The three types of normal sequential effect algebras

Abraham Westerbaan1, Bas Westerbaan1,2, and John van de Wetering1

1Radboud Universiteit Nijmegen
2University College London

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A sequential effect algebra (SEA) is an effect algebra equipped with a $\textit{sequential product}$ operation modeled after the Lüders product $(a,b)\mapsto \sqrt{a}b\sqrt{a}$ on C$^*$-algebras. A SEA is called $\textit{normal}$ when it has all suprema of directed sets, and the sequential product interacts suitably with these suprema. The effects on a Hilbert space and the unit interval of a von Neumann or JBW algebra are examples of normal SEAs that are in addition $\textit{convex}$, i.e. possess a suitable action of the real unit interval on the algebra. Complete Boolean algebras form normal SEAs too, which are convex only when $0=1$.

We show that any normal SEA $E$ splits as a direct sum $E= E_b\oplus E_c \oplus E_{ac}$ of a complete Boolean algebra $E_b$, a convex normal SEA $E_c$, and a newly identified type of normal SEA $E_{ac}$ we dub $\textit{purely almost-convex}$.

Along the way we show, among other things, that a SEA which contains only idempotents must be a Boolean algebra; and we establish a spectral theorem using which we settle for the class of normal SEAs a problem of Gudder regarding the uniqueness of square roots. After establishing our main result, we propose a simple extra axiom for normal SEAs that excludes the seemingly pathological a-convex SEAs. We conclude the paper by a study of SEAs with an associative sequential product. We find that associativity forces normal SEAs satisfying our new axiom to be commutative, shedding light on the question of why the sequential product in quantum theory should be non-associative.

To better understand the properties and foundations of quantum theory it is useful to contrast it with hypothetical alternative physical theories and mathematical abstractions. By studying these alternatives it becomes clearer which parts of quantum theory are special to it, and which are present in any reasonable physical theory.

A framework that has been used extensively to study such alternatives is that of generalised probabilistic theories. These have built into their definition the classical concepts of probability theory, and hence they fundamentally rely on properties of real numbers. While this is certainly a useful framework, it precludes the study of physical theories that have a more exotic notion of probability.
To study systems that allow a broader notion of probability, a more general structure is needed.
This paper is about effect algebras, which generalise and abstract the set of measurement effects in a quantum system.
The study of effect algebras has become a flourishing field on its own and covers a variety of topics.
In this paper we look at effect algebras that come equipped with a 'sequential product'. This is an operation that corresponds to the process of sequential measurement where we first observe one effect and then another.

Examples of sequential effect algebras are Boolean algebras, which model classical deterministic systems, and certain convex subspaces of vector spaces, which model standard probabilistic models. A third type is given by what we dub 'almost-convex' sequential effect algebras, which resemble convex models but are somewhat weirder.
Our main result is showing that for a particularly nice class of sequential effect algebras that we call 'normal' these three types are the only possibilities.

This result is surprising because it gives a way to retrieve a dichotomy between deterministic models and probabilistic models from an abstract starting point. It might give some insight into why our physical reality needs a continuum of real numbers to describe our fundamental theories.

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[1] Samson Abramsky and Adam Brandenburger. The sheaf-theoretic structure of non-locality and contextuality. New Journal of Physics, 13 (11): 113036, 2011. 10.1088/​1367-2630.

[2] Samson Abramsky and Achim Jung. Domain Theory, page 1–168. Oxford University Press, Inc., USA, 1995. ISBN 019853762X. 10.5555/​218742.218744.

[3] Erik M Alfsen and Frederic W Shultz. Geometry of State Spaces of Operator Algebras. Springer Science & Business Media, 2012. 10.1007/​978-1-4612-0019-2.

[4] Howard Barnum and Joachim Hilgert. Strongly symmetric spectral convex bodies are jordan algebra state spaces. Preprint, 2019.

[5] Howard Barnum, Markus P Müller, and Cozmin Ududec. Higher-order interference and single-system postulates characterizing quantum theory. New Journal of Physics, 16 (12): 123029, 2014. 10.1088/​1367-2630/​16/​12/​123029.

