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arXiv:1905.08877

Complete Positivity for Mixed Unitary Categories

Robin Cockett, Priyaa Varshinee Srinivasan

Published May 21, 2019• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1905.08877

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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 CP{\sf CP}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 CP{\sf CP}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 CP{\sf CP}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 CP{\sf CP}CP-infinity construction for mixed unitary categories is characterized by this generalized environment structure.
Monoidal categories have recently been seen to be appropriate structures for establishing abstract axioms for quantum physics. The tensor product in a monoidal category corresponds to the creation of a composite system obtained by joining two independent quantum systems. This correspondence is formalized in the work of Abramsky-Coecke and Selinger. From this point of view, the tensor product in a monoidal category is abstracting the tensor product of Hilbert spaces. The notion of a dagger compact closed category axiomatizes further structure in the Hilbert space category. In particular, the dagger is an abstraction of the familiar Hilbert adjoint. Peter Selinger associates to each dagger compact closed category C its category of completely positive maps, denoted CPM(C). He proves that the resulting category is again a dagger compact closed category. We seek a similar result for tensored †-categories equipped with nuclear ideals. We establish an appropriate notion of completely positive maps in a nuclear ideal. We then define a CPM construction for tensored †-categories equipped with nuclear ideals. Analogous to Selinger's construction, given a nuclear ideal N for a tensored †-category C, our construction yields its category of completely positive maps, CPMN(C). We prove that CPM N(C). is again a tensored †-category. In the process, we also verify that our completely positive maps properly extend Selinger's notion of CPMs. In particular, they preserve nuclear-positive matrices, which are generalizations of von Neumann's positive density operators. As well, we characterize the completely positive maps for the nuclear ideal of Hilbert-Schmidt operators for Hilb and the nuclear ideal of finite relations for LFR. Our construction abstracts the passage from finite-dimensional Hilbert spaces to the category of all Hilbert spaces.
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.
Dagger compact closed categories were recently introduced by Abramsky and Coecke, under the name “strongly compact closed categories”, as an axiomatic framework for quantum mechanics. We present a graphical language for dagger compact closed categories, and sketch a proof of its completeness for equational reasoning. We give a general construction, the CPM construction, which associates to each dagger compact closed category its “category of completely positive maps”, and we show that the resulting category is again dagger compact closed. We apply these ideas to Abramsky and Coecke's interpretation of quantum protocols, and to D'Hondt and Panangaden's predicate transformer semantics.
Dagger compact closed categories were studied by Abramsky and Coecke (under the name “strongly compact closed categories”) as an abstract presentation of the category of Hilbert spaces and linear maps, and as a framework in which to carry out the interpretation of quantum protocols. I subsequently showed that dagger compact closed categories can also describe mixed quantum computation, where the morphisms are completely positive maps. I introduced the CPM construction as a way to pass from the pure to the mixed setting. One technical detail of the CPM(C) construction is that it does not preserve biproducts. Therefore, to obtain an interpretation of classical types such as bit=I⊕I, one must work in the free biproduct completion CPM(C)⊕. In this paper, we show that there is another view of classical types, namely as splittings of self-adjoint idempotents on quantum types. We show that all the objects of CPM(C)⊕ arise as such splittings.
There exist several graphical languages for quantum information processing, like quantum circuits, ZX-Calculus, ZW-Calculus, etc. Each of these languages forms a dagger-symmetric monoidal category (dagger-SMC) and comes with an interpretation functor to the dagger-SMC of (finite dimension) Hilbert spaces. In the recent years, one of the main achievements of the categorical approach to quantum mechanics has been to provide several equational theories for most of these graphical languages, making them complete for various fragments of pure quantum mechanics. We address the question of the extension of these languages beyond pure quantum mechanics, in order to reason on mixed states and general quantum operations, i.e. completely positive maps. Intuitively, such an extension relies on the axiomatisation of a discard map which allows one to get rid of a quantum system, operation which is not allowed in pure quantum mechanics. We introduce a new construction, the discard construction, which transforms any dagger-symmetric monoidal category into a symmetric monoidal category equipped with a discard map. Roughly speaking this construction consists in making any isometry causal. Using this construction we provide an extension for several graphical languages that we prove to be complete for general quantum operations. However this construction fails for some fringe cases like the Clifford+T quantum mechanics, as the category does not have enough isometries.