  The symmetric logarithmic derivative Cram\'er-Rao bound (SLDCRB) provides a
fundamental limit to the minimum variance with which a set of unknown
parameters can be estimated in an unbiased manner. It is known that the SLDCRB
can be saturated provided the optimal measurements for the individual
parameters commute with one another. However, when this is not the case the
SLDCRB cannot be attained in general. In the experimentally relevant setting,
where quantum states are measured individually, necessary and sufficient
conditions for when the SLDCRB can be saturated are not known. In this setting
the SLDCRB is attainable provided the SLD operators can be chosen to commute on
an extended Hilbert space. However, beyond this relatively little is known
about when the SLD operators can be chosen in this manner. In this paper we
present explicit examples which demonstrate novel aspects of this condition.
Our examples demonstrate that the SLD operators commuting on any two of the
following three spaces: support space, support-kernel space and kernel space,
is neither a necessary nor sufficient condition for commutativity on the
extended space. We present a simple analytic example showing that the
Nagaoka-Hayashi Cram\'er-Rao bound is not always attainable. Finally, we
provide necessary and sufficient conditions for the attainability of the SLDCRB
in the case when the kernel space is one-dimensional. These results provide new
information on the necessary and sufficient conditions for the attainability of
the SLDCRB.
