Positive contraction mappings for classical and quantum Schrödinger systems
The classical Schrödinger 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 noncommutative 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 puremore »
 Authors:

^{[1]};
^{[2]}
 Department of Electrical and Computer Engineering, University of Minnesota, 200 Union Street S.E. Minneapolis, Minnesota 55455 (United States)
 Dipartimento di Matematica, Università di Padova, via Trieste 63, 35121 Padova (Italy)
 Publication Date:
 OSTI Identifier:
 22403123
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 3; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ENTROPY; MAPS; MARKOV PROCESS; MATHEMATICAL SOLUTIONS; METRICS; SPACE; TRANSFORMATIONS