Towards a Quantum Domain Theory: Order-enrichment and Fixpoints in W*-algebras

Rennela, Mathys

doi:10.1016/j.entcs.2014.10.016

Cited by 16 publications

(17 citation statements)

References 3 publications

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“…Domains have played a role in labelled Markov processes [17], [18], but not in terms of C*-algebras. As far as we know, the only work similar to the current one is in quantum computing: modeling weakest preconditions [19] and giving semantics for quantum programming languages [20] in an enriched category of certain C*-algebras, but with an alternative order.…”

Section: B Related Workmentioning

confidence: 99%

Domains of Commutative C*-Subalgebras

Heunen

Lindenhovius

2015

2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science

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Abstract-Operator algebras provide uniform semantics for deterministic, reversible, probabilistic, and quantum computing, where intermediate results of partial computations are given by commutative subalgebras. We study this setting using domain theory, and show that a given operator algebra is scattered if and only if its associated partial order is, equivalently: continuous (a domain), algebraic, atomistic, quasi-continuous, or quasialgebraic. In that case, conversely, we prove that the Lawson topology, modelling information approximation, allows one to associate an operator algebra to the domain.

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Section: B Related Workmentioning

confidence: 99%

Domains of Commutative C*-Subalgebras

Heunen

Lindenhovius

2015

2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science

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“…For f : A ⊕ C → B take f (a) = f (a, 0), and for g : A → B, take g(a, z) = g(a) + z · (1 − g (1)). When A, B are W * -algebras, the normal completely positive subunital maps A → B carry a dcpo structure, see [15,70], which can be used to interpret loops, like while or recursion, see also [71] for more information.…”

Section: Pred(x)mentioning

confidence: 99%

New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic

Jacobs¹

2015

Logical Methods in Computer Science

140

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Abstract. Intuitionistic logic, in which the double negation law ¬¬P = P fails, is dominant in categorical logic, notably in topos theory. This paper follows a different direction in which double negation does hold, especially in quantitative logics for probabilistic and quantum systems. The algebraic notions of effect algebra and effect module that emerged in theoretical physics form the cornerstone. It is shown that under mild conditions on a category, its maps of the form X → 1 + 1 carry such effect module structure, and can be used as predicates. Maps of this form X → 1 + 1 are identified in many different situations, and capture for instance ordinary subsets, fuzzy predicates in a probabilistic setting, idempotents in a ring, and effects (positive elements below the unit) in a C * -algebra or Hilbert space.In quantum foundations the duality between states and effects (predicates) plays an important role. This duality appears in the form of an adjunction in our categorical setting, where we use maps 1 → X as states. For such a state ω and a predicate p, the validity probability ω |= p is defined, as an abstract Born rule. It captures many forms of (Boolean or probabilistic) validity known from the literature.Measurement from quantum mechanics is formalised categorically in terms of 'instruments', using Lüders rule in the quantum case. These instruments are special maps associated with predicates (more generally, with tests), which perform the act of measurement and may have a side-effect that disturbs the system under observation. This abstract description of side-effects is one of the main achievements of the current approach. It is shown that in the special case of C * -algebras, side-effects appear exclusively in the noncommutative (properly quantum) case. Also, these instruments are used for test operators in a dynamic logic that can be used for reasoning about quantum programs/protocols. The paper describes four successive assumptions, towards a categorical axiomatisation of quantitative logic for probabilistic and quantum systems, in which the above mentioned elements occur.

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“…Our interpretation of quantum programs as maps between C*-algebras follows other recent work in this direction (e.g. [13,22,41]). …”

Section: Rudiments Of C*-algebrasmentioning

confidence: 73%

Algebraic Effects, Linearity, and Quantum Programming Languages

Staton

2015

Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

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We develop a new framework of algebraic theories with linear parameters, and use it to analyze the equational reasoning principles of quantum computing and quantum programming languages. We use the framework as follows:• we present a new elementary algebraic theory of quantum computation, built from unitary gates and measurement; • we provide a completeness theorem for the elementary algebraic theory by relating it with a model from operator algebra; • we extract an equational theory for a quantum programming language from the algebraic theory; • we compare quantum computation with other local notions of computation by investigating variations on the algebraic theory.

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Towards a Quantum Domain Theory: Order-enrichment and Fixpoints in W*-algebras

Cited by 16 publications

References 3 publications

Domains of Commutative C*-Subalgebras

Domains of Commutative C*-Subalgebras

New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic

Algebraic Effects, Linearity, and Quantum Programming Languages

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