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 discern 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) of any set, to a larger group 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”.
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 discern 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) of any set, to a larger group 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”.
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) of any set, to a larger group 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".
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.
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.
Abstract Monoidal category theory serves as a powerful framework for describing logical aspects of quantum theory, giving an abstract language for parallel and sequential composition and a conceptual way to understand many high-level quantum phenomena. Here, we lay the foundations for this categorical quantum mechanics, with an emphasis on the graphical calculus that makes computation intuitive. We describe superposition and entanglement using biproducts and dual objects, and show how quantum teleportation can be studied abstractly using these structures. We investigate monoids, Frobenius structures and Hopf algebras, showing how they can be used to model classical information and complementary observables. We describe the CP construction, a categorical tool to describe probabilistic quantum systems. The last chapter introduces higher categories, surface diagrams and 2-Hilbert spaces, and shows how the language of duality in monoidal 2-categories can be used to reason about quantum protocols, including quantum teleportation and dense coding. Previous knowledge of linear algebra, quantum information or category theory would give an ideal background for studying this text, but it is not assumed, with essential background material given in a self-contained introductory chapter. Throughout the text, we point out links with many other areas, such as representation theory, topology, quantum algebra, knot theory and probability theory, and present nonstandard models including sets and relations. All results are stated rigorously and full proofs are given as far as possible, making this book an invaluable reference for modern techniques in quantum logic, with much of the material not available in any other textbook.
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.
Abstract. Quantum logic aims to capture essential quantum mechanical structure in order-theoretic terms. The Achilles ’ heel of quantum logic is the absence of a canonical description of composite systems, given descriptions of their components. We introduce a framework in which order-theoretic structure comes with a primitive composition operation. The order is extracted from a generalisation of C*-algebra that applies to arbitrary dagger symmetric monoidal categories, which also provide the composition operation. In fact, our construction is entirely compositional, without any additional assumptions on limits or enrichment. Interpreted in the category of finite-dimensional Hilbert spaces, it yields the projection lattices of arbitrary finite-dimensional C*-algebras. Interestingly, there are models that falsify standardly assumed correspondences, most notably the correspondence between noncommutativity of the algebra and nondistributivity of the order. 1.