Related papers: Quantum sets
Q-system completion can be thought of as a notion of higher idempotent completion of C*-2-categories. We introduce a notion of quantum bi-elements, and study Q-system completion in the context of compact quantum groups. We relate our notion…
Quantum coherence quantifies the amount of superposition in a quantum system, and is the reason and resource behind several phenomena and technologies. It depends on the natural basis in which the quantum state of the system is expressed,…
Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFTs. Although other constructions have since been found,…
In this sequence of papers, noncommutative analysis is used to give a consistent axiomatic approach to a unified conceptual foundation of classical and quantum physics. The present Part I defines the concepts of observables, states and…
We introduce an algebraic framework for interacting quantum systems that enables studying complex phenomena, characterized by the coexistence and competition of various broken symmetry states of matter. The approach unveils the hidden unity…
A construction of the noncommutative-geometric counterparts of classical classifying spaces is presented, for general compact matrix quantum structure groups. A quantum analogue of the classical concept of the classifying map is introduced…
We consider in general terms dynamical systems with finite-dimensional, non-simply connected configuration-spaces. The fundamental group is assumed to be finite. We analyze in full detail those ambiguities in the quantization procedure that…
We introduce a construction that turns a category of pure state spaces and operators into a category of observable algebras and superoperators. For example, it turns the category of finite-dimensional Hilbert spaces into the category of…
Quantum fields are shown to provide an example of infinite-dimensional quantum groups. A dictionary is established between quantum field and quantum group concepts: the expectation value over the vacuum is the counit, Wick's theorem is the…
We discuss various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups. On the von Neumann algebra level we find an interesting duality for such objects. A definition of a quantum…
According to Cantor, a set is a collection into a whole of defined and separate (we shall say distinct) objects. So, a natural question is ``How to treat as `sets' collections of indistinguishable objects?". This is the aim of quasi-set…
Universality of quantum mechanics -- its applicability to physical systems of quite different nature and scales -- indicates that quantum behavior can be a manifestation of general mathematical properties of systems containing…
Quantum algorithms are sequences of abstract operations, performed on non-existent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke…
Within the context of an involutive monoidal category the notion of a comparison relation is identified. Instances are equality on sets, inequality on posets, orthogonality on orthomodular lattices, non-empty intersection on powersets, and…
Quantum mechanics is formulated as a geometric theory on a Hilbert manifold. Images of charts on the manifold are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations in this…
It is shown that quantum mechanics is noncontextual if quantum properties are represented by subspaces of the quantum Hilbert space (as proposed by von Neumann) rather than by hidden variables. In particular, a measurement using an…
This article provides a basic introduction to some concepts of non-commutative geometry. The importance of quantum groups and quantum spaces is stressed. Canonical non-commutativity is understood as an approximation to the quantum group…
We discuss the notion about physical quantities as having values represented by real numbers, and its limiting to describe nature to be understood in relation to our appreciation that the quantum theory is a better theory of natural…
Quantum mechanics is a special kind of description of motion. The concept of wave function itself implies the openness of quantum system. We show that quantum mechanics describes the quantum correlation, i.e., entanglement, and information…
Operating quantum sensors and quantum computers would make data in the form of quantum states available for purely quantum processing, opening new avenues for studying physical processes and certifying quantum technologies. In this…