol. 208(2790) (2010), pp. 10–11.Google Scholar
[6] M., Atiyah, Topological quantum field theories, Publications Mathématiques de l'IHÉS, vol. 68(1) (1988), pp. 175–186.Google Scholar
[7] J.C., Baez, Quantum quandaries: a category-theoretic perspective, The Structural Foundations of Quantum Gravity (D., Rickles, S., French, and J. T., Saatsi, editors), Oxford University Press, 2006, arXiv:quant-ph/0404040, pp. 240–266.
[8] J. C., Baez and J., Dolan, Higher-dimensional algebra and topological quantum field theory, Journal of Mathematical Physics, vol. 36 (1995), p. 6073, arXiv:q-alg/9503002.Google Scholar
[9] J. C., Baez and M., Stay, Physics, topology, logic and computation: a Rosetta stone, New Structures for Physics (B., Coecke, editor), Lecture Notes in Physics, Springer, 2011, pp. 95–172.
[10] Y., Bar-Hillel, A quasiarithmetical notation for syntactic description, Language, vol. 29 (1953), pp. 47–58.Google Scholar
[11] J., Benabou, Categories avec multiplication, Comptes Rendus des Séances de l'Académie des Sciences. Paris, vol. 256 (1963), pp. 1887–1890.Google Scholar
[12] C. H., Bennett, G., Brassard, C., Crepeau, R., Jozsa, A., Peres, and W. K., Wootters, Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels, Physical Review Letters, vol. 70(13) (1993), pp. 1895–1899.Google Scholar
[13] G., Birkhoff and J. von, Neumann, The logic of quantum mechanics, Annals of Mathematics, vol. 37 (1936), pp. 823–843.Google Scholar
[14] S., Boixo and C., Heunen, Entangled and sequential quantum protocols with dephasing, Physical Review Letters, vol. 108 (2012), p. 120402.Google Scholar
[15] A., Bundy, F., Cavallo, L., Dixon, M., Johansson, and R., McCasland, The Theory behind TheoryMine.
[16] A., Carboni and R. F. C., Walters, Cartesian bicategories I, Journal of Pure and Applied Algebra, vol. 49 (1987), pp. 11–32.Google Scholar
[17] G., Chiribella, G. M., D'Ariano, and P., Perinotti, Informational derivation of quantum theory, Physical Review A, vol. 84 (2011), no. 1, p. 012311.Google Scholar
[18] N., Chomsky, Tree models for the description of language, I.R.E.Transactions on Information Theory, vol. IT-2 (1956), pp. 113–124.Google Scholar
[19] S., Clark, B., Coecke, and M., Sadrzadeh, A compositional distributional model of meaning, Proceedings of the Second Quantum Interaction Symposium (QI-2008), 2008, pp. 133–140.
[20] B., Coecke, Kindergarten quantum mechanics — lecture notes, Quantum Theory: Reconsiderations of the Foundations III (A., Khrennikov, editor), AIP Press, 2005, arXiv:quant-ph/0510032, pp. 81–98.
[21] B., Coecke, Automated quantum reasoning: Non logic – semi-logic – hyper-logic, AAAI Spring Symposium: Quantum Interaction, AAAI, 2007, pp. 31–38.
[22] B., Coecke, Axiomatic description of mixed states from Selinger's CPM-construction, Electronic Notes in Theoretical Computer Science, vol. 210 (2008), pp. 3–13.Google Scholar
[23] B., Coecke, Quantum picturalism, Contemporary Physics, vol. 51 (2009), pp. 59–83, arXiv:0908.1787.Google Scholar
[24] B., Coecke and R., Duncan, Interacting quantum observables, Proceedings of the 37th International Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Computer Science, 2008.
[25] B., Coecke and R., Duncan, Interacting quantum observables: categorical algebra and diagrammatics, New Journal of Physics, vol. 13 (2011), p. 043016, arXiv:quant-ph/09064725.Google Scholar
[26] B., Coecke, R., Duncan, A., Kissinger, and Q., Wang, Strong complementarity and non-locality in categorical quantum mechanics, Proceedings of the 27th Annual IEEE Symposium on Logic in Computer Science (LICS), IEEE Computer Society, 2012, arXiv:1203.4988.
[27] B., Coecke, B., Edwards, and R. W., Spekkens, Phase groups and the origin of non-locality for qubits, Electronic Notes in Theoretical Computer Science, vol. 270(2) (2011), arXiv:1003.5005.Google Scholar
[28] B., Coecke and A., Kissinger, The compositional structure of multipartite quantum entanglement, Automata, Languages and Programming, Lecture Notes in Computer Science, Springer, 2010, Extended version: arXiv:1002.2540, pp. 297–308.
[29] B., Coecke and D. J., Moore, Operational Galois adjunctions, Current Research in Operational Quantum Logic: Algebras, Categories and Languages (D. J., Moore B., Coecke and A., Wilce, editors), Fundamental Theories of Physics, vol. 111, Springer-Verlag, 2000, pp. 195–218.
