  Morozova and Chentsov (Morozova and Chentsov 90) studied Riemannian metrics
on the set of probability measures. They showed that, up to a constant factor,
the Fisher information is the only Riemannian metric which is monotone under
stochastic transformation. Sarovar and Milburn (Sarovar and Milburn 06)
computed an upper bound on the Fisher information for one-parameter channels.
In (O'Loan 07) we extended their bound to an upper bound on the Fisher
information of multi-parameter families of states; we call this the SM quantum
information. Petz and Sud\'ar (Petz 95) characterized fully the set of monotone
metrics on the space of all density matrices. We analyse the SM quantum
information in light of their work. We show that the SM quantum information is
not a well-defined metric on the space of density matrices: different choices
of phase of the eigenvectors lead to different metrics. We define a new metric
C_L as a lower bound among the SM quantum informations. We look at properties
of C_L and show that it is invariant but not monotone.
