Strassen's theorem for quantum couplings

Strassen's theorem for probabilistic couplings is a fundamental theorem in probability theory that can be used to bound the probability of an event in a distribution by the probability of an event in another distribution coupled with the first. It has been widely applied in computer science for analysis of random algorithms, machine learning and verification of security and privacy protocols. We extend the coupling techniques in probability theory to quantum systems. A quantum generalisation of the notion of lifting, a coupling under certain constraints, is introduced. Several interesting examples and basic properties of quantum couplings and liftings are presented. Finally, a quantum extension of Strassen's theorem is established.