From the discrete Weyl -- Wigner formalism for symmetric ordering to a number -- phase Wigner function

TL;DR

This paper develops a discrete Weyl-Wigner formalism for symmetric operator ordering in finite-dimensional quantum systems, connecting it to the number-phase Wigner function in quantum optics and exploring its limits as dimension grows.

Contribution

It introduces a new discrete Wigner function for symmetric ordering in odd and even-dimensional Hilbert spaces, linking finite-dimensional formalism to continuous quantum optics.

Findings

Derived a discrete Wigner function for symmetric ordering in odd dimensions.

Showed the limit of this formalism converges to the number-phase Wigner function.

Compared their discrete Wigner functions with existing ones in literature.

Abstract

The general Weyl -- Wigner formalism in finite dimensional phase spaces is investigated. Then this formalism is specified to the case of symmetric ordering of operators in an odd -- dimensional Hilbert space. A respective Wigner function on the discrete phase space is found and the limit, when the dimension of Hilbert space tends to infinity, is considered. It is shown that this limit gives the number -- phase Wigner function in quantum optics. Analogous results for the `almost' symmetric ordering in an even -- dimensional Hilbert space are obtained. Relations between the discrete Wigner functions introduced in our paper and some other discrete Wigner functions appearing in literature are studied.