A recent proposal of Sjoqvist et.al. to extend Pancharatnam's criterion for phase difference between two different pure states to the case of mixed states in quantum mechanics is analyzed and the existence of phase singularities in the parameter space of an interference experiment with particles in mixed states pointed out. In the vicinity of such singular points the phase changes sharply and precisely at these points it becomes undefined. A closed circuit in the parameter space around such points results in a measurable phase shift equal to 2nπ, where n is an integer. Such effects have earlier been observed in interference experiments with pure polarization states of light, a system isomorphic to the spin-1/2 system in quantum mechanics. Implications of phase singularities for the interpretation of experiments with partially polarized and unpolarized neutrons are discussed. New kinds of topological phases involving variables representing decoherence (depolarization) of pure states are predicted and experiments to verify them suggested.

4 pages Latex, 1 postscript figure; submitted to Physical Review Letters 12 Dec 2000; Revised on 13 August 2001