Positive contraction mappings for classical and quantum Schrodinger systems

Understand

The classical Schrodinger 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, and the law dictates a controlled path that abides by the specified marginals.

• Schrodinger proved that the optimal steering of the density between the two end points is effected by a multiplicative functional transformation of the prior; this transformation represents an automorphism on the space of probability measures and has since 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 with the problem of steering a Markov chain between given marginals.