Weak measurement of berry's phase

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