Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat r...

We introduce the CP*–construction on a dagger compact closed category as a generalisation of Selinger’s CPM–construction. While the latter takes a dagger compact closed category and forms its category of “abstract matrix algebras” and completely positive maps, the CP*–construction forms its category of “abstract C*-algebras” and completely positive maps. This analogy is justified by the case of...

Linear algebra is considered an essential mathematical theory that has many engineering applications. While many theorem provers support linear spaces, they only consider finite dimensional spaces. In addition, available libraries only deal with real vectors, whereas complex vectors are extremely useful in many fields of engineering. In this paper, we propose a new linear space formalization wh...

In spite of all no-go theorems (e.g., von Neumann, Kochen and Specker,..., Bell,...) we constructed a realist basis of quantum mechanics. In our model both classical and quantum spaces b are rough images of the fundamental prespace. Quantum mechanics cannot be reduced to classical one. Both classical and quantum representations induce reductions of prespace information.

On classical phase spaces admitting just one complex–differentiable structure, there is no indeterminacy in the choice of the creation operators that create quanta out of a given vacuum. In these cases the notion of a quantum is universal, i.e., independent of the observer on classical phase space. Such is the case in all standard applications of quantum mechanics. However, recent developments ...

Framework for constructing Fock spaces associated either with certain solutions of the quantum Yang-Baxter equation or with infinite dimensional Hecke algebra is presented. For the former case, the quantum deformed oscillator algebra associated with the solution of the quantum Yang-Baxter equation is found. During last decade much attention have been paid to the quantum (q-deformed) groups and ...

We introduce and study the concept of a bornological quantum group. This generalizes the theory of algebraic quantum groups in the sense of van Daele from the algebraic setting to the framework of bornological vector spaces. Working with bornological vector spaces, the scope of the latter theory can be extended considerably. In particular, the bornological theory covers smooth convolution algeb...

Recent results by Krähmer [Israel J. Math. 189 (2012), 237–266] on smoothness of Hopf–Galois extensions and by Liu [arXiv:1304.7117] on smoothness of generalized Weyl algebras are used to prove that the coordinate algebras of the noncommutative pillow orbifold [Internat. J. Math. 2 (1991), 139–166], quantum teardrops O(WPq(1, l)) [Comm. Math. Phys. 316 (2012), 151–170], quantum lens spaces O(Lq...

Shor’s quantum factoring algorithm and the efficient simulation of quantum evolutions by quantum computers are compelling evidence that quantum physics will change the landscape of information science in the future. Despite great progress in the last two decades, the construction of a large scale quantum computer is still in its infancy, largely due to decoherence. A revolutionary idea that eme...