  We address in this paper the notion of relative phase shift for mixed quantum
systems. We study the Pancharatnam-Sjoeqvist phase shift for metaplectic
isotopies acting on Gaussian mixed states. We complete and generalize previous
results obtained by one of us while giving rigorous proofs. This gives us the
opportunity to review and complement the theory of the Conley-Zehnder index
which plays an essential role in the determination of phase shifts.
