Device-independent state estimation based on Bell’s inequalities

The only information available about an alleged source of entangled quantum states is the amount $S$ by which the Clauser-Horne-Shimony-Holt inequality is violated: nothing is known about the nature of the system or the measurements that are performed. We discuss how the quality of the source can be assessed in this black-box scenario, as compared to an ideal source that would produce maximally entangled states (more precisely, any state for which $S=2\sqrt{2}$). To this end, we present several inequivalent notions of fidelity, each one related to the use one can make of the source after having assessed it, and we derive quantitative bounds for each of them in terms of the violation $S$. We also derive a lower bound on the entanglement of the source as a function of $S$ only.

Figure 1

Device-independent state estimation. The quality of an unknown source of entangled pairs should be established using unknown measurement devices. The only available information is the statistics $P\left(a,b\mid A,B\right)$ of the outcomes $\left(a,b\right)$ for measurement settings $A,B$. This represents the particular case studied in this paper, where both the choice of measurement settings and the outputs are binary.

Figure 2

(Color online) Lower bounds on the fidelity as a function of the observed violation $S_{\text{obs}}$ of the CHSH inequality. From top to bottom: $F_{\text{MY}}$ assuming two qubits (10); $F_{\text{MY}}$ assuming pure states and the Gisin-Peres conjecture (16) equal to $F_{\text{LOCC}}$ (20); and $F_{\text{LO}}$ (19).