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

TL;DR

This paper develops the theory of quantum doubly stochastic operators on non-commutative Lp-spaces, including their properties, criteria for compactness, and applications to quantum majorization and interpolation stability.

Contribution

It introduces and systematically studies quantum doubly stochastic operators, providing new characterizations, examples, and applications in quantum information theory and operator algebras.

Findings

Characterized strict norm inequalities for quantum doubly stochastic operators.

Provided necessary and sufficient conditions for compactness in Schatten-ideals.

Presented new examples of such operators in finite and infinite dimensions.

Abstract

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.