1998/08/30 by Andrew Lesniewski, Mary Beth Ruskai · 174 citations
Mathematics · Physics and Astronomy · #Combinatorics #Convex function #Generalized relative entropy #Geodesic #Geometric Analysis and Curvature Flows #Kullback–Leibler divergence #Mathematical Inequalities and Applications #Mathematical analysis #Mathematics #Pure mathematics #Quantum #Quantum entanglement #Regular polygon #Statistical Mechanics and Entropy #math-ph #math.MP #math.OA #msc:47D45 #msc:53B20 #msc:82B10 #msc:94A17 #quant-ph
paper · pdf · doi:10.1063/1.533053
published in Journal of Mathematical Physics 40(11), 5702-5724 (American Institute of Physics) · 33 pages. Connections to Quantum Information Theory
arxiv created 1998/08/30 · openalex publication_date 1999/11/01 · arxiv updated 2015/06/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We use the relative modular operator to define a generalized relative entropy for any convex operator function g on (0,∞) 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 (parametrized 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.