This paper contains two new results: (i) We amend the notion of abstract base in a dagger symmetric monoidal category, as well as its corre- sponding graphical representation, in order to accommodate non-self-dual dagger compact structures; this is crucial for obtaining a planar diagrammatical representation of the induced dagger compact structure as well as for representing several incompatible bases within one diagrammatic calculus. (ii) We (crucially) rely on these base structures in a purely diagrammatic derivation of the quantum state transfer protocol; this derivation provides interesting insights in the distinct structural resources required for state-transfer and teleportation as models of quantum computing.
This paper contains two new results: (i) We amend the notion of abstract base in a dagger symmetric monoidal category, as well as its corre- sponding graphical representation, in order to accommodate non-self-dual dagger compact structures; this is crucial for obtaining a planar diagrammatical representation of the induced dagger compact structure as well as for representing several incompatible bases within one diagrammatic calculus. (ii) We (crucially) rely on these base structures in a purely diagrammatic derivation of the quantum state transfer protocol; this derivation provides interesting insights in the distinct structural resources required for state-transfer and teleportation as models of quantum computing.
This paper contains two new results: We amend the notion of abstract basis in a dagger symmetric monoidal category, as well as its corresponding graphical representation, in order to accommodate non-self-dual dagger compact structures; this is crucial for obtaining a planar diagrammatical representation of the induced dagger compact structure as well as for representing many complementary bases within one diagrammatic calculus. We (crucially) rely on these basis structures in a purely diagrammatic derivation of the quantum state transfer protocol; this derivation provides interesting insights in the distinct structural resources required for state-transfer and teleportation as models of quantum computing.
As shown by Abramsky and Coecke, quantum mechanics can be studied in terms of dagger compact closed categories with biproducts. Within this structure, many well-known quantum protocols can be described and their validity can be shown by establishing the commutativity of certain diagrams in that category. In this paper, we propose an explicit realisation of a category with enough structure to check the validity of a certain class of quantum protocols. In order to do this, we construct a category based on 1-dimensional cobordisms with attached elements of a certain group freely generated by a finite set. We use this category as a graphical language, and we show that it is dagger compact closed with biproducts. Then, relying on the coherence result for compact closed categories, proved by Kelly and Laplaza, we show the coherence result, which enables us to check the validity of quantum protocols just by drawing diagrams. In particular, we show the validity of quantum teleportation, entanglement swapping (as formulated in the work of Abramsky and Coecke) and superdense coding protocol.
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.
This is the first paper in a series where we generalize the Categorical Quantum Mechanics program (due to Abramsky, Coecke, et al) to braided systems. In our view a uniform description of quantum information for braided systems has not yet emerged. The picture is complicated by a diversity of examples that lacks a unifying framework for proving theorems and discovering new protocols. We use category theory to construct a high-level language that abstracts the quantum mechanical properties of braided systems. We exploit this framework to propose an axiomatic description of braided quantum information intended for topological quantum computation. In this installment we first generalize the primordial Abramsky-Coecke "quantum information flow" paradigm from compact closed categories to right-rigid strict monoidal categories. We then study dagger structures for rigid and/or braided categories and formulate a graphical dagger calculus. We then propose two generalizations of strongly compact closed categories. Finally we study partial traces in the context of dagger categories.
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.
We establish a rigorous connection between Relational Quantum Dynamics (RQD) and Categorical Quantum Mechanics (CQM) by formally showing that the category Q of quantum processes in RQD can be endowed with the structure of a dagger compact category. In particular, we demonstrate that Q satisfies the axioms of a dagger symmetric monoidal category with compact closure, mirroring the standard categorical formalism for quantum theory. Within this framework, key physical features of quantum mechanics, such as the transfer of awareness (information) in interactions, the no-cloning theorem, and the quantum teleportation protocol, are naturally expressed and interpreted. We define the categorical structures and illustrate the concepts with examples from quantum computing and quantum information. These examples highlight how RQD’s relational worldview aligns with CQM’s diagrammatic and algebraic characterization of quantum processes, thus providing a robust unified description. The results underscore that the RQD category Q not only has the formal properties of a dagger compact category but also that this structure provides clear insight into quantum informational tasks and no-go theorems, while incorporating the notion of observer-dependent reality in RQD via a functional mapping of awareness.