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Entropy 2015, 17(5), 2876-2894; doi:10.3390/e17052876

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

1
Lebedev Physical Institute, Leninskii Prospect 53, Moscow 119991, Russia
2
Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Moscow Region 141700, Russia
*
Author to whom correspondence should be addressed.
Academic Editor: Gerardo Adesso
Received: 30 December 2014 / Revised: 28 April 2015 / Accepted: 4 May 2015 / Published: 6 May 2015
(This article belongs to the Special Issue Entanglement Entropy)
View Full-Text   |   Download PDF [245 KB, uploaded 7 May 2015]

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. View Full-Text
Keywords: von Neumann entropy; information and entropic inequalities; spin tomography; single-qudit subadditivity condition von Neumann entropy; information and entropic inequalities; spin tomography; single-qudit subadditivity condition
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Man'ko, M.A.; Man'ko, V.I. Properties of Nonnegative Hermitian Matrices and New Entropic Inequalities for Noncomposite Quantum Systems. Entropy 2015, 17, 2876-2894.

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