[30] B., Coecke, D. J., Moore, and A., Wilce, Operational quantum logic: An overview, Current Research in Operational Quantum Logic: Algebras, Categories and Languages (B., Coecke, D. J., Moore, and A., Wilce, editors), Fundamental Theories of Physics, vol. 111, Springer-Verlag, 2000, arXiv:quant-ph/0008019, pp. 1–36.
[31] B., Coecke and E. O., Paquette, POVMs and Naimark's theorem without sums, Electronic Notes in Theoretical Computer Science, vol. 210 (2008), pp. 15–31, arXiv:quant-ph/0608072.Google Scholar
[32] B., Coecke and E. O., Paquette, Categories for the practicing physicist, New Structures for Physics (B., Coecke, editor), Lecture Notes in Physics, Springer, 2011, arXiv:0905.3010, pp. 167–271.
[33] B., Coecke, E. O., Paquette, and D., Pavlovic, Classical and quantum structuralism, Semantic Techniques in Quantum Computation (S., Gay and I., Mackie, editors), Cambridge University Press, 2010, arXiv:0904.1997, pp. 29–69.
[34] B., Coecke and D., Pavlovic, Quantum measurements without sums, Mathematics of Quantum Computing and Technology (L., Kauffman G., Chen and S., Lamonaco, editors), Taylor and Francis, 2007, arXiv:quant-ph/0608035, pp. 567–604.
[35] B., Coecke, D., Pavlovic, and J., Vicary, A new description of orthogonal bases, Mathematical Structures in Computer Science, 2011, to appear; arXiv:quant-ph/0810.1037.
[36] B., Coecke and S., Perdrix, Environment and classical channels in categorical quantum mechanics, Proceedings of the 19th EACSL Annual Conference on Computer Science Logic (CSL), Lecture Notes in Computer Science, vol. 6247, 2010, Extended version: arXiv:1004.1598, pp. 230– 244.Google Scholar
[37] B., Coecke, M., Sadrzadeh, and S., Clark, Mathematical foundations for a compositional distributional model of meaning, Linguistic Analysis, vol. 36 (2010), pp. 345–384.Google Scholar
[38] B., Coecke and R. W., Spekkens, Picturing classical and quantum Bayesian inference, Synthese, (2011), pp. 1–46, arXiv:1102.2368.
[39] P. A. M., Dirac, The principles of quantum mechanics (third edition), Oxford University Press, 1947.
[40] L., Dixon and R., Duncan, Graphical reasoning in compact closed categories for quantum computation, Annals of Mathematics and Artificial Intelligence, vol. 56(1) (2009), pp. 23–42.Google Scholar
[41] L., Dixon, R., Duncan, B., Frot, A., Merry, A., Kissinger, and M., Soloviev, quantomatic, 2011, http://dream.inf.ed.ac.uk/projects/quantomatic/.
[42] L., Dixon and A., Kissinger, Open graphs and monoidal theories, Mathematical Structures in Computer Science, 2011, to appear; arXiv:1011.4114.
[43] R., Duncan, Types for Quantum Computation, Ph.D. thesis, Oxford University, 2006.
[44] R., Duncan and S., Perdrix, Rewriting measurement-based quantum computations with generalised flow, Proceedings of ICALP, Lecture Notes in Computer Science, Springer, 2010, pp. 285–296.
[45] S., Eilenberg and S. Mac, Lane, General theory of natural equivalences, Transactions of the American Mathematical Society, vol. 58(2) (1945), p. 231.Google Scholar
[46] Cl-.A., Faure, D. J., Moore, and C., Piron, Deterministic evolutions and Schrödinger flows, Helvetica Physica Acta, vol. 68(2) (1995), pp. 150–157.Google Scholar
[47] D. J., Foulis and C. H., Randall, Operational statistics. I. Basic concepts, Journal of Mathematical Physics, vol. 13(11) (1972), pp. 1667–1675.Google Scholar
[48] G., Frege, Über Sinn und Bedeutung, Zeitschrift für Philosophie und Philosophische Kritik, vol. 1007 (1892), pp. 25–50.Google Scholar
[49] G. M., D'Ariano G., Chiribella and P., Perinotti, Probabilistic theories with purification, Physical Review A, vol. 81 (2010), no. 6, p. 062348.Google Scholar
[50] J.-Y., Girard, Linear logic, Theoretical Computer Science, vol. 50(1) (1987), pp. 1–101.Google Scholar
[51] E., Grefenstette and M., Sadrzadeh, Experimental support for a categorical compositional distributional model of meaning, EMNLP, ACL, 2011, pp. 1394–1404.
[52] H., Halvorson, Deep Beauty: Understanding the Quantum World Through Mathematical Innovation, Cambridge University Press, 2011.
