Abstract A Frobenius structure is a monoid together with a comonoid, which satisfies an interaction law. Frobenius structures have a powerful graphical calculus and we prove a normal form theorem that makes them easy to work with. The Frobenius law itself is justified as a coherence property between daggers and closure of a category. We prove classification theorems for dagger Frobenius structures: in Hilb in terms of operator algebras and in Rel in terms of groupoids. Of special interest is the commutative case—as for Hilbert spaces this corresponds to a choice of basis—and provides a powerful tool to model classical information. We discuss phase gates and the state transfer protocol—as well as modules for Frobenius structures—and show how we can use these to model measurement, controlled operations and quantum teleportation.
Abstract A Frobenius structure is a monoid together with a comonoid, which satisfies an interaction law. Frobenius structures have a powerful graphical calculus and we prove a normal form theorem that makes them easy to work with. The Frobenius law itself is justified as a coherence property between daggers and closure of a category. We prove classification theorems for dagger Frobenius structures: in Hilb in terms of operator algebras and in Rel in terms of groupoids. Of special interest is the commutative case—as for Hilbert spaces this corresponds to a choice of basis—and provides a powerful tool to model classical information. We discuss phase gates and the state transfer protocol—as well as modules for Frobenius structures—and show how we can use these to model measurement, controlled operations and quantum teleportation.
We extend categorical semantics of monadic programming to reversible computing, by considering monoidal closed dagger categories: the dagger gives reversibility, whereas closure gives higher-order expressivity. We demonstrate that Frobenius monads model the appropriate notion of coherence between the dagger and closure by reinforcing Cayley's theorem; by proving that effectful computations (Kleisli morphisms) are reversible precisely when the monad is Frobenius; by characterizing the largest reversible subcategory of Eilenberg–Moore algebras; and by identifying the latter algebras as measurements in our leading example of quantum computing. Strong Frobenius monads are characterized internally by Frobenius monoids.
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.
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.
In recent work, several researchers including the authors have developed a categorical formalization of quantum mechanics in terms of symmetric monoidal dagger categories. In this framework, classical data turned out to be represented by an algebraic structure, that of special commutative dagger Frobenius algebras. This structure captures the distinct capabilities that apply to classical data – that they can be copied and deleted. In the present paper, we provide categorical semantics and diagrammatic representations of deterministic, nondeterministic, and probabilistic operations over classical data represented in this way. Moreover, a combination of some fundamental categorical constructions (the Kleisli construction of the category of free algebras and the Grothendieck construction of the total category of an indexed category) with the specific categorical presentations of pure and mixed quantum states provides a resource-sensitive categorical account of classical control of quantum data and of classical data resulting from quantum measurements, as well as of the classical data processing that may happen in between measurements and controls. Along the way we also discover some apparently novel quantum typing structures. One of the salient features of categorical quantum mechanics is still its graphic calculus, which allows succinct presentations of diverse quantum protocols. The elements of an abstract stochastic calculus are beginning to emerge from it, pointing toward convenient refinements of resource-sensitive logics that are hoped to capture the probabilistic content and limited observability of quantum data.
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.
We show that an orthogonal basis for a finite-dimensional Hilbert space can be equivalently characterised as a commutative †-Frobenius monoid in the category FdHilb, which has finite-dimensional Hilbert spaces as objects and continuous linear maps as morphisms, and tensor product for the monoidal structure. The basis is normalised exactly when the corresponding commutative †-Frobenius monoid is special. Hence, both orthogonal and orthonormal bases are characterised without mentioning vectors, but just in terms of the categorical structure: composition of operations, tensor product and the †-functor. Moreover, this characterisation can be interpreted operationally, since the †-Frobenius structure allows the cloning and deletion of basis vectors. That is, we capture the basis vectors by relying on their ability to be cloned and deleted. Since this ability distinguishes classical data from quantum data, our result has important implications for categorical quantum mechanics.