An original presentation of Categorical Quantum Physics, in the line of Abramsky and Coecke, tries to introduce only objects and assumptions that are clearly relevant to Physics and does not assume compact closure. Adjoint arrows, tensor products and biproducts are the ingredients of this presentation. Tensor products are defined, up to a unitary arrow, by a universal property related to transformations of composite systems, not by assuming a monoidal structure. Entangled states of a tensor product define mixed states on the components of the tensor product. Coproducts that fit the adjoint structure are shown to be defined up to a unitary arrow and to provide biproducts. An abstract no-cloning result is proved.
An original presentation of Categorical Quantum Physics, in the line of Abramsky and Coecke, tries to introduce only objects and assumptions that are clearly relevant to Physics and does not assume compact closure. Adjoint arrows, tensor products and biproducts are the ingredients of this presentation. Tensor products are defined, up to a unitary arrow, by a universal property related to transformations of composite systems, not by assuming a monoidal structure. Entangled states of a tensor product define mixed states on the components of the tensor product. Coproducts that fit the adjoint structure are shown to be defined up to a unitary arrow and to provide biproducts. An abstract no-cloning result is proved.
Elaborating on our joint work with Abramsky in [S. Abramsky, B. Coecke, B. (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130, S. Abramsky, B. Coecke, (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124] we further unravel the linear structure of Hilbert spaces into several constituents. Some prove to be very crucial for particular features of quantum theory while others obstruct the passage to a formalism which is not saturated with physically insignificant global phases. First we show that the bulk of the required linear structure is purely multiplicative, and arises from the strongly compact closed tensor which, besides providing a variety of notions such as scalars, trace, unitarity, self-adjointness and bipartite projectors [S. Abramsky, B. Coecke, B. (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130, S. Abramsky, B. Coecke, (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124], also provides Hilbert-Schmidt norm, Hilbert-Schmidt inner-product, and in particular, the preparation-state agreement axiom which enables the passage from a formalism of the vector space kind to a rather projective one, as it was intended in the (in)famous Birkhoff & von Neumann paper [G. Birkhoff, J. von Neumann, (1936) The logic of quantum mechanics. Annals of Mathematics 37, 823–843]. Next we consider additive types which distribute over the tensor, from which measurements can be build, and the correctness proofs of the protocols discussed in [S. Abramsky, B. Coecke, (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130] carry over to the resulting weaker setting. A full probabilistic calculus is obtained when the trace is moreover linear and satisfies the diagonal axiom, which brings us to a second main result, characterization of the necessary and sufficient additive structure of a both qualitatively and quantitatively effective categorical quantum formalism without redundant global phases. Along the way we show that if in a category a (additive) monoidal tensor distributes over a strongly compact closed tensor, then this category is always enriched in commutative monoids.
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.
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.
These lecture notes survey some joint work with Samson Abramsky as it was presented by me at Mathematical Foundations of Programming Semantics XXI in Birmingham (2005).2 It concerns a categorical semantics for quantum mechanics introduced in [Abramsky, S. and Coecke, B. (2004) A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. An extended & improved version is available at http://arXiv.org/quant-ph/0402130; Abramsky, S. and Coecke, B. (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124. Available at www.tac.mta.ca/tac/volumes/14/6/14-06abs.html]. We present this semantics with a particular focus on the connection between categorical diagrams and certain pictures involving typed squares, triangles, diamonds and lines. Along the way we unravel the structural components which come with strong compact closure. We provide pointers to related literature. Keywords: Strong compact closure, semantics, quantum mechanics, quantum logic, quantum informatics.
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.
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.