Different quantum divergences in general von Neumann algebras

By Fumio Hiai (Tohoku University)
Abstract: Different quantum divergences, including standard f-divergences, maximal f-divergences, measured f-divergences, sandwiched R\'enyi divergences, $\alpha$-z-R\'enyi relative entropies, etc., have extensively been developed in these years, with various applications to quantum information, in particular, to the reversibility of quantum operations. However, quantum divergences in the von Neumann algebra setting have not been well developed yet, apart from the earlier work on quasi-entropies (whose special case is standard f-divergences) due to D. Petz and some others. In my talk I give a comprehensive survey on quantum divergences in general von Neumann algebras, based on Haagerup's theory of non-commutative $L^p$-spaces and Kosaki's complex interpolation theory of non-commutative $L^p$-spaces. Recent works on sandwiched R\'enyi divergences in von Neumann algebras due to Jencov\'a and Berta-Scholz-Tomamichel are referred to as well.