by R. F. Streater, to appear in Proc. of the RIMS Symposium on the Analytical Study of Quantum Information and Related Fields, Kyoto, 2000.
We survey the Fisher information and the Rao metric, and the Cramér-Rao inequality. The connections of Amari, Efron and Dawid are mentioned, and their quantum versions due to Hasegawa, Nagaoka and Petz. In infinite dimensions, Araki's expansionals for bounded perturbations of the KMS (=Kubo-Martin-Schwinger) Hamiltonian are generalised to perturbations in the sense of quadratic forms. Some analyticity of the free energy remains true for this wider class of states, under conditions made explicit in a subsequent paper.
The article contains a simple proof of the entropy inequality
S(ap+bq) leq aS(p) + bS(q) + a log 1/a + b log 1/b
where p, q are probabilities, and a,b are positive numbers such that a + b
= 1. I now see that the same proof is to be found in
Brattelli and Robinson, Operator Algebras and Quantum Statistical
Mechanics, Vol II, Springer-Verlag, 1981, Prop. 6.2.25.
Recent work on the relation between the curvature and the concept of being more mixed can be seen in a paper by Attila Andai in the archives.
The text of my paper can be viewed in the archives; the local preprint number is KCL-MTH-00-69
This work was also presented at the Nottingham conference, 3-4 May 2000.
Return to my HOME PAGE for links to all my books and papers on mathematical physics.
© by Ray Streater, 23/6/00.