{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:26:44Z","timestamp":1760243204076,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2014,11,6]],"date-time":"2014-11-06T00:00:00Z","timestamp":1415232000000},"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>A new quantum f-divergence for trace class operators in Hilbert Spaces is introduced. It is shown that for normalised convex functions it is nonnegative. Some upper bounds are provided. Applications for some classes of convex functions of interest are also given.<\/jats:p>","DOI":"10.3390\/e16115853","type":"journal-article","created":{"date-parts":[[2014,11,7]],"date-time":"2014-11-07T02:03:59Z","timestamp":1415325839000},"page":"5853-5875","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A New Quantum f-Divergence for Trace Class Operators in Hilbert Spaces"],"prefix":"10.3390","volume":"16","author":[{"given":"Silvestru","family":"Dragomir","sequence":"first","affiliation":[{"name":"Mathematics, School of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia"},{"name":"School of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa\u00a0"}]}],"member":"1968","published-online":{"date-parts":[[2014,11,6]]},"reference":[{"key":"ref_1","first-page":"85","article-title":"Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizit\u00e4t von Markoffschen Ketten","volume":"8","year":"1963","journal-title":"Magyar Tud. Akad. Mat. Kutat\u00f3 Int. K\u00f6zl"},{"key":"ref_2","unstructured":"Liese, F., and Vajda, I. (1987). Texte zur Mathematik, Band 95, Teubuer."},{"key":"ref_3","first-page":"745","article-title":"Bounds on extended f-divergences for a variety of classes","volume":"40","author":"Cerone","year":"2004","journal-title":"Kybernetika"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/j.mcm.2004.02.044","article-title":"Approximation of the integral mean divergence and f-divergence via mean results","volume":"42","author":"Cerone","year":"2005","journal-title":"Math. Comput. Modelling"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"35","DOI":"10.5036\/mjiu.33.35","article-title":"Some inequalities for (m, M)-convex mappings and applications for the Csisz\u00e1r \u03a6-divergence in information theory","volume":"33","author":"Dragomir","year":"2001","journal-title":"Math. J. 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Math"},{"key":"ref_10","first-page":"123","article-title":"New inequalities for Csisz\u00e1r divergence and applications","volume":"28","author":"Dragomir","year":"2003","journal-title":"Acta Math. Vietnam"},{"key":"ref_11","first-page":"61","article-title":"A generalized f-divergence for probability vectors and applications","volume":"13","author":"Dragomir","year":"2003","journal-title":"Panamer. Math. J"},{"key":"ref_12","first-page":"57","article-title":"Some inequalities for the Csisz\u00e1r \u03c6 -divergence when \u03c6 is an L-Lipschitzian function and applications","volume":"15","author":"Dragomir","year":"2004","journal-title":"Ital. J. Pure Appl. Math No."},{"key":"ref_13","first-page":"35","article-title":"A converse inequality for the Csisz\u00e1r \u03a6-divergence","volume":"20","author":"Dragomir","year":"2004","journal-title":"Tamsui Oxf. J. Math. Sci"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/s10474-005-0251-6","article-title":"Some general divergence measures for probability distributions","volume":"109","author":"Dragomir","year":"2005","journal-title":"Acta Math. Hung"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"153","DOI":"10.11650\/twjm\/1500405733","article-title":"A refinement of Jensen\u2019s inequality with applications for f-divergence measures","volume":"14","author":"Dragomir","year":"2010","journal-title":"Taiwan. J. Math"},{"key":"ref_16","first-page":"415","article-title":"On powers of f-divergence defining a distance","volume":"26","author":"Kafka","year":"1991","journal-title":"Stud. Sci. Math. Hung."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1007\/BF02517812","article-title":"A new class of metric divergences on probability spaces and its applicability in statistics","volume":"55","author":"Vajda","year":"2003","journal-title":"Ann. Inst. Stat. Math"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1007\/BF01645523","article-title":"Inequalities for traces on von Neumann algebras","volume":"26","author":"Ruskai","year":"1972","journal-title":"Commun. Math. Phys"},{"key":"ref_19","unstructured":"Simon, B. (1979). Trace Ideals and Their Applications, Cambridge University Press."