We prove a Tauberian theorem for singular values of noncommuting operators which allows us to prove exact asymptotic formulas in noncommutative geometry at a high degree of generality. We explain how, via the Birman--Schwinger principle, these asympt
otics imply that a semiclassical Weyl law holds for many interesting noncommutative examples. In Connes notation for quantized calculus, we prove that for a wide class of $p$-summable spectral triples $(mathcal{A},H,D)$ and self-adjoint $V in mathcal{A}$, there holds [lim_{hdownarrow 0} h^pmathrm{Tr}(chi_{(-infty,0)}(h^2D^2+V)) = int V_-^{frac{p}{2}}|ds|^p.] where $int$ is Connes noncommutative integral.
We provide a cohomological framework for contextuality of quantum mechanics that is suited to describing contextuality as a resource in measurement-based quantum computation. This framework applies to the parity proofs first discussed by Mermin, as w
ell as a different type of contextuality proofs based on symmetry transformations. The topological arguments presented can be used in the state-dependent and the state-independent case.
We study the effect of noncommutativity of space on the physics of a quantum interferometer located in a rotating disk in a gauge field background. To this end, we develop a path-integral approach which allows defining an effective action from which
relevant physical quantities can be computed as in the usual commutative case. For the specific case of a constant magnetic field, we are able to compute, exactly, the noncommutative Lagrangian and the associated shift on the interference pattern for any value of $theta$.
We study the relationship between POV-measures in quantum theory and asymptotic morphisms in the operator algebra E-theory of Connes-Higson. This is done by introducing the theory of asymptotic PV-measures and their integral correspondence with posit
ive asymptotic morphisms on locally compact spaces. Examples and applications involving various aspects of quantum physics, including quantum noise models, semiclassical limits, strong deformation quantizations, and pure half-spin particles, are also discussed.
We discuss various descriptions of a quantum particle on noncommutative space in a (possibly non-constant) magnetic field. We have tried to present the basic facts in a unified and synthetic manner, and to clarify the relationship between various app
roaches and results that are scattered in the literature.