Interplay of nonlocality and incompatibility breaking qubit channels

TL;DR

This paper explores the relationship between nonlocality and incompatibility breaking qubit channels, establishing conditions under which these quantum resources coincide or differ, especially in bipartite and tripartite scenarios.

Contribution

It demonstrates the equivalence between nonlocality breaking and incompatibility breaking channels in the CHSH scenario and investigates their interplay in multipartite systems.

Findings

Incompatibility breaking conjugate channels are nonlocality breaking in CHSH scenario

Equivalence between nonlocality and incompatibility breaking does not extend to multipartite systems

Identifies parameter ranges where incompatibility and nonlocality coexist or are broken by channels

Abstract

Incompatibility and nonlocality are not only of foundational interest but also act as important resources for quantum information theory. In the Clauser-Horne-Shimony-Holt (CHSH) scenario, the incompatibility 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 tripartite nonlocality scenario--while determining the interplay between nonlocality breaking qubit channels and incompatibility breaking qubit channels. In the Bell-CHSH scenario, we prove that if the conjugate of a channel is incompatibility breaking, then the channel is itself nonlocality breaking and vice versa. However, this equivalence is not straightforwardly generalized to multipartite systems, due to the absence of an equivalence relation between incompatibility and nonlocality in the multipartite scenario. We investigate this relation in the tripartite scenario by considering some well-known states like Greenberger-Horne-Zeilinger 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 incompatibility coexists with nonlocality. Further, we identify the set of unital qubit channels that is Mermin or Svetlichny nonlocality breaking irrespective of the input state.