Quantum Doubly Stochastic Operators on Non-commutative $L_p$-Spaces

We introduce and systematically develop the theory of \emph{quantum doubly stochastic operators}, i.e. positive, trace-preserving maps on non-commutative $L_p$-spaces associated to semifinite von Neumann algebras. After establishing basic norm and duality properties, we characterize strict norm inequalities, give necessary and sufficient criteria for compactness in the sense of Schatten-ideals, and exhibit a range of new examples in both finite and infinite dimensions. Applications to quantum majorization and stability under interpolation are also discussed.