A characterisation of ordered abstract probabilities

Westerbaan, Abraham; Westerbaan, Bas; Wetering, John van de

doi:10.1145/3373718.3394742

Cited by 9 publications

(22 citation statements)

References 38 publications

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“…It would be interesting to see if a similar characterization or decomposition is possible in the presence of just ω-completeness, that is, for what Gudder and Greechie dubbed σ-SEAs [20]. The characterisation of ω-complete effect monoids of [59] imposes considerable restrictions on the commutative subalgebras of such a σ-SEA, but it is not exactly clear how these can be brought to bear on the global structure of the σ-SEA.…”

Section: Discussionmentioning

confidence: 99%

“…The next five lemmas were originally proven in [59] for effect monoids, and we will need them for our results in the context of SEAs. Proof.…”

Section: Sequential Effect Algebrasmentioning

confidence: 99%

“…We will need the following representation theorem for directed-complete effect monoids from [59]. (In fact, a representation theory for the more general class of ω-directed complete effect monoids is established there, which we do not need here).…”

Section: Representation Theorem For Directed-complete Effect Monoidsmentioning

confidence: 99%

“…This work relies on recent advances made in the representation theory of directed-complete effect monoids [59]. An effect monoid is an effect algebra with an additional associative (not necessarily commutative) multiplication operation that is additive in both arguments.…”

Section: Introductionmentioning

confidence: 99%

“…Crucially, any commutative normal SEA is a directed-complete effect monoid. Section 2 contains the basic definitions and recalls the necessary results from [59]. Then in Section 3 we show that a (normal) SEA where every element is idempotent must be a (complete) Boolean algebra.…”

Section: Introductionmentioning

confidence: 99%

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The three types of normal sequential effect algebras

Westerbaan

Wetering

2020

Quantum

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A sequential effect algebra (SEA) is an effect algebra equipped with a sequential product operation modeled after the Lüders product (a,b)↦aba on C∗-algebras. A SEA is called 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 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=Eb⊕Ec⊕Eac of a complete Boolean algebra Eb, a convex normal SEA Ec, and a newly identified type of normal SEA Eac we dub 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.

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Section: Discussionmentioning

confidence: 99%

“…The next five lemmas were originally proven in [59] for effect monoids, and we will need them for our results in the context of SEAs. Proof.…”

Section: Sequential Effect Algebrasmentioning

confidence: 99%

Section: Representation Theorem For Directed-complete Effect Monoidsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The three types of normal sequential effect algebras

Westerbaan

Wetering

2020

Quantum

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Spectrality in Convex Sequential Effect Algebras

Jenčová,

Pulmannová

2023

Int J Theor Phys

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For convex and sequential effect algebras, we study spectrality in the sense of Foulis. We show that under additional conditions (strong archimedeanity, closedness in norm and a certain monotonicity property of the sequential product), such effect algebra is spectral if and only if every maximal commutative subalgebra is monotone $$\sigma $$ σ -complete. Two previous results on existence of spectral resolutions in this setting are shown to require stronger assumptions.

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A computer scientist’s reconstruction of quantum theory*

Westerbaan¹,

Wetering

2022

J. Phys. A: Math. Theor.

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The rather unintuitive nature of quantum theory has led numerous people to develop sets of (physically motivated) principles that can be used to derive quantum mechanics from the ground up, in order to better understand where the structure of quantum systems comes from. From a computer scientist’s perspective we would like to study quantum theory in a way that allows interesting transformations and compositions of systems and that also includes inﬁnite-dimensional datatypes. Here we present such a compositional reconstruction of quantum theory that includes inﬁnite-dimensional systems. This reconstruction is noteworthy for three reasons: it is only one of a few that includes no restrictions on the dimension of a system; it allows for both classical, quantum, and mixed systems; and it makes no a priori reference to the structure of the real (or complex) numbers. This last point is possible because we frame our results in the language of category theory, speciﬁcally the categorical framework of eﬀectus theory.

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A characterisation of ordered abstract probabilities

Cited by 9 publications

References 38 publications

The three types of normal sequential effect algebras

The three types of normal sequential effect algebras

Spectrality in Convex Sequential Effect Algebras

A computer scientist’s reconstruction of quantum theory*

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