NON-COMMUTATIVE EXTENSION OF INFORMATION GEOMETRY II

摘要

The Fisher information provides a canonical Riemannian metric in the geometric approach to classical statistics. It seems that the quantum analogue of the Fisher information is not uniquely defined, and it is necessary to study the possible candidates and to compare them on physical grounds. Description of monotone metrics under coarse graining has been given by Petz and this class of metrics fixes many candidates. Here we show that the skew information I_p(p,K) ≡ -1/2Tr[p~p,K}[p~(1-p),K] first introduced by Wigner, Yanase and Dyson (WYD) many years ago yields a monotone metric for all values of p; - 1 ≤ P ≤ 2(for p = 0,1 under a proper limiting procedure and beyond the limits with a change of the sign of I_p). Furthermore, we argue that the symmetry between I_p and I_(1-p) is identical to the quantum version of Amari's duality concept for smooth statistical manifolds.