[53] J., Harding, A link between quantum logic and categorical quantum mechanics, International Journal of Theoretical Physics, vol. 48(3) (2009), pp. 769–802.Google Scholar
[54] L., Hardy, A formalism-local framework for general probabilistic theories including quantum theory, arXiv:1005.5164, (2010).
[55] M., Hasegawa, M., Hofmann, and G. D., Plotkin, Finite dimensional vector spaces are complete for traced symmetric monoidal categories, Pillars of Computer Science (A., Avron, N., Dershowitz, and A., Rabinovich, editors), Lecture Notes in Computer Science, vol. 4800, Springer, 2008, pp. 367–385.
[56] C., Heunen and B., Jacobs, Quantum logic in dagger kernel categories, Order, vol. 27(2) (2010), pp. 177–212.Google Scholar
[57] C., Horsman, Quantum picturalism for topological cluster-state computing, New Journal of Physics, vol. 13 (2011), p. 095011, arXiv:1101.4722.Google Scholar
[58] M., Johansson, L., Dixon, and A., Bundy, Conjecture synthesis for inductive theories, Journal of Automated Reasoning, vol. 47(3) (2011), pp. 251–289.Google Scholar
[59] A., Joyal and R., Street, The geometry of tensor calculus I, Advances in Mathematics, vol. 88 (1991), pp. 55–112.Google Scholar
[60] G. M., Kelly and M. L., Laplaza, Coherence for compact closed categories, Journal of Pure and Applied Algebra, vol. 19 (1980), pp. 193–213.Google Scholar
[61] A., Kissinger, Synthesising graphical theories, arXiv:1202.6079, (2012).
[62] J., Kock, Frobenius Algebras and 2D Topological Quantum Field Theories, vol. 59, Cambridge University Press, 2004.
[63] S., Lack, Composing PROPs, Theory and Applications of Categories, vol. 13 (2004), pp. 147–163.Google Scholar
[64] J., Lambek, The mathematics of sentence structure, American Mathematics Monthly, vol. 65 (1958), pp. 154–170.Google Scholar
[65] J., Lambek, Type grammar revisited, Logical Aspects of Computational Linguistics, Lecture Notes in Computer Science, vol. 1582, 1999, pp. 1–27.Google Scholar
[66] J., Lambek and P. J., Scott, Introduction to Higher Order Categorical Logic, Cambridge University Press, 1988.
[67] S. Mac, Lane, Natural associativity and commutativity, The Rice University Studies, vol. 49(4) (1963), pp. 28–46.Google Scholar
[68] G. M., Mackey, The Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, New York, 1963.
[69] D. J., Moore, On state spaces and property lattices, Studies in History and Philosophy of Modern Physics, vol. 30(1) (March 1999), pp. 61–83.Google Scholar
[70] R., Penrose, Applications of negative dimensional tensors, Combinatorial Mathematics and its Applications, Academic Press, 1971, pp. 221–244.
[71] C., Piron, Foundations of Quantum Physics, W. A. Benjamin, 1976.
[72] A., Preller and M., Sadrzadeh, Bell states and negative sentences in the distributed model of meaning, Electronic Notes in Theoretical Computer Science, vol. 270(2) (2011), pp. 141–153.Google Scholar
[73] M., Redei, Why John von Neumann did not like the Hilbert space formalism of quantum mechanics (and what he liked instead), Studies in History and Philosophy of Modern Physics, vol. 27(4) (1996), pp. 493–510.Google Scholar
[74] E., Schrödinger, Discussion of probability relations between separated systems, Cambridge Philosophical Society, vol. 31 (1935), pp. 555–563.Google Scholar
[75] H., Schütze, Automatic word sense discrimination, Computational Linguistics, vol. 24(1) (1998), pp. 97–123.Google Scholar
[76] R. A. G., Seely, Linear logic, *-autonomous categories and cofree algebras, Contemporary Mathematics, vol. 92 (1989), pp. 371–382.Google Scholar
[77] P., Selinger, Dagger compact closed categories and completely positive maps, Electronic Notes in Theoretical Computer Science, vol. 170 (2007), pp. 139–163.Google Scholar
[78] P., Selinger, Finite dimensional Hilbert spaces are complete for dagger compact closed categories (extended abstract), Electronic Notes in Theoretical Computer Science, vol. 270(1) (2011), pp. 113– 119.Google Scholar
[79] P. W., Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Journal on Computing, vol. 26(5) (1997), pp. 1484–1509.Google Scholar
[80] L., Wittgenstein, Philosophical Investigations, Basil & Blackwell, 1972.
[81] M., Zukowski, A., Zeilinger, M. A., Horne, and A. K., Ekert, Event-ready-detectors Bell experiment via entanglement swapping, Physical Review Letters, vol. 71 (1993), pp. 4287–4290.Google Scholar

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