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arXiv:2207.09180

Polycategorical Constructions for Unitary Supermaps of Arbitrary Dimension

Matt Wilson, Giulio Chiribella

Published Jul 19, 2022• 2 citations
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Abstract

We provide a construction for holes into which morphisms of abstract symmetric monoidal categories can be inserted, termed the polyslot construction pslot[C], and identify a sub-class srep[C] of polyslots that are single-party representable. These constructions strengthen a previously introduced notion of locally-applicable transformation used to characterize quantum supermaps in a way that is sufficient to re-construct unitary supermaps directly from the monoidal structure of the category of unitaries. Both constructions furthermore freely reconstruct the enriched polycategorical semantics for quantum supermaps which allows to compose supermaps in sequence and in parallel whilst forbidding the creation of time-loops. By freely constructing key compositional features of supermaps, and characterizing supermaps in the finite-dimensional case, polyslots are proposed as a suitable generalization of unitary-supermaps to infinite dimensions and are shown to include canonical examples such as the quantum switch. Beyond specific applications to quantum-relevant categories, a general class of categorical structures termed path-contraction groupoids are defined on which the srep[C] and pslot[C] constructions are shown to coincide.

DOI
10.48550/arxiv.2207.09180

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We provide a construction for holes into which morphisms of abstract symmetric monoidal categories can be inserted, termed the polyslot construction pslot[C], and identify a sub-class srep[C] of polyslots that are single-party representable. These constructions strengthen a previously introduced notion of locally-applicable transformation used to characterize quantum supermaps in a way that is sufficient to re-construct unitary supermaps directly from the monoidal structure of the category of unitaries. Both constructions furthermore freely reconstruct the enriched polycategorical semantics for quantum supermaps which allows to compose supermaps in sequence and in parallel whilst forbidding the creation of time-loops. By freely constructing key compositional features of supermaps, and characterizing supermaps in the finite-dimensional case, polyslots are proposed as a suitable generalization of unitary-supermaps to infinite dimensions and are shown to include canonical examples such as the quantum switch. Beyond specific applications to quantum-relevant categories, a general class of categorical structures termed path-contraction groupoids are defined on which the srep[C] and pslot[C] constructions are shown to coincide.
We provide a new characterisation of quantum supermaps in terms of an axiom that refers only to sequential and parallel composition. Consequently, we generalize quantum supermaps to arbitrary monoidal categories and operational probabilistic theories. We do so by providing a simple definition of locally-applicable transformation on a monoidal category. The definition can be rephrased in the language of category theory using the principle of naturality, and can be given an intuitive diagrammatic representation in terms of which all proofs are presented. In our main technical contribution, we use this diagrammatic representation to show that locally-applicable transformations on quantum channels are in one-to-one correspondence with deterministic quantum supermaps. This alternative characterization of quantum supermaps is proven to work for more general multiple-input supermaps such as the quantum switch and on arbitrary normal convex spaces of quantum channels such as those defined by satisfaction of signaling constraints.
We observe that the existence of sequential and parallel composition supermaps in higher order theories of transformations can be formalised using enriched category theory. Encouraged by relevant examples such as unitary supermaps and layers within higher order causal categories (HOCCs), we treat the modelling of higher order physical theories with enriched monoidal categories in analogy with the modelling of physical theories with monoidal categories. We use the enriched monoidal setting to construct a suitable definition of structure preserving map between higher order physical theories via the Grothendieck construction. We then show that the convenient feature of currying in higher order physical theories can be seen as a consequence of combining the primitive assumption of the existence of parallel and sequential composition supermaps with an additional feature of linking. We then use our definition of structure preserving map to show that categories containing infinite towers of enriched monoidal categories with full and faithful structure preserving maps between them inevitably lead to closed monoidal structures. The aim of the proposed definitions is to step towards providing a broad framework for the study and comparison of novel causal structures in quantum theory, and, more broadly, a paradigm of physical theory where static and dynamical features are treated in a unified way.
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.
Categorical supermaps generalise higher-order quantum operations from finite-dimensional quantum theory to arbitrary circuit theories. In this paper, we establish the Yoneda lemma for categorical supermaps, which states that whenever a physical theory has a suitable notion of channel-state duality, then categorical supermaps on that theory can be concretely represented in terms of that duality. This lemma eliminates any guesswork or ambiguity when defining the appropriate notion of supermap for these theories. As a concrete application, we show that the categorical supermaps on Boxworld are in general characterised by a physically well-motivated weaking of the recently proposed NSWSE principle for higher-order processes on boxworld. Furthermore, via the same Yoneda lemma we put forward a stable definition for supermaps in real quantum theory.
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This paper explores the application of category theory to provide a semantic framework for quantum vector spaces, which are fundamental to quantum mechanics and quantum information. We move beyond the traditional algebraic and analytical descriptions of Hilbert spaces to adopt a more abstract, compositional, and graphical approach rooted in monoidal categories. This categorical perspective allows for a unified understanding of quantum phenomena, including superposition, entanglement, and measurement, by representing quantum systems as objects and processes as morphisms. We delve into the concept of compact closed categories and dagger symmetric monoidal categories as the appropriate mathematical structures to capture the essential features of quantum theory, such as duality, adjoints, and the tensor product for composite systems. Furthermore, the paper discusses how graphical calculi, arising from these categorical structures, offer an intuitive and rigorous method for reasoning about quantum protocols and computations, facilitating the design and verification of quantum algorithms. By providing a high-level, axiomatic framework, categorical semantics clarifies the foundational principles of quantum mechanics and opens new avenues for developing quantum programming languages and understanding the interplay between quantum information and computation.