Three characterisations of the sequential product

Wetering, John van de

doi:10.1063/1.5031089

Cited by 7 publications

(26 citation statements)

References 17 publications

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“…We completely characterize the pure -positive maps and show that they exactly correspond to a generalization of the sequential product maps b → √ ab √ a in von Neumann algebras, (and we'll deduce from this that pure -positivity and †-positivity coincide.) This result can be seen as a characterization of the sequential product like the ones given in [15,32,25].…”

Section: Introductionmentioning

confidence: 84%

Pure Maps between Euclidean Jordan Algebras

Westerbaan¹,

Westerbaan²,

Wetering³

2019

Electron. Proc. Theor. Comput. Sci.

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We propose a definition of purity for positive linear maps between Euclidean Jordan Algebras (EJA) that generalizes the notion of purity for quantum systems. We show that this definition of purity is closed under composition and taking adjoints and thus that the pure maps form a dagger category (which sets it apart from other possible definitions.) In fact, from the results presented in this paper, it follows that the category of EJAs with positive contractive linear maps is a †-effectus, a type of structure originally defined to study von Neumann algebras in an abstract categorical setting. In combination with previous work this characterizes EJAs as the most general systems allowed in a generalized probabilistic theory that is simultaneously a †-effectus. Using the dagger structure we get a notion of †-positive maps of the form f = g † • g. We give a complete characterization of the pure †-positive maps and show that these correspond precisely to the Jordan algebraic version of the sequential product (a, b) → √ ab √ a. The notion of †-positivity therefore characterizes the sequential product.Proposition 9. For any EJA E and a, b, c ∈ E, the following hold.

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Section: Introductionmentioning

confidence: 84%

Pure Maps between Euclidean Jordan Algebras

Westerbaan¹,

Westerbaan²,

Wetering³

2019

Electron. Proc. Theor. Comput. Sci.

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“…But F is also an inverse preserving COSEA under the standard sequential product •. It follows from Theorem 5.19 in [21] that J(a) · J(b) = J(a) • J(b). Hence, J : E → F is a COSEA isomorphism.…”

Section: E(ω)mentioning

confidence: 97%

“…A COSEA is state-unique if a is unique. Although it is not known whether an arbitrary COEA is state-unique, it is shown in [21,22] that every COSEA is state-unique.…”

Section: Convex Sequential Effect Algebrasmentioning

confidence: 99%

“…We then call b a pseudo-inverse for a. (A slightly different definition as well as a version of the next lemma are given in [21].) We denote the smallest nonzero eigenvalue of a by λ (a).…”

Section: E(ω)mentioning

confidence: 99%

“…Of course, these result in the traditional quantum formalism. There is some overlap of this paper and the work in [21,22]. However, our stress on contexts provides a different approach.…”

Section: Introductionmentioning

confidence: 97%

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Contexts in Convex and Sequential Effect Algebras

Gudder

2019

Electron. Proc. Theor. Comput. Sci.

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A convex sequential effect algebra (COSEA) is an algebraic system with three physically motivated operations, an orthogonal sum, a scalar product and a sequential product. The elements of a COSEA correspond to yes-no measurements and are called effects. In this work we stress the importance of contexts in a COSEA. A context is a finest sharp measurement and an effect will act differently according to the underlying context with which it is measured. Under a change of context, the possible values of an effect do not change but the way these values are obtained may be different. In this paper we discuss direct sums and the center of a COSEA. We also consider conditional probabilities and the spectra of effects. Finally, we characterize COSEA's that are isomorphic to COSEA's of positive operators on a complex Hilbert space. These result in the traditional quantum formalism. All of this work depends heavily on the concept of a context.

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Sequential product spaces are Jordan algebras

Wetering

2019

Journal of Mathematical Physics

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We show that finite-dimensional order unit spaces equipped with a continuous sequential product as defined by Gudder and Greechie are homogeneous and self-dual. As a consequence of the Koecher-Vinberg theorem these spaces therefore correspond to Euclidean Jordan algebras. We remark on the significance of this result in the context of reconstructions of quantum theory. In particular, we show that sequential product spaces that have locally tomographic tensor products, i.e. their vector space tensor products are also sequential product spaces, must be C * algebras. Finally we remark on a couple of ways these results can be extended to the infinite-dimensional setting of JB-and JBW-algebras and how changing the axioms of the sequential product might lead to a new characterisation of homogeneous cones.

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Three characterisations of the sequential product

Cited by 7 publications

References 17 publications

Pure Maps between Euclidean Jordan Algebras

Pure Maps between Euclidean Jordan Algebras

Contexts in Convex and Sequential Effect Algebras

Sequential product spaces are Jordan algebras

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