  We introduce a new kind of quantum measurement that is defined to be
symmetric in the sense of uniform Fisher information across a set of parameters
that injectively represent pure quantum states in the neighborhood of a
fiducial pure state. The measurement is locally informationally
complete---i.e., it uniquely determines these parameters, as opposed to
distinguishing two arbitrary quantum states---and it is maximal in the sense of
a multi-parameter quantum Cramer-Rao bound. For a $d$-dimensional quantum
system, requiring only local informational completeness allows us to reduce the
number of outcomes of the measurement from a minimum close to but below $4d-3$,
for the usual notion of global pure-state informational completeness, to
$2d-1$.
