99-59 Andrew Lesniewski, Mary Beth Ruskai

Monotone Riemannian Metrics and Relative Entropy on Non-Commutative Probability Spaces (80K, latex) Feb 21, 99

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Abstract. We use the relative modular operator to define a generalized relative entropy for any convex operator function $g$ on $(0,\infty)$ satisfying $g(1) = 0$. We show that these convex operator functions can be partitioned into convex subsets each of which defines a unique symmetrized relative entropy, a unique family (parameterized by density matrices) of continuous monotone Riemannian metrics, a unique geodesic distance on the space of density matrices, and a unique monotone operator function satisfying certain symmetry and normalization conditions. We describe these objects explicitly in several important special cases, including $g(w) = - \log w$ which yields the familiar logarithmic relative entropy. The relative entropies, Riemannian metrics, and geodesic distances obtained by our procedure all contract under completely positive, trace-preserving maps. We then define and study the maximal contraction associated with these quantities.

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