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Categories for Quantum TheoryAn Introduction$
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Chris Heunen and Jamie Vicary

Print publication date: 2019

Print ISBN-13: 9780198739623

Published to Oxford Scholarship Online: January 2020

DOI: 10.1093/oso/9780198739623.001.0001

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Monoids and Comonoids

Monoids and Comonoids

(p.127) 4 Monoids and Comonoids
Categories for Quantum Theory

Chris Heunen

Jamie Vicary

Oxford University Press

The tensor product of a monoidal category allows us to consider multiplications on its objects, leading to an abstract notion of monoid. This chapter investigates monoids and their relation to dual objects, as well as comonoids, which axiomatize copying and deleting operations. These play a major role in the categorical description of classical information, since classical information can be copied and deleted, while quantum information cannot. We prove categorical no-deleting and no-cloning theorems, showing that if these structures are able to copy and delete every state of the system, then the category collapses. We also characterize when a tensor product is a categorical product.

Keywords:   Monoid, Comonoid, Uniform deleting, Uniform copying, Categorical product

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