[6] Jonathan Barrett. Information processing in generalized probabilistic theories. Physical Review A, 75 (3): 032304, 2007. 10.1103/​PhysRevA.75.032304.

[7] Ivan Chajda, Radomir Halaš, and Jan Kühr. Every effect algebra can be made into a total algebra. Algebra universalis, 61 (2): 139, 2009. 10.1007/​s00012-009-0010-6.

[8] Kenta Cho. Effectuses in Categorical Quantum Foundations. PhD thesis, Radboud Universiteit Nijmegen, 2019. URL: https:/​/​​abs/​1910.12198.

[9] Kenta Cho, Bart Jacobs, Bas E Westerbaan, and Abraham A Westerbaan. An introduction to effectus theory. 2015. URL: https:/​/​​abs/​1512.05813.

[10] Anatolij Dvurečenskij. Every state on interval effect algebra is integral. Journal of Mathematical Physics, 51 (8): 083508, 2010. 10.1063/​1.3467463.

[11] Anatolij Dvurecenskij and Sylvia Pulmannová. New trends in quantum structures, volume 516. Springer Science & Business Media, 2013. ISBN 978-94-017-2422-7. 10.1007/​978-94-017-2422-7.

[12] Pau Enrique Moliner, Chris Heunen, and Sean Tull. Space in monoidal categories. In Bob Coecke and Aleks Kissinger, editors, Proceedings 14th International Conference on Quantum Physics and Logic, Nijmegen, The Netherlands, 3-7 July 2017, volume 266 of Electronic Proceedings in Theoretical Computer Science, pages 399–410. Open Publishing Association, 2018. 10.4204/​EPTCS.266.25.

[13] David J Foulis and Mary K Bennett. Effect algebras and unsharp quantum logics. Foundations of physics, 24 (10): 1331–1352, 1994. 10.1007/​BF02283036.

[14] David J Foulis and Sylvia Pulmannová. Type-decomposition of an effect algebra. Foundations of Physics, 40 (9-10): 1543–1565, 2010. 10.1007/​s10701-009-9344-3.

[15] Leonard Gillman and Meyer Jerison. Rings of continuous functions. Springer, 2013. 10.1007/​978-1-4615-7819-2.

[16] Zahra Eslami Giski and Mohamad Ebrahimi. Entropy of countable partitions on effect algebra with Riesz decomposition property and weak sequential effect algebra. Çankaya Üniversitesi Bilim ve Mühendislik Dergisi, 12 (1), 2015. ISSN 1309-6788. URL: https:/​/​​en/​pub/​cankujse/​issue/​33130/​368655.

[17] Zahra Eslami Giski and Abolfazl Ebrahimzadeh. An introduction of logical entropy on sequential effect algebra. Indagationes Mathematicae, 28 (5): 928–937, 2017. 10.1016/​j.indag.2017.06.007.

[18] Stan Gudder. Open problems for sequential effect algebras. International Journal of Theoretical Physics, 44 (12): 2199–2206, 2005. 10.1007/​s10773-005-8015-1.

[19] Stan Gudder. Convex and sequential effect algebras. 2018. URL: https:/​/​​abs/​1802.01265.

[20] Stan Gudder and Richard Greechie. Sequential products on effect algebras. Reports on Mathematical Physics, 49 (1): 87–111, 2002. 10.1016/​S0034-4877(02)80007-6.

[21] Stan Gudder and Richard Greechie. Uniqueness and order in sequential effect algebras. International Journal of Theoretical Physics, 44 (7): 755–770, 2005. 10.1007/​s10773-005-7054-y.

[22] Stan Gudder and Frédéric Latrémolière. Characterization of the sequential product on quantum effects. Journal of Mathematical Physics, 49 (5): 052106, 2008. 10.1063/​1.2904475.

[23] Stan Gudder and Gabriel Nagy. Sequential quantum measurements. Journal of Mathematical Physics, 42 (11): 5212–5222, 2001. 10.1063/​1.1407837.

[24] Stanley Gudder. Examples, problems, and results in effect algebras. International Journal of Theoretical Physics, 35 (11): 2365–2376, 1996. 10.1007/​bf02302453.

