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arXiv:1803.00707 · Electronic Proceedings in Theoretical Computer Science

A Shortcut from Categorical Quantum Theory to Convex Operational Theories

Alexander Wilce

Published Feb 27, 2018• 4 citations
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Abstract

This paper charts a very direct path between the categorical approach to quantum mechanics, due to Abramsky and Coecke, and the older convex-operational approach based on ordered vector spaces (recently reincarnated as "generalized probabilistic theories"). In the former, the objects of a symmetric monoidal category C are understood to represent physical systems and morphisms, physical processes. Elements of the monoid C(I,I) are interpreted somewhat metaphorically as probabilities. Any monoid homomorphism from the scalars of a symmetric monoidal category C gives rise to a covariant functor V_o from C to a category of dual-pairs of ordered vector spaces. Specifying a natural transformation u from V_o to 1 (where 1 is the trivial such functor) allows us to identify normalized states, and, thus, to regard the image category V_o(C) as consisting of concrete operational models. In this case, if A and B are objects in C, then V_o(A x B) defines a non-signaling composite of V_o(A) and V_o(B). Provided either that C satisfies a "local tomography" condition, or that C is compact closed, this defines a symmetric monoidal structure on the image category, and makes V_o a (strict) monoidal functor.

DOI
10.4204/eptcs.266.15

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This paper charts a very direct path between the categorical approach to quantum mechanics, due to Abramsky and Coecke, and the older convex-operational approach based on ordered vector spaces (recently reincarnated as "generalized probabilistic theories"). In the former, the objects of a symmetric monoidal category C are understood to represent physical systems and morphisms, physical processes. Elements of the monoid C(I,I) are interpreted somewhat metaphorically as probabilities. Any monoid homomorphism from the scalars of a symmetric monoidal category C gives rise to a covariant functor V_o from C to a category of dual-pairs of ordered vector spaces. Specifying a natural transformation u from V_o to 1 (where 1 is the trivial such functor) allows us to identify normalized states, and, thus, to regard the image category V_o(C) as consisting of concrete operational models. In this case, if A and B are objects in C, then V_o(A x B) defines a non-signaling composite of V_o(A) and V_o(B). Provided either that C satisfies a "local tomography" condition, or that C is compact closed, this defines a symmetric monoidal structure on the image category, and makes V_o a (strict) monoidal functor.
We exhibit a functor from the category OUS of order unit spaces and positive, unit-preserving mappings into the category $\Prob$ of probabilistic models (test spaces with designated state spaces) and morphisms thereof. Restricted to any subcategory of OUS monoidal with respect to a positive, normalized, bilinear composition rule, our functor is also monoidal. This shows that the convex-operational approach to physical theories can be subsumed by the test-space approach, without resort to ``generalized test spaces''. A second construction, equipping a probabilistic model with tests representing ``weighted coins'', also sheds light on the nature of unsharp observables.
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.
Given a monoidal probabilistic theory -- a symmetric monoidal category C\mathcal{C}C of systems and processes, together with a functor V\mathbf{V}V assigning concrete probabilistic models to objects of C\mathcal{C}C -- we construct a locally tomographic probabilistic theory LT(C,V)(\mathcal{C},\mathbf{V})(C,V) -- the locally tomographic shadow of (C,V)(\mathcal{C},\mathbf{V})(C,V) -- describing phenomena observable by local agents controlling systems in C\mathcal{C}C, and able to pool information about joint measurements made on those systems. Some globally distinct states become locally indistinguishable in LT(C,V)(\mathcal{C},\mathbf{V})(C,V), and we restrict the set of processes to those that respect this indistinguishability. This construction is investigated in some detail for real quantum theory.
This paper proposes a comprehensive categorical framework for unifying classical and quantum probabilistic logics. We leverage the inherent compositional and structural advantages of category theory to construct a common mathematical language capable of expressing both Kolmogorovian probability and the non-Boolean structures characteristic of quantum mechanics. By treating systems and processes as objects and morphisms within suitable monoidal categories, we demonstrate how core concepts like states, observables, measurements, and conditioning can be abstractly represented and seamlessly transitioned between classical and quantum domains. The framework utilizes probability monads for classical uncertainty and dagger compact closed categories for quantum processes, showing how these distinct structures can be seen as instances of a more general categorical paradigm. This unification not only provides a deeper foundational understanding of uncertainty across different physical theories but also opens new avenues for developing hybrid probabilistic-logical systems in fields such as artificial intelligence and quantum computation.
From every pair of adjoint functors it is possible to produce a (possibly trivial) equivalence of categories by restricting to the subcategories where the unit and counit are isomorphisms. If we do this for the adjunction between effect algebras and abstract convex sets, we get the surprising result that the equivalent subcategories consist of reflexive order-unit spaces and reflexive base-norm spaces, respectively. These are the convex sets that can occur as state spaces in generalized probabilistic theories satisfying both the no-restriction hypothesis and its dual. The linearity of the morphisms is automatic. If we add a compact topology to either the states or the effects, we can obtain a duality for all Banach order-unit spaces or all Banach base-norm spaces, but not both at the same time.
The past decade has seen a remarkable resurgence of the old programme of finding more or less a priori axioms for the mathematical framework of quantum mechanics. The new impetus comes largely from quantum information theory; in contrast to work in the older tradition, which tended to concentrate on structural features of individual quantum systems, the newer work is marked by an emphasis on systems in interaction. Within this newer work, one can discerne two distinct approaches: one is "top-down", and attempts to capture in category-theoretic terms what is distinctive about quantum information processing. The other is "bottom up", attempting to construct non-classical models and theories by hand, as it were, and then characterizing those features that mark out quantum-like behavior. This paper blends these approaches. We present a constructive, bottom-up recipe for building probabilistic theories having strong symmetry properties, using as data any uniform enlargement of the symmetric group S(E)S(E)S(E) of any set, to a larger group G(E)G(E)G(E). Subject to some natural conditions, our construction leads to a monoidal category of fully symmetric test spaces, in which the monoidal product is "non-signaling".