Geodesic
On Schr\"odinger's bridge problem
Sbornik. Mathematics, Tome 208 (2017) no. 11, pp. 1705-1721

Voir la notice de l'article provenant de la source Math-Net.Ru

In the first part of this paper we generalize Georgiou-Pavon's result that a positive square matrix can be scaled uniquely to a column stochastic matrix which maps a given positive probability vector to another given positive probability vector. In the second part we prove that a positive quantum channel can be scaled to another positive quantum channel which maps a given positive definite density matrix to another given positive definite density matrix using Brouwer's fixed point theorem. This result proves the Georgiou-Pavon conjecture for two positive definite density matrices, made in their recent paper. We show that the fixed points are unique for certain pairs of positive definite density matrices. Bibliography: 15 titles.
Keywords: scaling of matrices, scaling of quantum channels, Schrödinger's bridge problem, fixed points. 
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     author = {Sh. Friedland},
     title = {On {Schr\"odinger's} bridge problem},
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Sh. Friedland. On Schr\"odinger's bridge problem. Sbornik. Mathematics, Tome 208 (2017) no. 11, pp. 1705-1721. http://geodesic.mathdoc.fr/item/SM_2017_208_11_a6/