  Open questions from Sarovar and Milburn (2006 J.Phys. A: Math. Gen. 39 8487)
are answered. Sarovar and Milburn derived a convenient upper bound for the
Fisher information of a one-parameter quantum channel. They showed that for
quasi-classical models their bound is achievable and they gave a necessary and
sufficient condition for positive operator-valued measures (POVMs) attaining
this bound. They asked (i) whether their bound is attainable more generally,
(ii) whether explicit expressions for optimal POVMs can be derived from the
attainability condition. We show that the symmetric logarithmic derivative
(SLD) quantum information is less than or equal to the SM bound, i.e.\
$H(\theta) \leq C_{\Upsilon}(\theta)$ and we find conditions for equality. As
the Fisher information is less than or equal to the SLD quantum information,
i.e. $F_M(\theta) \leq H(\theta)$, we can deduce when equality holds in
$F_M(\theta) \leq C_{\Upsilon}(\theta)$. Equality does not hold for all
channels. As a consequence, the attainability condition cannot be used to test
for optimal POVMs for all channels. These results are extended to
multi-parameter channels.
