 9959 Andrew Lesniewski, Mary Beth Ruskai
 Monotone Riemannian Metrics and Relative Entropy
on NonCommutative Probability Spaces
(80K, latex)
Feb 21, 99

Abstract ,
Paper (src),
View paper
(auto. generated ps),
Index
of related papers

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,
tracepreserving maps.
We then define and study the maximal contraction associated
with these quantities.
 Files:
9959.src(
9959.keywords ,
andylesfin.tex )