Interplay of nonlocality and incompatibility breaking qubit channels

Abstract

Incompatibility and nonlocality are not only of foundational interest but also act as important resources for quantum information theory. In CHSH (Clauser--Horne--Shimony--Holt) scenario, the incompatiblity of a pair of observables is known to be equivalent to Bell nonlocality. Here, we investigate these notions in the context of qubit channels. The Bell-CHSH inequality has a greater perspective -- compared to any genuine tri-partite nonlocality scenario -- while determining about the interplay between nonlocality breaking qubit channels and incompatibility breaking qubit channels. In Bell CHSH scenario, we prove that if the conjugate of a channel is incompatibility breaking, then the channel is itself nonlocality breaking and the converse also holds provided the channel is unital. However, this equivalence is not straightforwardly generalized to multi-partite systems, due to the absence of an equivalence relation between incompatiblity and nonlocality in the multi-partite scenario. We investigate this relation in tripartite scenario by considering some well known states like GHZ and W states and using the notion of Mermin and Svetlichny nonlocality. By subjecting the parties in question to unital qubit channels, we identify the range of state and channel parameters for which incompatiblity coexists with nonlocality. Further, we identify the set of unital qubit channels that is Mermin/Svetlichny nonlocality breaking \emph{irrespective} of the input state.