Relative Phase Shifts for Metaplectic Isotopies Acting on Mixed Gaussian States

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.