Monotone Riemannian metrics and relative entropy on noncommutative probability spaces

@article{Lesniewski1999MonotoneRM,
  title={Monotone Riemannian metrics and relative entropy on noncommutative probability spaces},
  author={Andrew S. Lesniewski and Mary Beth Ruskai},
  journal={Journal of Mathematical Physics},
  year={1999},
  volume={40},
  pages={5702-5724}
}
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… 
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