Extended Cahill-Glauber formalism for finite-dimensional spaces: I. Fundamentals

The search for discrete quantum phase-space quasiprobability distribution functions is a subject of continuous and growing interest in the literature [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 26, 27]. The possibility of representing quantum systems characterized by a finite-dimensional state space by such discrete quasidistributions lays the ground for interesting developments and fruitful applications on quantum computation and quantum information theory [12, 13, 14, 15, 16, 17, 18, 19]. It is well known that, as a well established counterpart to the discrete case, a huge variety of quasiprobabilty distribution functions can be defined upon continuous phase-space [20]. In this sense, the Cahill-Glauber (CG) approach [21] to the subject has proved to be a powerful mapping technique that provides a general class of quasiprobability distribution functions, where the Wigner, Glauber-Sudarshan and Husimi functions appear as particular cases. Therefore, it might be considered as a wide-range phase-space approach to quantum mechanics regarding degrees of freedom with classical counterparts. The aim of this paper is to present a discrete extension of the CG approach. Such extension is not obtained from that approach but, instead, properly constructed out of the finite dimensional context. Furthermore, this ab initio construction inherently embodies the discrete analogues of the desired properties of the CG formalism. In particular, discrete Wigner, Husimi and Glauber-Sudarshan quasiprobability distribution functions are obtained. Thus, besides the theoretical interest of its own, such extension has direct applications in quantum information processing, quantum tomography and quantum teleportation, which are explored in a following work [22]. This work is organized as follows: In the next section we briefly outline the CG approach, setting the stage for section III, where our proposal for a discrete extension of the CG mapping kernel is presented. In section IV basic properties of the mapping technique are discussed, and the continuum limit is carried out on section V. Finally, section VI contains our summary and conclusions. Also, important calculations are detailed in the Appendix.