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Quantum and classical structures in nondeterministic computation
 Proceedings of Quanum Interaction 2009, Lecture
"... Abstract. In categorical quantum mechanics, classical structures characterize the classical interfaces of quantum resources on one hand, while on the other hand giving rise to some quantum phenomena. In the standard Hilbert space model of quantum theories, classical structures over a space correspon ..."
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Abstract. In categorical quantum mechanics, classical structures characterize the classical interfaces of quantum resources on one hand, while on the other hand giving rise to some quantum phenomena. In the standard Hilbert space model of quantum theories, classical structures over a space correspond to its orthonormal bases. In the present paper, we show that classical structures in the category of relations correspond to direct sums of abelian groups. Although relations are, of course, not an interesting model of quantum computation, this result has some interesting computational interpretations. If relations are viewed as denotations of nondeterministic programs, it uncovers a wide variety of nonstandard quantum structures in this familiar area of classical computation. Ironically, it also opens up a version of what in philosophy of quantum mechanics would be called an onticepistemic gap, as it provides no interface to these nonstandard quantum structures. 1
Geometry of abstraction in quantum computation
"... Quantum algorithms are sequences of abstract operations, performed on nonexistent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke and Selinger. In particular, we analyze function abstraction i ..."
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Quantum algorithms are sequences of abstract operations, performed on nonexistent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke and Selinger. In particular, we analyze function abstraction in quantum computation, which turns out to characterize its classical interfaces. Some quantum algorithms provide feasible solutions of important hard problems, such as factoring and discrete log (which are the building blocks of modern cryptography). It is of a great practical interest to precisely characterize the computational resources needed to execute such quantum algorithms. There are many ideas how to build a quantum computer. Can we prove some necessary conditions? Categorical semantics help with such questions. We show how to implement an important family of quantum algorithms using just abelian groups and relations.
Fractional Types
"... Abstract. In previous work, we developed a firstorder, informationpreserving, and reversible programming language Π founded on type isomorphisms. Being restricted to firstorder types limits the expressiveness of the language: it is not possible, for example, to abstract common program fragments in ..."
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Abstract. In previous work, we developed a firstorder, informationpreserving, and reversible programming language Π founded on type isomorphisms. Being restricted to firstorder types limits the expressiveness of the language: it is not possible, for example, to abstract common program fragments into a higherlevel combinator. In this paper, we introduce a higherorder extension of Π based on the novel concept of fractional types 1/b. Intuitively, a value of a fractional type 1/v represents negative information. A function is modeled by a pair (1/v1, v2) with 1/v1 representing the needed argument and v2 representing the result. Fractional values are firstclass: they can be freely propagated and transformed but must ultimately — in a complete program — be offset by the corresponding amount of positive information. 1
CATEGORIES OF QUANTUM AND CLASSICAL CHANNELS
"... Abstract. We introduce a construction that turns a category of pure state spaces and operators into a category of observable algebras and superoperators. ..."
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Abstract. We introduce a construction that turns a category of pure state spaces and operators into a category of observable algebras and superoperators.
Operational Theories and Categorical Quantum Mechanics
, 2013
"... A central theme in current work in quantum information and quantum foundations is to see quantum mechanics as occupying one point in a space of possible theories, and to use this perspective to understand the special features and properties which single it out, and the possibilities for alternative ..."
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A central theme in current work in quantum information and quantum foundations is to see quantum mechanics as occupying one point in a space of possible theories, and to use this perspective to understand the special features and properties which single it out, and the possibilities for alternative theories. Two formalisms which have been used in this context are operational theories, and categorical quantum mechanics. The aim of the present paper is to establish strong connections between these two formalisms. We show how models of categorical quantum mechanics have representations as operational theories. We then show how nonlocality can be formulated at this level of generality, and study a number of examples from this point of view, including Hilbert spaces, sets and relations, and stochastic maps. The local, quantum, and nosignalling models are characterized in these terms. 1
CHARACTERIZATIONS OF CATEGORIES OF COMMUTATIVE C*SUBALGEBRAS
"... commutative C*subalgebras of various C*algebras, namely C*algebras ..."
Languages, Theory
"... Every functional programmer knows about sum and product types, a+b and a×b respectively. Negative and fractional types, a−b and a/b respectively, are much less known and their computational interpretation is unfamiliar and often complicated. We show that in a programming model in which information i ..."
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Every functional programmer knows about sum and product types, a+b and a×b respectively. Negative and fractional types, a−b and a/b respectively, are much less known and their computational interpretation is unfamiliar and often complicated. We show that in a programming model in which information is preserved (such as the model introduced in our recent paper on Information Effects), these types have particularly natural computational interpretations. Intuitively, values of negative types are values that flow “backwards ” to satisfy demands and values of fractional types are values that impose constraints on their context. The combination of these negative and fractional types enables greater flexibility in programming by breaking global invariants into local ones that can be autonomously satisfied by a subcomputation. Theoretically, these types give rise to two function spaces and to two notions of continuations, suggesting that the previously observed duality of computation conflated two orthogonal notions: an additive duality that corresponds to backtracking and a multiplicative duality that corresponds to constraint propagation.