Related papers: Completeness of dagger-categories and the complex …
The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory,…
Some aspects of basic category theory are developed in a finitely complete category $\C$, endowed with two factorization systems which determine the same discrete objects and are linked by a simple reciprocal stability law. Resting on this…
In the past decades, quantum entanglement has been recognized to be the basic resource in quantum information theory. A fundamental need is then the understanding its qualification and its quantification: Is the quantum state entangled, and…
This manuscript presents a novel framework that integrates higher-order symmetries and category theory into machine learning. We introduce new mathematical constructs, including hyper-symmetry categories and functorial representations, to…
We classify which dual functors on a unitary multitensor category are compatible with the dagger structure in terms of groupoid homomorphisms from the universal grading groupoid to $\mathbb{R}_{>0}$ where the latter is considered as a…
Graded rings provide a natural algebraic framework for encoding symmetry via decompositions into homogeneous components indexed by a group, together with multiplication rules reflecting the group operation. Among graded rings, strongly…
A certain amount of category theory is developed in an arbitrary finitely complete category with a factorization system on it, playing the role of the comprehensive factorization system on Cat. Those aspects related to the concepts of…
In this work, we present a logical formalism for reasoning about quantum systems in finite dimension. Contrary to the usual approach in quantum logic, our formalism is based classical first-order logic, which allows us to use the tools of…
This is the first part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. This theory generalizes the tensor category theory for…
There exist several graphical languages for quantum information processing, like quantum circuits, ZX-Calculus, ZW-Calculus, etc. Each of these languages forms a dagger-symmetric monoidal category (dagger-SMC) and comes with an…
Recent work in set theory indicates that there are many different notions of 'set', each captured by a different collection of axioms, as proposed by J. Hamkins in [Ham11]. In this paper we strive to give one class theory that allows for a…
In this paper, we demonstrate the equivalence between the complex Hilbert space and real Kahler space formulations of quantum mechanics. Complex numbers play an important role in the traditional formulation of quantum mechanics in complex…
We obtain two related characterizations of discrete quantum groups and discrete quantum groups of Kac type as allegorical group objects in the symmetric monoidal dagger category of quantum sets and relations, of interest to quantum…
Ongoing work in quantum information emphasises the need for a structural understanding of quantum speedups: in this work, we focus on the quantum Fourier transform and the structures in quantum theory that enable it. We elucidate a general…
This article introduces Hilbert $*$-categories: an abstraction of categories with similar algebraic and analytic properties to the categories of real, complex, and quaternionic Hilbert spaces and bounded linear maps. Other examples include…
Quantum categories were introduced in [4] as generalizations of both bi(co)algebroids and small categories. We clarify details of that work. In particular, we show explicitly how the monadic definition of a quantum category unpacks to a set…
We argue that category theory should become a part of the daily practice of the physicist, and more specific, the quantum physicist and/or informatician. The reason for this is not that category theory is a better way of doing mathematics,…
This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented…
A denotational semantics of quantum Turing machines having a quantum control is defined in the dagger compact closed category of finite dimensional Hilbert spaces. Using the Moore-Penrose generalized inverse, a new additive trace is…
We introduce a hierarchical classification of theories that describe systems with fundamentally limited information content. This property is introduced in an operational way and gives rise to the existence of mutually complementary…