Abstract
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 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.
Highlights
In 1931 Erwin Schrodinger published a manuscript on “the reversal of the laws of nature” (“Uber die Umkehrung der Naturgesetze”)
A similar approach allows a constructive proof for matching uniform marginal density matrices via a multiplicative functional transformation of any given prior Kraus map
The standard model for a quantum experiment as well as for a quantum channel is provided by a linear trace-preserving completely-positive (TPCP) map between density matrices
Summary
In 1931 Erwin Schrodinger published a manuscript on “the reversal of the laws of nature” (“Uber die Umkehrung der Naturgesetze”). A similar approach allows a constructive proof for matching uniform marginal density matrices via a multiplicative functional transformation of any given prior Kraus map In essence, this result extends to the non-commutative case a result of Sinkhorn [28, 29] that any strictly positive stochastic matrix can be transformed into a doubly stochastic matrix via a multiplicative functional transformation. The mathematical statement for this special case represents a generalization of a result of Sinkhorn on the existence of doubly stochastic maps to a corresponding quantum probabilistic analog This result is followed by a discussion and a conjecture about the general quantum Schrodinger problem, namely, that a fixed point of a certain map which is used to construct doubly stochastic maps, suitably modified, has a fixed point for general marginal density matrices as well
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