Fumio Hiai - Quantum f-Divergences in von Neumann Algebras: Reversibility of Quantum Operations

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After I wrote in 1981 the joint paper [60] with M. Tsukada and M. Ohya on sufficiency of von Neumann subalgebras and the relative entropy in a specialized situation, I wanted to extend the sufficiency notion to general von Neumann subalgebras. But I felt some difficulty in doing that because of the lack of the conditional expectation onto a general von Neumann subalgebra. In the meantime, I was shocked by D. Petz remarkable paper [105] in 1986, where he settled the problem mentioned above by making use of the generalized conditional expectation due to L. Accardi and C. Cecchini [1] in 1982. In his 1988 paper [106] Petz followed the same idea to introduce the notion of sufficient (or reversible) channels (more precisely, unital normal 2-positive maps) between von Neumann algebras, and he further collaborated with A. Jenov on the subject in [68, 69].

After the joint work of the 2011 paper [62], I thought that it would be nice to extend the materials for [62] to the von Neumann algebra setting, including a renewal of Petz theory on sufficiency and reversibility in von Neumann algebras. But that was not undertaken due to my laziness before I looked at the papers [20, 66], where the sandwiched Rnyi divergence, a new Rnyi divergence extensively developed in recent years, was extended to the von Neumann algebra setting. Those papers strongly motivated me to develop a general theory of quantum f-divergences in von Neumann algebras with applications to the sufficiency and reversibility problem. I planned to first write three papers on three different types of f-divergences (the standard f-divergences, the maximal f-divergences, and the measured f-divergences) in von Neumann algebras. The first two papers were published in [53, 54], which are surveyed in Chaps. , the main part of the monograph, I revisit the work [68, 105, 106, 108] with slight extensions and some modifications.