Purity through Factorisation

Cunningham, Oscar; Heunen, Chris

doi:10.4204/eptcs.266.20

Cited by 9 publications

(13 citation statements)

References 17 publications

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“…For example, the identity map on a direct sum is not atomic and therefore not pure in the sense of [7]. Also, the adjoint of a pure map in the sense of [12,23] on a direct sum need not be pure.…”

Section: Resultsmentioning

confidence: 99%

“…This definition is very general in that it can be defined for any generalized probabilistic theory [4], but it has the drawback that even a canonical map like the identity will not always be pure. Purity can also be defined in terms of leaks [23], or using orthogonal factorization [12]. These definitions work well when considering finite dimensional spaces with a well-behaved sequential product, but when considering more general quantum systems like von Neumann algebras, they fail to reproduce many desirable properties.…”

Section: Introductionmentioning

confidence: 99%

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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: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Pure Maps between Euclidean Jordan Algebras

Westerbaan¹,

Westerbaan²,

Wetering³

2019

Electron. Proc. Theor. Comput. Sci.

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“…[21,7], and [9, Ch. 6], but also [10,11,16,17]). Effectus theory is a bridge between logic and CQM, and affine monoidal categories play a key role there (e.g.…”

Section: Discussion and Outlookmentioning

confidence: 99%

Universal Properties in Quantum Theory

Huot¹,

Staton²

2019

Electron. Proc. Theor. Comput. Sci.

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We argue that notions in quantum theory should have universal properties in the sense of category theory. We consider the completely positive trace preserving (CPTP) maps, the basic notion of quantum channel. Physically, quantum channels are derived from pure quantum theory by allowing discarding. We phrase this in category theoretic terms by showing that the category of CPTP maps is the universal monoidal category with a terminal unit that has a functor from the category of isometries. In other words, the CPTP maps are the affine reflection of the isometries.

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“…Other definitions of purity are those given by leaks [41], orthogonal factorizations [16] or dilations [45]. Without going into the details, these definitions of purity are in general not closed under a dagger operation.…”

Section: Puritymentioning

confidence: 99%

An effect-theoretic reconstruction of quantum theory

Wetering

2019

Compositionality

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An often used model for quantum theory is to associate to every physical system a C∗-algebra. From a physical point of view it is unclear why operator algebras would form a good description of nature. In this paper, we find a set of physically meaningful assumptions such that any physical theory satisfying these assumptions must embed into the category of finite-dimensional C∗-algebras. These assumptions were originally introduced in the setting of effectus theory, a categorical logical framework generalizing classical and quantum logic. As these assumptions have a physical interpretation, this motivates the usage of operator algebras as a model for quantum theory.In contrast to other reconstructions of quantum theory, we do not start with the framework of generalized probabilistic theories and instead use effect theories where no convex structure and no tensor product needs to be present. The lack of this structure in effectus theory has led to a different notion of pure maps. A map in an effectus is pure when it is a composition of a compression and a filter. These maps satisfy particular universal properties and respectively correspond to `forgetting' and `measuring' the validity of an effect.We define a pure effect theory (PET) to be an effect theory where the pure maps form a dagger-category and filters and compressions are adjoint. We show that any convex finite-dimensional PET must embed into the category of Euclidean Jordan algebras. Moreover, if the PET also has monoidal structure, then we show that it must embed into either the category of real or complex C∗-algebras, which completes our reconstruction.

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Purity through Factorisation

Cited by 9 publications

References 17 publications

Pure Maps between Euclidean Jordan Algebras

Pure Maps between Euclidean Jordan Algebras

Universal Properties in Quantum Theory

An effect-theoretic reconstruction of quantum theory

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