Geodesic
On Schrödinger's bridge problem
Sbornik. Mathematics, Tome 208 (2017) no. 11, pp. 1705-1721 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{SM_2017_208_11_a6,
     author = {Sh. Friedland},
     title = {On {Schr\"odinger's} bridge problem},
     journal = {Sbornik. Mathematics},
     pages = {1705--1721},
     year = {2017},
     volume = {208},
     number = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2017_208_11_a6/}
}
TY  - JOUR
AU  - Sh. Friedland
TI  - On Schrödinger's bridge problem
JO  - Sbornik. Mathematics
PY  - 2017
SP  - 1705
EP  - 1721
VL  - 208
IS  - 11
UR  - http://geodesic.mathdoc.fr/item/SM_2017_208_11_a6/
LA  - en
ID  - SM_2017_208_11_a6
ER  - 
%0 Journal Article
%A Sh. Friedland
%T On Schrödinger's bridge problem
%J Sbornik. Mathematics
%D 2017
%P 1705-1721
%V 208
%N 11
%U http://geodesic.mathdoc.fr/item/SM_2017_208_11_a6/
%G en
%F SM_2017_208_11_a6
Sh. Friedland. On Schrödinger's bridge problem. Sbornik. Mathematics, Tome 208 (2017) no. 11, pp. 1705-1721. http://geodesic.mathdoc.fr/item/SM_2017_208_11_a6/

[1] G. Birkhoff, “Extensions of Jentzsch's theorem”, Trans. Amer. Math. Soc., 85:1 (1957), 219–227 | DOI | MR | Zbl

[2] R. A. Brualdi, “Convex sets of non-negative matrices”, Canad. J. Math, 20 (1968), 144–157 | DOI | MR | Zbl

[3] R. A. Brualdi, S. V. Parter, H. Schneider, “The diagonal equivalence of a nonnegative matrix to a stochastic matrix”, J. Math. Anal. Appl., 16:1 (1966), 31–50 | DOI | MR | Zbl

[4] S. Friedland, Matrices – algebra, analysis and applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016, xii+582 pp. | MR | Zbl

[5] S. Friedland, S. Karlin, “Some inequalities for the spectral radius of non-negative matrices and applications”, Duke Math. J., 42:3 (1975), 459–490 | DOI | MR | Zbl

[6] T. T. Georgiou, M. Pavon, “Positive contraction mappings for classical and quantum Schrödinger systems”, J. Math. Phys., 56:3 (2015), 033301, 24 pp. | DOI | MR | Zbl

[7] L. Gurvits, “Classical complexity and quantum entanglement”, J. Comput. System Sci., 69:3 (2004), 448–484 | DOI | MR | Zbl

[8] C. B. Mendl, M. M. Wolf, “Unital quantum channels – convex structure and revivals of Birkhoff's theorem”, Comm. Math. Phys., 289:3 (2009), 1057–1086 | DOI | MR | Zbl

[9] M. V. Menon, “Matrix links, an extremization problem, and the reduction of a non-negative matrix to one with prescribed row and column sums”, Canad. J. Math, 20 (1968), 225–232 | DOI | MR | Zbl

[10] J. W. Milnor, Topology from the differentiable viewpoint, Princeton Landmarks Math., 2nd rev. ed., Princeton Univ. Press, Princeton, NJ, 1997, xii+64 pp. | MR | MR | Zbl | Zbl

[11] H. Perfect, L. Mirsky, “The distribution of positive elements in doubly-stochastic matrices”, J. London Math. Soc., 40 (1965), 689–698 | DOI | MR | Zbl

[12] E. Schrödinger, “Über die Umkehrung der Naturgesetze”, Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Kl., 1931:8-9 (1931), 144–153 | Zbl

[13] E. Schrödinger, “Sur la théorie relativiste de l'électron et l'interprétation de la mécanique quantique”, Ann. Inst. H. Poincaré, 2:4 (1932), 269–310 | MR | Zbl

[14] R. Sinkhorn, “A relationship between arbitrary positive matrices and doubly stochastic matrices”, Ann. Math. Statist., 35:2 (1964), 876–879 | DOI | MR | Zbl

[15] R. D. Sinkhorn, P. J. Knopp, “Concerning nonnegative matrices and doubly stochastic matrices”, Pacific J. Math., 21:2 (1967), 343–348 | DOI | MR | Zbl