Paschke Dilations

Westerbaan, Abraham; Westerbaan, Bas

doi:10.4204/eptcs.236.15

Cited by 14 publications

(13 citation statements)

References 15 publications

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“…For instance, other properties from effectus theory could be added to the present framework, such as images, comprehension, quotients, see [10], or subcategories of pure maps with daggers [64]. Also, more probability theory can be lifted to the abstract categorical level, where regular conditionals are of immediate interest, see [14].…”

Section: Discussionmentioning

confidence: 99%

From probability monads to commutative effectuses

Jacobs

2018

Journal of Logical and Algebraic Methods in Programming

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Effectuses have recently been introduced as categorical models for quantum computation, with probabilistic and Boolean (classical) computation as special cases. These 'probabilistic' models are called commutative effectuses, and are the focus of attention here. The paper describes the main known 'probability' monads: the monad of discrete probability measures, the Giry monad, the expectation monad, the probabilistic power domain monad, the Radon monad, and the Kantorovich monad. It also introduces successive properties that a monad should satisfy so that its Kleisli category is a commutative effectus. The main properties are: partial additivity, strong affineness, and commutativity. It is shown that the resulting commutative effectus provides a categorical model of probability theory, including a logic using effect modules with parallel and sequential conjunction, predicate-and state-transformers, normalisation and conditioning of states.

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Section: Discussionmentioning

confidence: 99%

From probability monads to commutative effectuses

Jacobs

2018

Journal of Logical and Algebraic Methods in Programming

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“…See for instance, [ For diagrammatic argument, see [9,Ch. 6]; for an alternative universal uniqueness condition and generalizations, see [27,Prop. 13]; for an analysis from the perspective of probabilistic-theories, see [5].…”

Section: Main Theoremmentioning

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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“…Let A ⊗ K denote the vector space tensor product of A with K. In particular, elements of A ⊗ K are finite sums of tensor products of vectors in A and vectors in K (in fact, all sums that follow are finite). To avoid any abusive notation, the inner product on 2 The vertical concatenation is the vertical composition of natural transformations. This notation was used in [1].…”

Section: Show Thatmentioning

confidence: 99%

“…Neither of our results subsume the other but are complementary. In [2], the authors describe a universal property for minimal Paschke dilations for normal completely positive maps between von Neumann algebras but do not discuss the functoriality of Paschke dilations under changes of the C * -algebras. In the present work, we do not consider this more general class of completely positive maps.…”

Section: Introduction and Outlinementioning

confidence: 99%

Stinespring's construction as an adjunction

Parzygnat

2019

Compositionality

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Given a representation of a C * -algebra together with an isometry, one obtains an operator state on the algebra via restriction using the adjoint action given by the isometry. We show that Stinespring's construction furnishes a left adjoint of this restriction. This is an extension of previous work that showed the Gelfand-Naimark-Segal construction can be viewed as a 2-categorical adjunction. We apply this perspective to analyze Kraus decompositions and properties of the Radon-Nikodym derivative.

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Paschke Dilations

Cited by 14 publications

References 15 publications

From probability monads to commutative effectuses

From probability monads to commutative effectuses

Universal Properties in Quantum Theory

Stinespring's construction as an adjunction

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