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For the purpose of this task, we generalise the notion of Bregman entropy to the infinite-dimensional non-commutative case using the Legendre\u2013Fenchel duality. In addition, we use the Falcone\u2013Takesaki duality to extend the duality between coarse-grainings and Markov maps to the infinite-dimensional non-commutative case. Following the recent result of Amari for the Zhu\u2013Rohwer entropies, we conjecture that the proposed family of relative entropies is uniquely characterised by the Markov monotonicity and the Legendre\u2013Fenchel duality. The role of these results in the foundations and applications of quantum information theory is discussed.<\/jats:p>","DOI":"10.1142\/s1230161211000133","type":"journal-article","created":{"date-parts":[[2011,6,22]],"date-time":"2011-06-22T11:49:21Z","timestamp":1308743361000},"page":"191-221","source":"Crossref","is-referenced-by-count":0,"title":["The General Form of \u03b3-Family of Quantum Relative Entropies"],"prefix":"10.1142","volume":"18","author":[{"given":"Ryszard Pawe\u0142","family":"Kostecki","sequence":"first","affiliation":[{"name":"Institute of Theoretical Physics, University of Warsaw, Ho\u017ca 69, 00\u2013681 Warszawa, Poland"}]}],"member":"219","published-online":{"date-parts":[[2011,11,21]]},"reference":[{"volume-title":"On measures of information and their characterizations","year":"1975","author":"Acz\u00e9l J.","key":"rf1"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5056-2"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2009.2030485"},{"key":"rf6","doi-asserted-by":"crossref","unstructured":"H.\u00a0Araki, Relative Hamiltonian for faithful normal states of von Neumann algebra\u00a09 (Publ. 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