Monotonicity properties of certain measures over the two-level quantum systems

We demonstrate - using the case of the two-dimensional quantum systems - th at the 'natural measure in the space of density matrices rho describing N-d imensional quantum systems' proposed by Zyczkowski et al. [Phys. Rev. A 58 (1998), 883] does not belong to the class of normalized volume elements of monotone metrics on the quantum systems. Such metrics possess the statistic ally important property of being decreasing under stochastic mappings (coar se-grainings). We do note that the proposed natural measure (and certain ev ident variations upon it) exhibit quite specific monotonicity characteristi cs, but not of the form required for membership in that distinguished class .