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2 - Classical and Quantum Structuralism

Published online by Cambridge University Press:  05 July 2014

By
Bob Coecke ,
Éric Oliver Paquette  and
Dusko Pavlovic
Edited by
Simon Gay  and
Ian Mackie
Bob Coecke
Affiliation:
Oxford University Computing Laboratory
Éric Oliver Paquette
Affiliation:
Oxford University Computing Laboratory
Dusko Pavlovic
Affiliation:
Oxford University Computing Laboratory
Simon Gay
Affiliation:
University of Glasgow
Ian Mackie
Affiliation:
Imperial College London
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Summary

Abstract

In recent work, several researchers including the authors have developed a categorical formalization of quantum mechanics in terms of symmetric monoidal dagger categories. In this framework, classical data turned out to be represented by an algebraic structure, that of special commutative dagger Frobenius algebras. This structure captures the distinct capabilities that apply to classical data – that they can be copied and deleted. In the present paper, we provide categorical semantics and diagrammatic representations of deterministic, nondeterministic, and probabilistic operations over classical data represented in this way.

Moreover, a combination of some fundamental categorical constructions (the Kleisli construction of the category of free algebras and the Grothendieck construction of the total category of an indexed category) with the specific categorical presentations of pure and mixed quantum states provides a resource-sensitive categorical account of classical control of quantum data and of classical data resulting from quantum measurements, as well as of the classical data processing that may happen in between measurements and controls. Along the way we also discover some apparently novel quantum typing structures.

One of the salient features of categorical quantum mechanics is still its graphic calculus, which allows succinct presentations of diverse quantum protocols. The elements of an abstract stochastic calculus are beginning to emerge from it, pointing toward convenient refinements of resource-sensitive logics that are hoped to capture the probabilistic content and limited observability of quantum data.

Type
Chapter
Information
Semantic Techniques in Quantum Computation , pp. 29 - 69
Publisher: Cambridge University Press
Print publication year: 2009

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