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In the present paper we show a link between bistochastic quantum channels andclassical maps. The primary goal of this work is to analyse the multiplicativestructure of the Birkhoff polytope of order 3 (the simplest non-trivial case).A suitable complex parametrization of the Birkhoff polytope is proposed, whichreveals several its symmetries and characteristics, in particular: (i) thestructure of Markov semigroups inside the Birkhoff polytope, (ii) the relationbetween the set of Markov time evolutions, the set of positive definitematrices and the set of divisible matrices. A condition for Markov timeevolution of semigroups in the set of symmetric bistochastic matrices is thenderived, which leads to an universal conserved quantity for all Markovevolutions. Finally, the complex parametrization is extended to the Birkhoffpolytope of order 4.

Dynamical semigroups in the Birkhoff polytope of order 3 as a tool for analysis of quantum channels