  We show how preparation uncertainty relations that are formulated as sums of
variances may be tightened by using the quantum Fisher information to quantify
quantum fluctuations. We apply this to derive stronger angular momentum
uncertainty relations, which in the case of spin-$1/2$ turn into equalities
involving the purity. Using an analogy between pure-state decompositions in the
Bloch sphere and the moment of inertia of rigid bodies, we identify optimal
decompositions that achieve the convex- and concave-roof decomposition of the
variance. Finally, we illustrate how these results may be used to identify the
classical and quantum limits on phase estimation precision with an unknown
rotation axis.
