Related papers: Supermaps on generalised theories

To implement any quantum operation (a.k.a. ``superoperator'' or ``CP map'') on a d-dimensional quantum system, it is enough to apply a suitable overall unitary transformation to the system and a d^2-dimensional environment which is…

We consider generalizations of Reed-Muller codes, toric codes, and codes from certain plane curves, such as those defined by norm and trace functions on finite fields. In each case we are interested in codes defined by evaluating arbitrary…

The study of abstraction and composition - the focus of category theory - naturally leads to sophisticated diagrams which can encode complex algebraic semantics. Consequently, these diagrams facilitate a clearer visual comprehension of…

We extend the usual process-theoretic view on locality and causality in subsystems (based on the tensor product case) to general quantum systems (i.e.\ possibly non-factor, finite-dimensional von Neumann algebras). To do so, we introduce a…

We present a domain-specific type theory for constructions and proofs in category theory. The type theory axiomatizes notions of category, functor, profunctor and a generalized form of natural transformations. The type theory imposes an…

These are expanded lecture notes from lectures given at the Workshop on higher structures at MATRIX Melbourne. These notes give an introduction to Feynman categories and their applications. Feynman categories give a universal categorical…

Category theory has become central to certain aspects of theoretical physics. Bain [Synthese, 190:1621--1635 (2013)] has recently argued that this has significance for ontic structural realism. We argue against this claim. In so doing, we…

Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional…

In category theory, the use of string diagrams is well known to aid in the intuitive understanding of certain concepts, particularly when dealing with adjunctions and monoidal categories. We show that string diagrams are also useful in…

The study of complex systems through the lens of category theory consistently proves to be a powerful approach. We propose that cognition deserves the same category-theoretic treatment. We show that by considering a highly-compact cognitive…

A generalized channel is a completely positive map that preserves trace on a given subspace. We find conditions under which a generalized channel with respect to a positively generated subspace J is an extreme point in the set of all such…

These notes were originally developed as lecture notes for a category theory course. They should be well-suited to anyone that wants to learn category theory from scratch and has a scientific mind. There is no need to know advanced…

The purpose of this dissertation is to set up a theory of generalized operads and multicategories, and to use it as a language in which to propose a definition of weak n-category. Included is a full explanation of why the proposed…

There are two ways to turn a categorical model for pure quantum theory into one for mixed quantum theory, both resulting in a category of completely positive maps. One has quantum systems as objects, whereas the other also allows classical…

A new and extensive formalism is developed for monads and galaxies in non-standard enlargements. It is shown that monads and galaxies can be manipulated using order-preserving and order-reversing set-to-set maps, and that set properties…

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…

No quantum circuit can turn a completely unknown unitary gate into its coherently controlled version. Yet, coherent control of unknown gates has been realised in experiments, making use of a different type of initial resources. Here, we…

We introduce the notion of virtual ultracategory. From a topological point of view, this notion can be seen as a categorification of relational $\beta$-algebras. From a categorical point of view, virtual ultracategories generalize…

In this paper we redevelop the foundations of the category theory of quasi-categories (also called infinity-categories) using 2-category theory. We show that Joyal's strict 2-category of quasi-categories admits certain weak 2-limits, among…

The questions we raise in this letter are as follows: What is the most general representation of a quantum state at a single point in time? Can we adapt the current formalisms to situations where the order of quantum operations is…