  The quantum Cram\'{e}r-Rao bound (QCRB) as the ultimate lower bound for
precision in quantum parameter estimation is only known to be saturable in the
multiparameter setting in special cases and under conditions such as full or
average commutavity of the symmetric logarithmic derivatives (SLDs) associated
with the parameters. Moreover, for general mixed states, collective
measurements over infinitely many identical copies of the quantum state are
generally required to attain the QCRB. In the important and experimentally
relevant single-copy scenario, a necessary condition for saturating the QCRB in
the multiparameter setting for general mixed states is the so-called partial
commutativity condition on the SLDs. However, it is not known if this condition
is also sufficient. This paper establishes necessary and sufficient conditions
for saturability of the multiparameter QCRB in the single-copy setting in terms
of the commutativity of a set of projected SLDs and the existence of a unitary
solution to a system of nonlinear partial differential equations. New necessary
conditions that imply partial commutativity are also obtained, which together
with another condition become sufficient. Moreover, when the sufficient
conditions are satisfied an optimal measurement saturating the QCRB can be
chosen to be projective and explicitly characterized. An example is developed
to illustrate the case of a multiparameter quantum state where the conditions
derived herein are satisfied and can be explicitly verified.
