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We use the Falcone-Takesaki non-commutative flow of weights and the resultingtheory of non-commutative Lp spaces in order to define the family of relativeentropy functionals that naturally generalise the quantum relative entropies ofJencova-Ojima and the classical relative entropies of Zhu-Rohwer, and belong toan intersection of families of Petz relative entropies with Bregman relativeentropies. For the purpose of this task, we generalise the notion of Bregmanentropy to the infinite-dimensional non-commutative case using theLegendre-Fenchel duality. In addition, we use the Falcone-Takesaki duality toextend the duality between coarse-grainings and Markov maps to theinfinite-dimensional non-commutative case. Following the recent result of Amarifor the Zhu-Rohwer entropies, we conjecture that the proposed family ofrelative entropies is uniquely characterised by the Markov monotonicity and theLegendre-Fenchel duality. The role of these results in the foundations andapplications of quantum information theory is discussed.

The general form of gamma-family of quantum relative entropies