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arXiv:1408.0049 · Electronic Proceedings in Theoretical Computer Science

Categories of Quantum and Classical Channels (extended abstract)

Bob Coecke, Chris Heunen, Aleks Kissinger

Published Jul 29, 2014• 6 citations

Abstract

We introduce the CP*-construction on a dagger compact closed category as a generalisation of Selinger's CPM-construction. While the latter takes a dagger compact closed category and forms its category of "abstract matrix algebras" and completely positive maps, the CP*-construction forms its category of "abstract C*-algebras" and completely positive maps. This analogy is justified by the case of finite-dimensional Hilbert spaces, where the CP*-construction yields the category of finite-dimensional C*-algebras and completely positive maps. The CP*-construction fully embeds Selinger's CPM-construction in such a way that the objects in the image of the embedding can be thought of as "purely quantum" state spaces. It also embeds the category of classical stochastic maps, whose image consists of "purely classical" state spaces. By allowing classical and quantum data to coexist, this provides elegant abstract notions of preparation, measurement, and more general quantum channels.

DOI: 10.4204/eptcs.158.1

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We introduce the CP*-construction on a dagger compact closed category as a generalisation of Selinger's CPM-construction. While the latter takes a dagger compact closed category and forms its category of "abstract matrix algebras" and completely positive maps, the CP*-construction forms its category of "abstract C*-algebras" and completely positive maps. This analogy is justified by the case of finite-dimensional Hilbert spaces, where the CP*-construction yields the category of finite-dimensional C*-algebras and completely positive maps. The CP*-construction fully embeds Selinger's CPM-construction in such a way that the objects in the image of the embedding can be thought of as "purely quantum" state spaces. It also embeds the category of classical stochastic maps, whose image consists of "purely classical" state spaces. By allowing classical and quantum data to coexist, this provides elegant abstract notions of preparation, measurement, and more general quantum channels.

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.

CP

^{\infty}

and beyond: 2-categorical dilation theory

The problem of extending the insights and techniques of categorical quantum mechanics to infinite-dimensional systems was considered in (Coecke and Heunen, 2016). In that work the

\mathrm{CP}^{\infty}

-construction, which recovers the category of Hilbert spaces and quantum operations from the category of Hilbert spaces and bounded linear maps, was defined. Here we show that by a `horizontal categorification' of the

\mathrm{CP}^{\infty}

-construction, one can recover the category of all von Neumann algebras and channels (normal unital completely positive maps) from the 2-category

[W^*]

of von Neumann algebras, bimodules and intertwiners. As an application, we extend Choi's characterisation of extremal channels between finite-dimensional matrix algebras to a characterisation of extremal channels between arbitrary von Neumann algebras.

The problem of extending the insights and techniques of categorical quantum mechanics to infinite-dimensional systems was considered in (Coecke and Heunen, 2016).In that work the CP ∞ -construction, which recovers the category of Hilbert spaces and quantum operations from the category of Hilbert spaces and bounded linear maps, was defined.Here we show that by a 'horizontal categorification' of the CP ∞construction, one can recover the category of all von Neumann algebras and channels (normal unital completely positive maps) from the 2-category of von Neumann algebras, bimodules and intertwiners.As an application, we extend Choi's characterisation of extremal channels between finite-dimensional matrix algebras to a characterisation of extremal channels between arbitrary von Neumann algebras.

Coecke and Heunen described completely positive maps in dagger monoidal categories and the {\sf CP}-infinity construction on these categories in order to construct a category of arbitrary dimensional quantum processes. This article generalizes the

{\sf CP}

-infinity construction of dagger monoidal categories to mixed unitary categories. Mixed unitary categories, on the one hand, generalize the (compact) dagger monoidal categories, and on the other hand, accommodate arbitrary dimensional quantum processes, both without sacrificing the notion of dual objects. This means that the

{\sf CP}

-infinity construction for mixed unitary categories provides a suitable semantics for higher-order quantum programming languages which employ arbitrary dimensional structures. The existing results for the

{\sf CP}

-infinity construction are shown to generalize to the new setting. In particular, the notion of environment structures generalizes to mixed unitary categories and it is shown that the

{\sf CP}

-infinity construction for mixed unitary categories is characterized by this generalized environment structure.