Detecting nonlocality of noisy multipartite states with the Clauser-Horne-Shimony-Holt inequality

The Clauser-Horne-Shimony-Holt inequality was originally proposed as a Bell inequality to detect nonlocality in bipartite systems. However, it can also be used to certify the nonlocality of multipartite quantum states. We apply this to study the nonlocality of multipartite Greenberger-Horne-Zeilinger (GHZ), W, and graph states under local decoherence processes. We derive lower bounds on the critical local-noise strength tolerated by the states before becoming local. In addition, for the whole noisy dynamics, we derive lower bounds on the corresponding nonlocal content for the three classes of states. All the bounds presented can be calculated efficiently and, in some cases, provide significantly tighter estimates than with any other known method. For example, they reveal that $N$-qubit GHZ states undergoing local dephasing are, for all $N$, nonlocal throughout all the dephasing dynamics.

Figure 1

Lower bounds on the local content of the GHZ state under parallel dephasing, for $N=3$. (Red) The bound obtained from the MK inequality [38]; (blue) the new bound (19) from the CHSH method; and (black dashed) the value obtained through a numerical optimization described in the main text. For $p>0.18$, the nonlocal content is better described by the bound from the CHSH method.

Figure 2

Lower bounds on the local content of the GHZ state under transversal dephasing. (Blue) Lower bound from the MK inequality of $N=3$; (red) idem for $N=7$; (purple) idem for $N=9$; (black dashed) lower bound (20) from the CHSH method. The bounds from the MK inequality were obtained through numerical optimization over all possible projective measurements. The CHSH bound is analytical and independent of $N$.

Figure 3

Lower bounds on the nonlocal content for star graph state of $N$ qubits and $\left|I\right|=N-1$ (see Appendix for details) and under parallel local dephasing. (Red and blue) The bounds obtained from the inequalities of Ref. [45], for $N=3$ and $N=100$, respectively; (black dashed) the new bound given by the CHSH method. The CHSH bound is size independent and offers an exponentially tighter estimate as $N$ increases.