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In the first part of this paper we generalize the result of Georgiou-Pavonthat a positive square matrix can be scaled uniquely to a column stochasticmatrix which maps a given positive probability vector to another given positiveprobability vector. In the second part of this paper we prove that a positivequantum channel can be scaled to another positive quantum channel which maps agiven positive definite density matrix to another given positive definitedensity matrix using Brower's fixed point theorem. This result proves theGeorgiou-Pavon conjecture for two positive definite density matrices in theirrecent paper. We show uniqueness of fixed points for certain two positivedefinite density matrices.

On Schrodinger's bridge problem