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We study the globalization of partial actions on sets and topological spaces and of partial coactions on algebras by applying the general theory of globalization for geometric partial comodules, as previously developed by the authors. We…

In this work, we develop a novel mathematical framework for universal digital quantum computation using algebraic probability theory. We rigorously define quantum circuits as finite sequences of elementary quantum gates and establish their…

We propose models of quantum neural networks through Clifford algebras, which are capable of capturing geometric features of systems and to produce entanglement. Due to their representations in terms of Pauli matrices, the Clifford algebras…

This paper shows how numerical methods on a regular grid in a box can be used to generate numerical schemes for problems in general smooth domains contained in the box with no need for a domain specific discretization. The focus is mainly…

We characterize those varieties of universal algebras where every split epimorphism considered as a map of sets is a product projection. In addition we obtain new characterizations of protomodular, unital and subtractive varieties as well…

The theory of algebraic extensions of Banach algebras is well established, and there are many constructions which yield interesting extensions. In particular, Cole's method for extending uniform algebras by adding square roots of functions…

We consider a quaternionic quantum formalism for the description of quantum states and quantum dynamics. We prove that generalized quantum measurements on physical systems in quaternionic quantum theory can be simulated by usual quantum…

Higher-order quantum theory is an extension of quantum theory where one introduces transformations whose input and output are transformations, thus generalizing the notion of channels and quantum operations. The generalization then goes…

We describe a universal factorization for a functor with values in finite-dimensional measured algebras. More precisely we contruct the quantum automorphism group of this functor. This general recontruction result allows us to recapture a…

We propose the extension of the complex numbers to be the new domain where new concepts, like negative and imaginary probabilities, can be defined. The unit of the new space is defined as the solution of the unsolvable equation in the…

We present a generalization of the notion of an algebra norm relevant to real finite-dimensional unital associative algebras. Among other things, this leads to a novel set of algebra isomorphism invariants, some of which are computationally…

Generalized dualities had an intriguing incursion into Double Field Theory (DFT) in terms of local $O(d,d)$ transformations. We review this idea and use the higher derivative formulation of DFT to compute the first order corrections to…

We investigate the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps. We show the existence of 'indivisible'…

Using the known possibility to associate the completely positive maps with density matrices and recent results on expressing the density matrices with sets of classical probability distributions of dichotomic random variables we construct…

In this paper, we study the unitarizations in the spaces of holomorphic sections of equivariant holomorphic line bundles over a bounded homogeneous domain under the action of a connected algebraic group acting transitively on the domain. We…

Generalized cycles can be thought of as the extension of form-cycle duality between holomorphic forms and cycles, to meromorphic forms and generalized cycles. They appeared as an ubiquitous tool in the study of spectral curves and…

Bilinear maps and their classifying tensor products are well-known in the theory of linear algebra, and their generalization to algebras of commutative monads is a classical result of monad theory. Motivated by constructions needed in…

We propose a new formalism of quantum subsystems which allows to unify the existing and new methods of reduced description of quantum systems. The main mathematical ingredients are completely positive maps and correlation functions. In this…

In the present paper we generate binary pseudorandom sequences using generalized polynomials. A generalized polynomial is a function in whose description we not only allow addition and product (as it is the case in usual polynomials) but…

Quantum channels can be mathematically represented as completely positive trace-preserving maps that act on a density matrix. A general quantum channel can be written as a convex sum of `extremal' channels. We show that for an $N$-level…