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Categories of quantum and classical channels

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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. For example, it turns the category of finite-dimensional Hilbert spaces into the category of finite-dimensional C*-algebras and completely positive maps. In particular, the new category contains both quantum and classical channels, providing elegant abstract notions of preparation and measurement. We also consider nonstandard models that can be used to investigate which notions from algebraic quantum information theory are operationally justifiable.

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Notes

  1. By an abstract C*-algebra, we mean an object in a monoidal category satisfying certain requirements. By a concrete one, we mean an object satisfying those requirements in the category of (finite-dimensional) Hilbert spaces. This is not to be confused with terminology from functional analysis. There, a concrete C*-algebra is a *-subalgebra of the algebra \(\mathcal {B}(H)\) of bounded operators on a Hilbert space \(H\) that is uniformly closed, whereas an abstract C*-algebra is any Banach algebra with an involution satisfying \(\Vert a^*a\Vert =\Vert a\Vert ^2\); these notions are equivalent by the Gelfand–Naimark–Segal construction; see e.g.  [16, Theorem I.9.12].

  2. There is a closely related notion called specialness. A dagger Frobenius algebra is normal if and only if it is special and symmetric. In \(\mathbf {FHilb} \), normal and special coincide for dagger Frobenius algebras.

  3. Commutativity might be too strong a notion of “completely classical” system in the abstract. A weaker notion of broadcastability, that coincides with commutativity in \(\mathbf {FHilb} \), seems more reasonable. Subsequent work will investigate such more operational notions of classicality.

  4. Notice also that \(\mathbb {R}_{\ge 0}\) is not a quantale under its usual ordering.

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Acknowledgments

This research was supported by the Engineering and Physical Sciences Research Council Fellowship EP/L002388/1, and the John Templeton Foundation.

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  1. Department of Computer Science, University of Oxford, Oxford, UK

    Bob Coecke, Chris Heunen & Aleks Kissinger

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  1. Bob Coecke
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  2. Chris Heunen
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  3. Aleks Kissinger
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Correspondence to Chris Heunen.

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Coecke, B., Heunen, C. & Kissinger, A. Categories of quantum and classical channels. Quantum Inf Process 15, 5179–5209 (2016). https://doi.org/10.1007/s11128-014-0837-4

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