Andrew Lesniewski, Mary Beth Ruskai
Monotone Riemannian Metrics and Relative Entropy
on Non-Commutative Probability Spaces
(80K, latex)
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.