Abstract
Quantum systems with a finite number of states at all times have been a primary element of many physical models in nuclear and elementary particle physics, as well as in condensed matter physics. Today, however, due to a practical demand in the area of developing quantum technologies, a whole set of novel tasks for improving our understanding of the structure of finite-dimensional quantum systems has appeared. In the present article we will concentrate on one aspect of such studies related to the problem of explicit parameterization of state space of an N-level quantum system. More precisely, we will discuss the problem of a practical description of the unitary SU(N)-invariant counterpart of the N-level state space P_N, i.e., the unitary orbit space P_N/SU(N). It will be demonstrated that the combination of well-known methods of the polynomial invariant theory and convex geometry provides useful parameterization for the elements of P_N/SU(N). To illustrate the general situation, a detailed description of P_N/SU(N) for low-level systems: qubit (N=2), qutrit (N=3), quatrit (N=4) - will be given.