Purification of quantum trajectories in infinite dimensions

In this work we exhibit a class of examples that show that the characterization of purification of quantum trajectories in terms of ‘dark’ subspaces that was proved for finite dimensional systems ( Infin. Dimens. Anal. Quantum Probab. Relat. Top. , 06(02), 223-243, 2003 and IMS Lectures Notes-Monograph Series , 48, 252-261, 2006) fails to hold in infinite dimensional ones. Moreover, we prove that the new phenomenon emerging in our class of models and preventing purification to happen is the only new possibility that appears in infinite dimensional systems. Our proof strategy points out that the presence of new phenomena in infinite dimensional systems is due to the fact that the set of orthogonal projections is not sequentially compact. Having in mind this insight, we are able to prove that the finite dimensional result extends to a class of infinite dimensional models.