[25] Stanley Gudder. Convex structures and effect algebras. International Journal of Theoretical Physics, 38 (12): 3179–3187, 1999. 10.1023/​A:1026678114856.

[26] Stanley Gudder and Sylvia Pulmannová. Representation theorem for convex effect algebras. Commentationes Mathematicae Universitatis Carolinae, 39 (4): 645–660, 1998. ISSN 0010-2628. URL: https:/​/​​handle/​10338.dmlcz/​119041.

[27] Stanley P Gudder and Richard Greechie. Effect algebra counterexamples. Mathematica Slovaca, 46 (4): 317–325, 1996. ISSN 0139-9918. URL: https:/​/​​handle/​10338.dmlcz/​129156.

[28] Eissa D Habil. Tensor product of distributive sequential effect algebras and product effect algebras. International Journal of Theoretical Physics, 47 (1): 280–290, 2008. 10.1007/​s10773-007-9472-5.

[29] Harald Hanche-Olsen and Erling Størmer. Jordan Operator Algebras, volume 21. Pitman Advanced Pub. Program, 1984. ISBN 978-0273086192.

[30] Teiko Heinosaari, Leevi Leppäjärvi, and Martin Plávala. No-free-information principle in general probabilistic theories. Quantum, 3: 157, 7 2019. ISSN 2521-327X. 10.22331/​q-2019-07-08-157. URL https:/​/​​10.22331/​q-2019-07-08-157.

[31] Samuel S Holland. The current interest in orthomodular lattices. In The Logico-Algebraic Approach to Quantum Mechanics, pages 437–496. 1975. 10.1007/​978-94-010-1795-4_25.

[32] Bart Jacobs. Probabilities, distribution monads, and convex categories. Theoretical Computer Science, 412 (28): 3323–3336, 2011. 10.1016/​j.tcs.2011.04.005.

[33] Bart Jacobs and Jorik Mandemaker. Coreflections in algebraic quantum logic. Foundations of physics, 42 (7): 932–958, 2012. 10.1007/​s10701-012-9654-8.

[34] Bart Jacobs and Abraham A Westerbaan. An effect-theoretic account of Lebesgue integration. Electronic Notes in Theoretical Computer Science, 319: 239–253, 2015. 10.1016/​j.entcs.2015.12.015.

[35] Gejza Jenča. Blocks of homogeneous effect algebras. Bulletin of the Australian Mathematical Society, 64 (1): 81–98, 2001. 10.1017/​s0004972700019705.

[36] Anna Jenčová and Martin Plávala. On the properties of spectral effect algebras. Quantum, 3: 148, June 2019. ISSN 2521-327X. 10.22331/​q-2019-06-03-148.

[37] Pascual Jordan. Über Verallgemeinerungsmöglichkeiten des Formalismus der Quantenmechanik. Weidmann, 1933.

[38] Shen Jun and Wu Junde. Sequential product on standard effect algebra. Journal of Physics A: Mathematical and Theoretical, 42 (34): 345203, 2009. 10.1088/​1751-8113/​42/​34/​345203.

[39] Shen Jun and Junde Wu. Remarks on the sequential effect algebras. Reports on Mathematical Physics, 63 (3): 441–446, 2009. 10.1016/​s0034-4877(09)90015-5.

[40] Richard V Kadison. Operator algebras with a faithful weakly-closed representation. Annals of mathematics, pages 175–181, 1956. 10.2307/​1969954.

[41] Richard V Kadison and Gert Kjærgaard Pedersen. Equivalence in operator algebras. Mathematica Scandinavica, 27 (2): 205–222, 1971. 10.7146/​math.scand.a-10999.

[42] Max Koecher. Positivitatsbereiche im $R^n$. American Journal of Mathematics, pages 575–596, 1957.

[43] Lynn H Loomis. The lattice theoretic background of the dimension theory of operator algebras, volume 18. American Mathematical Soc., 1955.

[44] Shûichirô Maeda et al. Dimension functions on certain general lattices. Journal of Science of the Hiroshima University, Series A (Mathematics, Physics, Chemistry), 19 (2): 211–237, 1955.

[45] Kuppusamy Ravindran. On a Structure Theory of Effect Algebras. PhD thesis, Kansas State University, 1996.

