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Types, Abstraction, and Parametric Polymorphism, Part 2
, 1991
"... The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logicalrelations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesianclosedcategory models of the simply typed lambda calculus and P ..."
Abstract

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The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logicalrelations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesianclosedcategory models of the simply typed lambda calculus and PLcategory models of the polymorphic typed lambda calculus. Treatments of Kripke relations and of complete relations on domains are included.
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"... The origin of this apparent ability to signal is postselection, i.e. conditioning on the outcome of the quantum measurement, which is easily seen to be a (virtual) resource that enables signaling. Hence ‘quantum processes in a causal universe ’ cannot form a dagger or a compact symmetric monoidal c ..."
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The origin of this apparent ability to signal is postselection, i.e. conditioning on the outcome of the quantum measurement, which is easily seen to be a (virtual) resource that enables signaling. Hence ‘quantum processes in a causal universe ’ cannot form a dagger or a compact symmetric monoidal category, since both the dagger and the transpose (induced by the compact structure) of a state yield a postselected effect. In order to retain compatibility of quantum mechanics with relativity one needs to exclude postselection and only consider processes with a certain ‘overall ’ probability; that is, one always needs to consider all possible measurement outcomes together. Formally, still in terms of monoidal categories, this can be achieved by using internal dagger Frobenius structures to index over all possible outcome scenarios as in [8]. In that case, an ‘indexchannel’ prevents signaling in the teleportation protocol: