{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T12:44:37Z","timestamp":1773665077782,"version":"3.50.1"},"reference-count":39,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2015,5,6]],"date-time":"2015-05-06T00:00:00Z","timestamp":1430870400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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\u2013Lieb 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.<\/jats:p>","DOI":"10.3390\/e17052876","type":"journal-article","created":{"date-parts":[[2015,5,6]],"date-time":"2015-05-06T10:58:41Z","timestamp":1430909921000},"page":"2876-2894","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":35,"title":["Properties of Nonnegative Hermitian Matrices and New Entropic Inequalities for Noncomposite Quantum Systems"],"prefix":"10.3390","volume":"17","author":[{"given":"Margarita","family":"Man'ko","sequence":"first","affiliation":[{"name":"Lebedev Physical Institute, Leninskii Prospect 53, Moscow 119991, Russia"}]},{"given":"Vladimir","family":"Man'ko","sequence":"additional","affiliation":[{"name":"Lebedev Physical Institute, Leninskii Prospect 53, Moscow 119991, Russia"},{"name":"Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Moscow Region 141700, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2015,5,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/BF01609834","article-title":"Relative entropy and the Wigner\u2013Yanase\u2013Dyson\u2013Lieb concavity in an interpolatation","volume":"54","author":"Ulhmann","year":"1977","journal-title":"Commun. Math. Phys"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/S0024-3795(03)00386-0","article-title":"Partial traces and entropy inequalities","volume":"370","author":"Bhatia","year":"2003","journal-title":"Linear Algebra Appl"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1501","DOI":"10.1109\/18.104312","article-title":"Information theoretic inequalities","volume":"37","author":"Dembo","year":"1991","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_4","unstructured":"Nielsen, M.A., and Chuang, I.L. (2000). Quantum Computation and Quantum Information, Cambridge University Press."},{"key":"ref_5","first-page":"245","article-title":"Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik","volume":"11","year":"1927","journal-title":"Nach. Ges. Wiss. G\u00f6tt"},{"key":"ref_6","unstructured":"Von Neumann, J. (1932). Mathematische Grundlagen der Quantummechanik, Springer."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"430","DOI":"10.1007\/BF01343064","article-title":"The damping problem in wave mechanics","volume":"45","author":"Landau","year":"1927","journal-title":"Z. Phys"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1016\/S0375-9601(97)00199-0","article-title":"Positive distribution description for spin states","volume":"229","author":"Dodonov","year":"1997","journal-title":"Phys. Lett. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"430","DOI":"10.1134\/1.558326","article-title":"Spin state tomography","volume":"85","year":"1997","journal-title":"J. Exp. Theor. Phys"},{"key":"ref_10","unstructured":"R\u00e9nyi, A. (1970). Probability Theory, North-Holland."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Abe, S., and Okamoto, Y. (2001). Nonextensive Statistical Mechanics and Its Applications, Springer.","DOI":"10.1007\/3-540-40919-X"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1002\/j.1538-7305.1948.tb00917.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1103\/RevModPhys.81.299","article-title":"Continuous-variable optical quantum-state tomography","volume":"81","author":"Lvovsky","year":"2009","journal-title":"Rev. Mod. Phys"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1007\/BF01120338","article-title":"Correlated states in quantum electronics (resonant circuit)","volume":"10","author":"Dodonov","year":"1989","journal-title":"J. Sov. Laser Res"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"164036","DOI":"10.1088\/1751-8113\/41\/16\/164036","article-title":"Nonstationary boundary effect for a quantum flux in superconducting nanocircuits","volume":"41","author":"Takashima","year":"2008","journal-title":"J. Phys. A Math. Theor"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"174521","DOI":"10.1103\/PhysRevB.84.174521","article-title":"Quantum circuit analog of the dynamical Casimir effect","volume":"84","author":"Fujii","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_17","unstructured":"Nielsen, M.A., and Petz, D. (2004). A simple proof of the strong subadditivity inequality, arXiv:quant-ph\/0408130."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1007\/s11005-008-0223-1","article-title":"A Minkowski-type trace inequality and strong subadditivity of quantum entropy II: Convexity and concavity","volume":"83","author":"Carlen","year":"2008","journal-title":"Lett. Math. Phys"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1007\/BF01646092","article-title":"Entropy inequalities","volume":"18","author":"Araki","year":"1970","journal-title":"Commun. Math. Phys"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0002-9904-1975-13621-4","article-title":"Some convexity and subadditivity properties of entropy","volume":"81","author":"Lieb","year":"1975","journal-title":"Bull. Am. Math. Soc"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4358","DOI":"10.1063\/1.1497701","article-title":"Inequalities for quantum entropy: A review with conditions for equality","volume":"43","author":"Ruskai","year":"2002","journal-title":"J. Math. Phys"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Carlen, E.A., and Lieb, E.H. (2014). Remainder terms for some quantum entropy inequalities, arXiv:1402.3840 [quant-ph].","DOI":"10.1063\/1.4871575"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Rastegin, A.E. (2014). Formulation of Leggett-Garg inequalities in terms of q-entropies, arXiv:1403.6945v3 [quant-ph].","DOI":"10.1088\/0253-6102\/62\/3\/05"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s10946-013-9343-x","article-title":"Inequalities for nonnegative numbers and information properties of qudit tomograms","volume":"34","year":"2013","journal-title":"J. Russ. Laser Res"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"014030","DOI":"10.1088\/0031-8949\/2014\/T160\/014030","article-title":"Quantum strong subadditivity condition for systems without subsystems","volume":"T160","year":"2014","journal-title":"Phys. Scr"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1007\/s10946-013-9367-2","article-title":"Generalized qubit portrait of the qutrit state density matrix","volume":"34","author":"Chernega","year":"2013","journal-title":"J. Russ. Laser Res"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1007\/s10946-014-9427-2","article-title":"Maps of matrices and portrait maps of the density operators of composite and noncomposite systems","volume":"35","year":"2014","journal-title":"J. Russ. Laser Res"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s10946-014-9397-4","article-title":"Tomographic and improved subadditivety conditions for two qubits and qudit with j = 3\/2","volume":"35","author":"Chernega","year":"2014","journal-title":"J. Russ. Laser Res"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1007\/s10946-014-9454-z","article-title":"Deformed subadditivity condition for qudit states and hybrid positive maps","volume":"35","year":"2014","journal-title":"J. Russ. Laser Res"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1007\/s10946-015-9486-z","article-title":"Minkowski-type inequality for arbitrary density matrix of composite and noncomposite systems","volume":"36","author":"Chernega","year":"2015","journal-title":"J. Russ. Laser Res"},{"key":"ref_31","first-page":"110","article-title":"New Minkowski-type inequalities and entropic inequalities for quantum states of qudits","volume":"36","author":"Markovich","year":"2015","journal-title":"J. Russ. Laser Res"},{"key":"ref_32","first-page":"156006","article-title":"Entanglement and other quantum correlations of a single qudit state","volume":"12","year":"2014","journal-title":"Int. J. Quantum Inf"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Chernega, V.N., Man\u2019ko, O.V., and Man\u2019ko, V.I. (2015). Deformed entropy and information relations for composite and noncomposite systems. Found. Phys.","DOI":"10.1007\/s10701-015-9890-9"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Chernega, V.N., and Man\u2019ko, O.V. (2015). No-signaling and strong subadditivity condition for tomographic q-entropy of single qudit states, arXiv:1504.03858 [quant-ph]; Phys. Scr. 2015, in press.","DOI":"10.1088\/0031-8949\/90\/7\/074052"},{"key":"ref_35","unstructured":"Suhov, V., and Zohren, S. (2014). Quantum weighted entropy and its properties, arXiv:1411.0892v1."},{"key":"ref_36","first-page":"163","article-title":"Introduction to tomography, classical and quantum","volume":"36","author":"Marmo","year":"2013","journal-title":"Nuovo Cimento C"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"042312","DOI":"10.1103\/PhysRevA.91.042312","article-title":"Multilevel superconducting circuits as two-qubit systems: Operations, state preparation, and entropic inequalities","volume":"91","author":"Fedorov","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"062322","DOI":"10.1103\/PhysRevA.66.062322","article-title":"Implementation schemes in NMR of quantum processors and the Deutsch-Jozsa algorithm by using virtual spin representation","volume":"66","author":"Kessel","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1409","DOI":"10.1016\/j.physleta.2015.03.023","article-title":"Single qudit realization of the Deutsch algorithm using superconducting many-level quantum circuits","volume":"379","author":"Kiktenko","year":"2015","journal-title":"Phys. Lett. A"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/5\/2876\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:45:52Z","timestamp":1760215552000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/5\/2876"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,5,6]]},"references-count":39,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2015,5]]}},"alternative-id":["e17052876"],"URL":"https:\/\/doi.org\/10.3390\/e17052876","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,5,6]]}}}