[46] Mathys Rennela. Towards a quantum domain theory: order-enrichment and fixpoints in W*-algebras. Electronic Notes in Theoretical Computer Science, 308: 289–307, 2014. 10.1016/​j.entcs.2014.10.016.

[47] Frank Roumen. Cohomology of effect algebras. In QPL 2016, volume 236 of EPTCS, pages 174–202, 2016. 10.4204/​eptcs.236.12.

[48] Leonardo Santos and Barbara Amaral. On possibilistic conditions to contextuality and nonlocality. arXiv preprint arXiv:2011.04111, 2020.

[49] Jun Shen and Junde Wu. Not each sequential effect algebra is sharply dominating. Physics Letters A, 373 (20): 1708–1712, 2009. 10.1016/​j.physleta.2009.02.073.

[50] Jun Shen and Junde Wu. The $n$th root of sequential effect algebras. Journal of mathematical physics, 51 (6): 063514, 2010. 10.1063/​1.3431627.

[51] Sam Staton and Sander Uijlen. Effect algebras, presheaves, non-locality and contextuality. Information and Computation, 261: 336–354, 2018. 10.1016/​j.ic.2018.02.012.

[52] Josef Tkadlec. Atomic sequential effect algebras. International Journal of Theoretical Physics, 47 (1): 185–192, 2008. 10.1007/​s10773-007-9492-1.

[53] John van de Wetering. Three characterisations of the sequential product. Journal of Mathematical Physics, 59 (8), 2018. 10.1063/​1.5031089.

[54] John van de Wetering. An effect-theoretic reconstruction of quantum theory. Compositionality, 1: 1, 12 2019a. ISSN 2631-4444. 10.32408/​compositionality-1-1.

[55] John van de Wetering. Sequential product spaces are Jordan algebras. Journal of Mathematical Physics, 60 (6): 062201, 2019b. 10.1063/​1.5093504.

[56] John van de Wetering. Commutativity in Jordan operator algebras. Journal of Pure and Applied Algebra, page 106407, 2020. ISSN 0022-4049. 10.1016/​j.jpaa.2020.106407.

[57] Ernest B Vinberg. Theory of homogeneous convex cones. Trans. Moscow Math. Soc., 12: 303–368, 1967.

[58] Liu Weihua and Wu Junde. A uniqueness problem of the sequence product on operator effect algebra E(H). Journal of Physics A: Mathematical and Theoretical, 42 (18): 185206, 2009. 10.1088/​1751-8113/​42/​18/​185206.

[59] Abraham Westerbaan, Bas Westerbaan, and John van de Wetering. A characterisation of ordered abstract probabilities. In Proceedings of the 35th Annual ACM/​IEEE Symposium on Logic in Computer Science, LICS ’20, page 944–957, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450371049. 10.1145/​3373718.3394742.

[60] Abraham A Westerbaan. The Category of Von Neumann Algebras. PhD thesis, Radboud Universiteit Nijmegen, 2019a. https:/​/​​abs/​1804.02203.

[61] Bas E Westerbaan. Sequential product on effect logics. Master's thesis, Radboud University Nijmegen, 2013. Available at https:/​/​​publish/​pages/​813276/​masterscriptie_bas_westerbaan.pdf.

[62] Bas E Westerbaan. Dagger and Dilations in the Category of von Neumann Algebras. PhD thesis, Radboud Universiteit Nijmegen, 2019b. https:/​/​​abs/​1803.01911.

Cited by

[1] Anna Jenčová and Sylvia Pulmannová, "Spectral resolutions in effect algebras", Quantum 6, 849 (2022).

[2] John van de Wetering, "A Categorical Construction of the Real Unit Interval", Electronic Proceedings in Theoretical Computer Science 372, 43 (2022).

[3] Martin Plávala, "One Measurement, Two Measurements: from Sequential Products to Convexity", Quantum Views 5, 49 (2021).

[4] Bas Westerbaan and John van de Wetering, "A computer scientist’s reconstruction of quantum theory* ", Journal of Physics A: Mathematical and Theoretical 55 38, 384002 (2022).

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