Positive contraction mappings for classical and quantum Schrödinger systems

The classical Schroedinger bridge seeks the most likely probability law for a diffusion process, in path space, that matches marginals at two end points in time; the likelihood is quantified by the relative entropy between the sought law and a prior. Jamison proved that the new law is obtained through a multiplicative functional transformation of the prior. This transformation is characterised by an automorphism on the space of endpoints probability measures, which has been studied by Fortet, Beurling, and others. A similar question can be raised for processes evolving in a discrete time and space as well as for processes defined over non-commutative probability spaces. The present paper builds on earlier work by Pavon and Ticozzi and begins by establishing solutions to Schrödinger systems for Markov chains. Our approach is based on the Hilbert metric and shows that the solution to the Schrödinger bridge is provided by the fixed point of a contractive map. We approach, in a similar manner, the steering of a quantum system across a quantum channel. We are able to establish existence of quantum transitions that are multiplicative functional transformations of a given Kraus map for the cases where the marginals are either uniform or pure states. As in the Markov chain case, and for uniform density matrices, the solution of the quantum bridge can be constructed from the fixed point of a certain contractive map. For arbitrary marginal densities, extensive numerical simulations indicate that iteration of a similar map leads to fixed points from which we can construct a quantum bridge. For this general case, however, a proof of convergence remains elusive.