Covariant Integral Quantization of the Semi-Discrete Cylinder

Investigating the usefulness of the semi-discrete cylinder (Γ = Z χ S1) as a phase space, we follow the work of Gazeau and Murenzi (Quantum Rep. 2022). We analyze the Weyl-Heisenberg/Gabor transformation in the context of quantum information and signal analysis. Using key examples of fiducial vectors such as the Von Mises, Wrapped (cyclic) Gaussian, and Fejér vectors, we derive relevant coherent states and reproducing kernels. Through the aforementioned Gabor transformation, wavefunctions or signals defined on the circle are obtained. Functions defined on Γ have promising applications in quantum algorithmic detection, quantum information processing, and other fields that utilize circular data. Additionally, using covariant integral quantization of functions on the semi-discrete cylinder, we derive novel quantum operators. We provide examples of density operators (i.e. quantum states) on the circle through quantizing density distributions defined on Γ using coherent state weights ω, which are related to a given fiducial vector Φ. The resulting quantized operator will be a density operator, whose degree of "quantumness" is related to the number and distribution of zeros of the specified density distribution. One such density distribution of interest is the stellar representation of a quantum state.