Related papers: Categories of Quantum and Classical Channels
We employ an algebraic procedure based on quantum mechanics to propose a `quantum number theory' (QNT) as a possible extension of the `classical number theory'. We built our QNT by defining pure quantum number operators ($q$-numbers) of a…
A quantum theory in a finite-dimensional Hilbert space can be geometrically formulated as a proper Hamiltonian theory as explained in [2, 3, 7, 8]. From this point of view a quantum system can be described in a classical-like framework…
We reconsider differential geometry from the point of view of the quantum theory of non-relativistic spinning particles, which provides examples of supersymmetric quantum mechanics. This enables us to encode geometrical structure in…
If $H_1$ and $H_2$ are finite-dimensional Hilbert spaces, a channel from $H_1$ to $H_2$ is a completely positive, linear map $\mathcal{I}$ that takes the set of states $\mathcal{S}(H_1)$ for $H_1$ to the set of states $\mathcal{S}(H_2)$ for…
Deterministic dynamical models are discussed which can be described in quantum mechanical terms. In particular, a local quantum field theory is presented which is a supersymmetric classical model. -- The Hilbert space approach of Koopman…
We introduce the category $\mathsf{NCP}$, whose objects are pairs of W$^\ast$-algebras and normal states and whose morphisms are state-preserving unital completely positive (CPU) maps, as a common stage for classical and quantum information…
This survey article is concerned with the modeling of the kinematical structure of quantum systems in an algebraic framework which eliminates certain conceptual and computational difficulties of the conventional approaches. Relying on the…
The well-behaved representations of the coordinate algebra of a 2-dimensional quantum complex plane are classified and a C*-algebra is defined which can be viewed as the algebra of continuous functions on the 2-dimensional quantum complex…
Quantum computation can be formulated through various models, each highlighting distinct structural and resource-theoretic aspects of quantum computational power. This paper develops a unified categorical framework that encompasses these…
Interest in combinatorial interpretations of mathematical entities stems from the convenience of the concrete models they provide. Finding a bijective proof of a seemingly obscure identity can reveal unsuspected significance to it. Finding…
We unravel a deep connection between limits of real numbers and limits in category theory. Using a new variant of the classical characterisation of the real numbers, we characterise the category of finite-dimensional Hilbert spaces and…
In a recent paper, a realizability technique has been used to give a semantics of a quantum lambda calculus. Such a technique gives rise to an infinite number of valid typing rules, without giving preference to any subset of those. In this…
We provide a new characterisation of quantum supermaps in terms of an axiom that refers only to sequential and parallel composition. Consequently, we generalize quantum supermaps to arbitrary monoidal categories and operational…
We explore the possibility of extending Mardare et al. quantitative algebras to the structures which naturally emerge from Combinatory Logic and the lambda-calculus. First of all, we show that the framework is indeed applicable to those…
Hilbert space operators may be mapped onto a space of ordinary functions (operator symbols) equipped with an associative (but noncommutative) star-product. A unified framework for such maps is reviewed. Because of its clear probabilistic…
Enormous activity in the Quantum Computing area has resulted in considering them to solve different difficult problems, including those of applied nature, together with classical computers. An attempt is made in this work to nail down a…
Bicommutant categories are higher categorical analogs of von Neumann algebras that were recently introduced by the first author. In this article, we prove that every unitary fusion category gives an example of a bicommutant category. This…
Familiar formulations of classical and quantum mechanics are shown to follow from a general theory of mechanics based on pure states with an intrinsic probability structure. This theory is developed to the stage where theorems from quantum…
A new approach to normal operators in real Hilbert spaces is discussed, and a spectral representation is obtained, derived directly from the complex case. The results are then applied to quaternionic normal operators, regarded as a special…
Categories of relations over a regular category form a family of models of quantum theory. Using regular logic, many properties of relations over sets lift to these models, including the correspondence between Frobenius structures and…