  Quantum measurements can be generalized to include complex quantities. It is
possible to relate the quantum weak values of projection operators to the third
order Bargmann invariants. The argument of the weak value becomes, up to a
sign, equal to the Berry's phase associated with the three state vectors. In
case of symmetric informationally complete, positive operator valued measures
(SIC-POVMs), this relation takes a particularly simple form. Alternating strong
and weak measurements can be used to determine Berry's phase directly, which
demonstrates that not only their real and imaginary parts but also moduli and
arguments of weak values have a physical significance. For an arbitrary
projection operator, weak value is real when the projector, pre- and
post-selected states lie on a so-called null phase curve which includes the
geodesic containing the three states as a special case.
