Interplay of nonlocality and incompatibility breaking qubit channels

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.

Figure 1

The region below the dashed (red) and solid (black) curve in (a), (b), (c), and (d) corresponds to M-NBC and S-NBC given by Eqs. (35), (37), (39), and (41), respectively, plotted against the (dimensionless) state coefficients. The pairwise incompatibility breaking condition (9) pertains to all points below the horizontal dashed line.

Figure 2

The range of channel parameter $\eta \in (\frac{1}{\sqrt{2}},\frac{3}{\sqrt{17}})$ in which it is S-NBC (for the $W$ state) but not CHSH-NBC.

Figure 3

Singular values corresponding to a sample of $5\times {10}^6$ randomly generated states of the form $\left(47\right)$ plotted with respect to the dimensionless parameter $\eta =\sqrt{{\eta }_x^2+{\eta }_y^2+{\eta }_z^2}$. In terms of these singular values, the conditions for establishing the Mermin and Svetlichny nonlocal correlations are given in Definition 3 which demands that the largest singular value be greater than $1/\sqrt{2}$ and 1, respectively. The corresponding bounds on the unsharpness parameters ${\eta }_M$ and ${\eta }_S$ trivially follow and are summarized in the text.