Related papers: Concrete Foundations for Categorical Quantum Physi…
Physical theories can be characterized in terms of their state spaces and their evolutive equations. The kinematical structure and the dynamical structure of finite dimensional quantum theory are, in light of the Choi-Jamio{\l}kowski…
We introduce a tensor product for symmetric monoidal categories with the following properties. Let SMC denote the 2-category with objects small symmetric monoidal categories, arrows symmetric monoidal functors and 2-cells monoidal natural…
We define an action of the braid group (associated with a simple Lie algebra) on the space of $n$-tuples of power series in an indeterminate u, with constant term zero. Using this, we give a sufficient condition for a tensor product of…
We describe a general framework for notions of commutativity based on enriched category theory. We extend Eilenberg and Kelly's tensor product for categories enriched over a symmetric monoidal base to a tensor product for categories…
A quantum set is defined to be simply a set of nonzero finite-dimensional Hilbert spaces. Together with binary relations, essentially the quantum relations of Weaver, quantum sets form a dagger compact category. Functions between quantum…
We prove that every locally Cartesian closed $\infty$-category with subobject classifier has a strict initial object and disjoint and universal binary coproducts.
We reconstruct finite-dimensional quantum theory from categorical principles. That is, we provide properties ensuring that a given physical theory described by a dagger compact category in which one may `discard' objects is equivalent to a…
We revise our "Physical Traces" paper in the light of the results in "A Categorical Semantics of Quantum Protocols". The key fact is that the notion of a strongly compact closed category allows abstract notions of adjoint, bipartite…
This is an expository paper on tensor products where the standard approaches for constructing concrete instances of algebraic tensor products of linear spaces, via quotient spaces or via linear maps of bilinear maps, are reviewed by…
The purpose of this note is to work out the details of the concrete incarnation of a few categorical constructions (products, coproducts, pullbacks, pushouts, equalizers, coequalizers, and exponentials) in some useful and basic categories:…
We construct the categories of standard vector bundles over schemes and define direct sum and tensor product. These categories are equivalent to the usual categories of vector bundles with additional properties. The tensor product is…
A graded tensor category over a group $G$ will be called a crossed product tensor category if every homogeneous component has at least one multiplicatively invertible object. Our main result is a description of the crossed product tensor…
Unitary Ribbon Fusion Categories (URFC) formalize anyonic theories. It has been widely assumed that the same category formalizes a topological quantum computing model. However, in previous work, we addressed and resolved this confusion and…
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…
A common feature of the extended phase space of gauge theory, the crossed product of quantum theory, and quantum reference frames (QRFs) is the adjoining of degrees of freedom followed by a constraining procedure for the resulting total…
Many insights into the quantum world can be found by studying it from amongst more general operational theories of physics. In this thesis, we develop an approach to the study of such theories purely in terms of the behaviour of their…
In this work we present an intuitive construction of the quantum logical axiomatic system provided by George Mackey. The goal of this work is a detailed discussion of the results from the paper 'Physical justification for using the tensor…
The quantum process framework is devised as a generalization of quantum theory to incorporate indefinite causal structure. In this short paper we point out a restriction in forming products of processes: there exist both classical and…
The quantum theory of ur objects postulates that all existing physical objects and their properties are constructed from fundamental objects called ur objects being described by an element of a two dimensional complex Hilbert space. This…
A group-category is an additively semisimple category with a monoidal product structure in which the simple objects are invertible. For example in the category of representations of a group, 1-dimensional representations are the invertible…