We introduce a novel compositional description of Feynman diagrams, with well-defined categorical semantics as morphisms in a dagger-compact category. Our chosen setting is suitable for infinite-dimensional diagrammatic reasoning, generalising the ZX calculus and other algebraic gadgets familiar to the categorical quantum theory community. The Feynman diagrams we define look very similar to their traditional counterparts, but are more general: instead of depicting scattering amplitude, they embody the linear maps from which the amplitudes themselves are computed, for any given initial and final particle states. This shift in perspective reflects into a formal transition from the syntactic, graph-theoretic compositionality of traditional Feynman diagrams to a semantic, categorical-diagrammatic compositionality. Because we work in a concrete categorical setting -- powered by non-standard analysis -- we are able to take direct advantage of complex additive structure in our description. This makes it possible to derive a particularly compelling characterisation for the sequential composition of categorical Feynman diagrams, which automatically results in the superposition of all possible graph-theoretic combinations of the individual diagrams themselves.
We introduce a novel compositional description of Feynman diagrams, with well-defined categorical semantics as morphisms in a dagger-compact category. Our chosen setting is suitable for infinite-dimensional diagrammatic reasoning, generalising the ZX calculus and other algebraic gadgets familiar to the categorical quantum theory community. The Feynman diagrams we define look very similar to their traditional counterparts, but are more general: instead of depicting scattering amplitude, they embody the linear maps from which the amplitudes themselves are computed, for any given initial and final particle states. This shift in perspective reflects into a formal transition from the syntactic, graph-theoretic compositionality of traditional Feynman diagrams to a semantic, categorical-diagrammatic compositionality. Because we work in a concrete categorical setting -- powered by non-standard analysis -- we are able to take direct advantage of complex additive structure in our description. This makes it possible to derive a particularly compelling characterisation for the sequential composition of categorical Feynman diagrams, which automatically results in the superposition of all possible graph-theoretic combinations of the individual diagrams themselves.
Categorical quantum mechanics exploits the dagger compact closed structure of finite dimensional Hilbert spaces, and uses the graphical calculus of string diagrams to facilitate reasoning about finite dimensional processes. A significant portion of quantum physics, however, involves reasoning about infinite dimensional processes, and it is well-known that the category of all Hilbert spaces is not compact closed. Thus, a limitation of using dagger compact closed categories is that one cannot directly accommodate reasoning about infinite dimensional processes. A natural categorical generalization of compact closed categories, in which infinite dimensional spaces can be modelled, is *-autonomous categories and, more generally, linearly distributive categories. This article starts the development of this direction of generalizing categorical quantum mechanics. An important first step is to establish the behaviour of the dagger in these more general settings. Thus, these notes simultaneously develop the categorical semantics of multiplicative dagger linear logic. The notes end with the definition of a mixed unitary category. It is this structure which is subsequently used to extend the key features of categorical quantum mechanics.
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.
Categorical quantum mechanics and the Wolfram model offer distinct but complementary approaches to studying the relationship between diagrammatic rewriting systems over combinatorial structures and the foundations of physics; the objective of the present article is to begin elucidating the formal correspondence between the two methodologies in the context of the ZX-calculus formalism of Coecke and Duncan for reasoning diagrammatically about linear maps between qubits. After briefly summarizing the relevant formalisms, and presenting a categorical formulation of the Wolfram model in terms of adhesive categories and double-pushout rewriting systems, we illustrate how the diagrammatic rewritings of the ZX-calculus can be embedded and realized within the broader context of Wolfram model multiway systems, and illustrate some of the capabilities of the software framework (ZXMultiwaySystem) that we have developed specifically for this purpose. Finally, we present a proof (along with an explicitly computed example) based on the methods of Dixon and Kissinger that the multiway evolution graphs and branchial graphs of the Wolfram model are naturally endowed with a monoidal structure based on rulial composition that is, furthermore, compatible with the monoidal product of ZX-diagrams.
This paper has two tightly intertwined aims: (i) to introduce an intuitive and universal graphical calculus for multi-qubit systems, the ZX -calculus, which greatly simplifies derivations in the area of quantum computation and information. (ii) To axiomatize complementarity of quantum observables within a general framework for physical theories in terms of dagger symmetric monoidal categories. We also axiomatize phase shifts within this framework. Using the well-studied canonical correspondence between graphical calculi and dagger symmetric monoidal categories, our results provide a purely graphical formalisation of complementarity for quantum observables. Each individual observable, represented by a commutative special dagger Frobenius algebra, gives rise to an Abelian group of phase shifts, which we call the phase group. We also identify a strong form of complementarity, satisfied by the Z - and X -spin observables, which yields a scaled variant of a bialgebra.
In this paper, we extend previous work carried out on the application of the mathematics of category theory to quantum information science.Specifically, we present a realization of a dagger-compact category that can model finitedimensional quantum systems and explicitly allows for the interaction of systems of arbitrary, possibly unequal, dimensions.Hence, our framework can handle generic tensor network states, including matrix product states.Our categorical model subsumes the traditional quantum circuit model while remaining directly and easily applicable to problems stated in the language of quantum information science.The circuit diagrams themselves now become morphisms in a category, making quantum circuits a special case of a much more general mathematical framework.We introduce the key algebraic properties of our tensor calculus diagrammatically and show how they can be applied to solve problems in the field of quantum information.
We introduce a family of diagrammatical equational theories unifying two research programs: categorical quantum mechanics and graphical linear algebra. We prove their completeness with respect to denotational semantics described in terms of relations between vector spaces equipped with symplectic structure. This provides versatile graphical languages encompassing both affinely constrained classical mechanical systems, as well as odd-prime-dimensional stabiliser and Gaussian quantum circuits. Terms are described by labelled graphs with input and output interfaces, and the languages are equipped with equational theories amenable to standard graph rewriting techniques. In order to reason about large composite systems, we introduce a compact scalable notation where the vertices are themselves labelled by graphs. This notation allows us to state new and powerful rewrite rules which operate on diagrams at a large scale. We also show how this notation neatly captures some important constructions, such as graph states of quantum computing and the impedance and admittance matrices of electrical networks.