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Entropy

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

Margarita A. Man’ko, Vladimir I. Man’ko

Published May 6, 2015• 37 citations
Publisher

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.

DOI
10.3390/e17052876

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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.
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.
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 discuss some inequalities for N nonnegative numbers. We use these inequalities to obtain known inequalities for probability distributions and new entropic and information inequalities for quantum tomograms of qudit states. The inequalities characterize the degree of quantum correlations in addition to noncontextuality and quantum discord. We use the subadditivity and strong subadditivity conditions for qudit tomographic-probability distributions depending on the unitary-group parameters in order to derive new inequalities for Shannon, Renyi, and Tsallis entropies of spin 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.
The probability distribution determining the quantum states of photons and qudits are reviewed. Shannon, Rényi and Tsallis entropies associated with the probability distributions are discussed. Shannon information associated with quantum states in the probability representation is considered. Known inequalities for the classical joint probability distributions determining quantum states of multipartite systems are discussed in detail and the relationship to the inequalities known for quantum von Neumann entropy of the states is presented. Properties of subadditivity and strong subadditivity of the von Neumann entropy of two-partite and multipartite qudit states are considered in view of the subadditivity and strong subadditivity properties of Shannon entropies associated with classical joint probability distributions determining the multiqudit quantum states. The new entropic uncertainty relationships for optical tomograms are suggested as a test for accuracy of the homodyne reconstructing the photon state.