On a discrete version of the position-momentum commutation relation

The usual position-momentum commutation relation plays a fundamental role in the mathematical description of continuous-variable quantum systems. In the case of a qudit described by a Hilbert space of a high enough dimension, there exists a discrete version of the position-momentum commutation relation satisfied with approximation of a large part of the pure quantum states. Our purpose is to explore in more details the set of these states. We show that it contains a family of discrete-variable Gaussian states depending on a continuous parameter and certain discrete coherent states. It also contains various discrete-variable versions of the Hermite-Gauss states, defined either as eigenstates of certain discrete versions of the harmonic oscillator Hamiltonian or generated by using a discrete version of the creation or annihilation operator. The presented results suggest a possible way to experimentally obtain some of the investigated states.