Titlebook: Quantum f-Divergences in von Neumann Algebras; Reversibility of Qua Fumio Hiai Book 2021 The Editor(s) (if applicable) and The Author(s), u

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Standard ,-Divergences,Let . be a general von Neumann algebra, and . be the positive cone of the predual .. consisting of normal positive linear functionals on .. Basics of von Neumann algebras are given in Sect. A.1.

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Maximal ,-Divergences,Let . be a von Neumann algebra with its standard form . as before. Throughout this chapter, we assume that . is an operator convex function on (0, .).

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Measured ,-Divergences,Let . be a convex function on (0, .), not necessarily operator convex unless we specify that. We use the convention in (.). Let . be a general von Neumann algebra. A . . in . is given by . for some ., where .. ∈ .. for 1 ≤ . ≤ . and .. The measurement . is identified with a unital positive map ., determined by .(..) := .., 1 ≤ . ≤ ..

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Reversibility and Quantum Divergences,Let . and . be von Neumann algebras, whose standard forms are . and ., respectively. For convenience, we first summarize basic properties of positive linear maps between von Neumann algebras, although those have already been used in previous chapters.

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Preservation of Maximal ,-Divergences,In this chapter we will characterize the preservation of . under a unital normal positive map ., i.e., the equality case in the monotonicity inequality ..

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