Scalable Bell Inequalities for Qubit Graph States and Robust Self-Testing

Bell inequalities constitute a key tool in quantum information theory: they not only allow one to reveal nonlocality in composite quantum systems, but, more importantly, they can be used to certify relevant properties thereof. We provide a general construction of Bell inequalities that are maximally violated by the multiqubit graph states and can be used for their robust self-testing. Apart from their theoretical relevance, our inequalities offer two main advantages from an experimental viewpoint: (i) they present a significant reduction of the experimental effort needed to violate them, as the number of correlations they contain scales only linearly with the number of observers; (ii) numerical results indicate that the self-testing statements for graph states derived from our inequalities tolerate noise levels that are met by present experimental data. We also discuss possible generalizations of our approach to entangled states whose stabilizers are not tensor products of Pauli matrices. Our work introduces a promising approach for the certification of complex many-body quantum states.

Figure 1

Two examples of graphs: (a) the star graph and (b) the ring graph.

Figure 2

Numerical estimations of the fidelity with the target graph state as a function of the relative observed violation $(\beta -{\beta }_C)/({\beta }_Q-{\beta }_C)$ of the corresponding Bell inequalities (6) and (7). The plots show the case of a GHZ state (left) and ring graph state (right) of $N=3,\dots ,7$ particles (see also Appendix C of the Supplemental Material [34] for more details on the numerical procedure). The numerical results are plotted until the highest value of fidelity with a separable $N$-partite state, taken from Ref. [41]. Indeed, after that threshold, the device-independent fidelity estimation would not witness any nonclassicality in the state.