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1006\/jmaa.1999.6433","article-title":"A matrix trace inequality for products of Hermitian matrices","volume":"237","author":"Chang","year":"1999","journal-title":"J. Math. Anal. Appl"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"999","DOI":"10.1006\/jmaa.1994.1475","article-title":"On matrix trace inequalities and related topics for products of Hermitian matrix","volume":"188","author":"Coop","year":"1994","journal-title":"J. Math. Anal. Appl"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1016\/0022-247X(92)90344-D","article-title":"A matrix trace inequality","volume":"166","author":"Neudecker","year":"1992","journal-title":"J. Math. Anal. Appl"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1016\/0022-247X(88)90423-4","article-title":"A matrix trace inequality","volume":"133","author":"Yang","year":"1988","journal-title":"J. Math. Anal. Appl"},{"key":"ref_24","first-page":"33","article-title":"Matrix Young inequalities","volume":"75","author":"Ando","year":"1995","journal-title":"Oper. Theory Adv. Appl"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Beckenbach, E.F. (1980). General Inequalities 2, Birkh\u00e4user Basel.","DOI":"10.1007\/978-3-0348-6324-7"},{"key":"ref_26","first-page":"5","article-title":"A generalization of a trace inequality for positive definite matrices","volume":"7","author":"Belmega","year":"2010","journal-title":"Aust. J. Math. Anal. Appl."},{"key":"ref_27","first-page":"4","article-title":"Refinements of the trace inequality of Belmega, Lasaulce and Debbah","volume":"7","author":"Furuichi","year":"2010","journal-title":"Aust. J. Math. Anal. Appl."},{"key":"ref_28","first-page":"565","article-title":"On some matrix inequalities","volume":"16","author":"Lee","year":"2008","journal-title":"Korean J. Math"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1484","DOI":"10.1016\/j.jmaa.2006.04.092","article-title":"A trace class operator inequality","volume":"328","author":"Liu","year":"2007","journal-title":"J. Math. Anal. Appl"},{"key":"ref_30","first-page":"1","article-title":"Operator norm inequalities of Minkowski type","volume":"9","author":"Shebrawi","year":"2008","journal-title":"J. Inequal. Pure Appl. Math"},{"key":"ref_31","first-page":"1\u2013201486","article-title":"On some matrix trace inequalities","volume":"201486","year":"2010","journal-title":"J. Inequal. Appl"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/s11117-006-2054-6","article-title":"H\u00f6lder and Young inequalities for the trace of operators","volume":"11","author":"Manjegani","year":"2007","journal-title":"Positivity"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"525","DOI":"10.2977\/prims\/79","article-title":"From quasi-entropy to various quantum information quantities","volume":"48","author":"Hiai","year":"2012","journal-title":"Publ. Res. Inst. Math. Sci"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1142\/S0129055X11004412","article-title":"Quantum f-divergences and error correction","volume":"23","author":"Hiai","year":"2011","journal-title":"Rev. Math. Phys"},{"key":"ref_35","first-page":"81","article-title":"From quasi-entropy","volume":"55","author":"Petz","year":"2012","journal-title":"Ann. Univ. Sci. Bp. E\u00f6tv\u00f6s Sect. Math"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"304","DOI":"10.3390\/e12030304","article-title":"From f-divergence to quantum quasi-entropies and their use","volume":"12","author":"Petz","year":"2010","journal-title":"Entropy"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1017\/S000497270004051X","article-title":"Bounds for the normalized Jensen functional","volume":"74","author":"Dragomir","year":"2006","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_38","unstructured":"De Barra, G. (1981). Measure Theory and Integration, Ellis Horwood Ltd."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/0024-3795(92)90253-7","article-title":"Inequalities for singular values and traces","volume":"171","author":"Chen","year":"1992","journal-title":"Linear Algebra Appl"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/11\/5853\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:09:00Z","timestamp":1760216940000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/11\/5853"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,11,6]]},"references-count":39,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2014,11]]}},"alternative-id":["e16115853"],"URL":"https:\/\/doi.org\/10.3390\/e16115853","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2014,11,6]]}}}