Parameterizing qudit states

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 NN-level quantum system. More precisely, we will discuss the problem of a practical description of the unitary SU(N){SU(N)}-invariant counterpart of the NN-level state space BN{\mathcal{B}_N}, i.e., the unitary orbit space BN/SU(N){B_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 BN/SU(N){B_N/SU(N)}. To illustrate the general situation, a detailed description ofBN/SU(N){B_N/SU(N)} for low-level systems: qubit (N=2{N= 2}), qutrit (N=3{N=3}), quatrit (N=4{N= 4}) - will be given.