Abstract
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.
Highlights
Quantum states are characterized by entropies, which have the properties associated with well-known purely mathematical properties of nonnegative Hermitian matrices
We show that the known entropic inequalities valid for multipartite systems are valid for the systems without subsystems; they characterize correlations of the degrees of freedom of, e.g., only one single qudit state
We present an analog of inequality (30) for the density matrix ρ of the two-qubit state; in this case, the density matrix has matrix elements ρm1 m2,m01 m02 ), where m1 m2 (m01 m02 ) take values ±1/2, and the tomogram w(m1, m2 |~n1, ~n2 ) satisfies the known entropic inequality:
Summary
Quantum states are characterized by entropies, which have the properties associated with well-known purely mathematical properties of nonnegative Hermitian matrices (see, for example, [1,2]). The aim of this paper is to review a recent approach employed in [25,26,27,28,29,30,31,32,33,34] to study the possibility of finding such new entropic inequalities like the subadditivity and strong subadditivity conditions for noncomposite quantum systems and obtaining some other entropic and information equalities and inequalities for conditional and relative entropies and q-entropies known for composite systems, which can be introduced for the systems without subsystems like, e.g., a single qudit.
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