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arXiv:1409.0475 · International Journal of Quantum Information

New Minkowski type inequalities and entropic inequalities for quantum states of qudits

Vladimir I. Man’ko, Liubov A. Markovich

Published Nov 1, 2014• 5 citations
arXivHTMLPDF

Abstract

The two-parameter Minkowski like inequality written for composite quantum system state is obtained for arbitrary Hermitian non-negative matrix with trace equal to unity. The inequality can be used as entropic and information inequality for density matrix of noncomposite finite quantum system, e.g. for a single qudit state. The analogs of strong subadditivity condition for the single qudit is discussed in context of obtained Minkowski like inequality.

DOI
10.1142/s0219749915600217

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The two-parameter Minkowski like inequality written for composite quantum system state is obtained for arbitrary Hermitian non-negative matrix with trace equal to unity. The inequality can be used as entropic and information inequality for density matrix of noncomposite finite quantum system, e.g. for a single qudit state. The analogs of strong subadditivity condition for the single qudit is discussed in context of obtained Minkowski like inequality.
We consider information characteristics of single qudit state (spin j=9/2), such as von Neumann entropy, von Neumann mutual information. We review different mathematical properties of these information characteristics: subadditivity and strong subadditivity conditions, Araki-Lieb inequality. The inequalities are entropic inequalities for composite systems (bipartite, tripartite), but they can be written for noncomposite systems. Using the density matrix, describing the noncomposite qudit system state in explicit matrix form we proved new entropic inequalities for single qudit state (spin j=9/2). In addition, we also consider the von Neumann information of a qudit toy model as a function of a real parameter. The obtained inequalities describe the quantum hidden correlations in the single qudit system.
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.
We obtain a new quantum entropic inequality for the states of a system of n ≥ 1 qudits. The inequality has the form of the quantum subadditivity condition of a bipartite qudit system and coincides with the subadditivity condition for the system of two qudits. We formulate a general statement on the existence of the subadditivity condition for an arbitrary probability distribution and an arbitrary qudit-system tomogram. We discuss the nonlinear quantum channels creating the entangled states from separable states.
For noncomposite systems in classical and quantum domains, we obtain new inequalities such as the subadditivity and strong subadditivity conditions for Shannon entropies and information determined by the probability distributions and for von Neumann entropies of quantum states determined by the density operators. We extend the relations of Shannon and Tsallis entropies to the entropies of conditional probability distributions known for composite systems to the case of noncomposite systems. We give a review of the approach to construct the tomographic-probability distributions for qudit systems and present the entropic and information inequalities for spin tomograms, as well as the subadditivity and strong subadditivity conditions for tomograms of the both noncomposite and composite system states.
The single qudit state tomograms are shown to have the no signaling property. The known and new entropic and information inequalities for Shannon, von Neumann and q-entropies of the composite and noncomposite systems characterizing correlations in these systems are discussed. The spin tomographic probability distributions determining the single qudit states are demonstrated to satisfy the strong subadditivity condition for Tsallis q-entropy. Examples of the new entropic inequalities for q-entropy are considered for qudits with j = 5=2, j = 7=2.
We show that the density-matrix states of noncomposite qudit systems satisfy entropic and information relations like the subadditivity condition, strong subadditivity condition, and Araki-Lieb inequality, which characterize hidden quantum correlations of observables associated with these indivisible systems. We derive these relations employing a specific map of the entropic inequalities known for density matrices of multiqudit systems to the inequalities for density matrices of single-qudit systems. We present the obtained relations in the form of mathematical inequalities for arbitrary Hermitian N × N -matrices. We consider examples of superconducting qubits and qudits. We discuss the hidden correlations in single- qudit states as a new resource for quantum technologies analogous to the known resource in correlations associated with the entanglement in multiqudit systems.