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We consider the probability distributions, spin (qudit)-state tomograms and density matrices of quantum states, and their information characteristics, such as Shannon and von Neumann entropies and q-entropies, from the viewpoints of both well-known purely mathematical features of nonnegative numbers and nonnegative matrices and their physical characteristics, such as entanglement and other quantum correlation phenomena. We review entropic inequalities such as the Araki–Lieb inequality and the subadditivity and strong subadditivity conditions known for bipartite and tripartite systems, and recently obtained for single qudit states. We present explicit matrix forms of the known and some new entropic inequalities associated with quantum states of composite and noncomposite systems. We discuss the tomographic probability distributions of qudit states and demonstrate the inequalities for tomographic entropies of the qudit states. In addition, we mention a possibility to use the discussed information properties of single qudit states in quantum technologies based on multilevel atoms and quantum circuits produced of Josephson junctions

Properties of Nonnegative Hermitian Matrices and New Entropic Inequalities for Noncomposite Quantum Systems