Covariant Integral Quantization methods on the discrete Torus with applications in Quantum Information and Quantum Cryptography

The discrete torus, Z_d Χ Z_d, represents an intriguing phase space for quantum and signal systems with configuration space Z_d. Utilizing the Weyl-Heisenberg/Gabor transform on Z_d, we derive the discrete torus as a phase space. An integral Weyl-Heisenberg covariant quantization, labeled by the so-called “weight,” is derived and applied to the quantization of functions and distributions defined on the discrete torus phase space. When these functions are non-negative distributions on the phase space and a coherent state weight is used, covariant integral quantization leads to the fabrication of density matrix operators (i.e. quantum states) on the discrete circle, Z_d. This results in two separate outcomes: first, the derived quantum operators are used to design example algorithms for feature extraction in signals defined on Z_d; and second, the derived density operators are applied in the context of quantum information, quantum encryption, quantum computing algorithm design, and periodic cryptanalysis in signals.