We fully describe the doubly stochastic orbit of a self-adjoint element in noncommutative $$L_1$$ -space affiliated with semifinite von Neumann algebra, which answers problem posed by Alberti and Uhlmann (Stochasticity partial order: maps unitary mixing. VEB Deutscher Verlag der Wissenschaften, Berlin, 1982) 1980s, extending several results literature. It follows further from our methods that, ...