Experimenter’s freedom in Bell’s theorem and quantum cryptography

Bell’s theorem states that no local realistic explanation of quantum mechanical predictions is possible, in which the experimenter has a freedom to choose between different measurement settings. Within a local realistic picture the violation of Bell’s inequalities can only be understood if this freedom is denied. We determine the minimal degree to which the experimenter’s freedom has to be abandoned, if one wants to keep such a picture and be in agreement with the experiment. Furthermore, the freedom in choosing experimental arrangements may be considered as a resource, since its lacking can be used by an eavesdropper to harm the security of quantum communication. We analyze the security of quantum key distribution as a function of the (partial) knowledge the eavesdropper has about the future choices of measurement settings which are made by the authorized parties (e.g., on the basis of some quasi-random generator). We show that the equivalence between the violation of Bell’s inequality and the efficient extraction of a secure key—which exists for the case of complete freedom (no setting knowledge)—is lost unless one adapts the bound of the inequality according to this lack of freedom.

Figure 1

Settings in the BBM-CHSH protocol. Alice chooses between two orthogonal measurement directions ${\alpha }_1$ and ${\alpha }_2$, whereas Bob has four different possibilities, namely the same directions as Alice, i.e., ${\beta }_1$ and ${\beta }_2$, as well as two directions rotated by $\pi ∕4$, i.e., ${\beta }_3$ and ${\beta }_4$.

Figure 2

(a) The (measured) CHSH expression $S$ as a function of Eve’s setting knowledge $Q$ (solid line). The CHSH inequality (8) with classical bound $S_{\mathrm{cl}}=2$ (dotted line) is violated for every setting knowledge $Q>Q_{\mathrm{cl}}\approx 0.44$. The quantum bound $S_{\mathrm{qm}}=1+\sqrt{2}$ is also indicated. (b) The mutual information between Alice and Bob $I_{\mathrm{AB}}$ (dashed line) and the actual mutual information between Bob and Eve $I_{\mathrm{BE}}$ (solid line), which is always smaller than (or equal to) the Alice-Eve mutual information. For every setting knowledge $Q$ one has $I_{\mathrm{BE}}⩾I_{\mathrm{AB}}$ and thus Alice and Bob can never extract a secret key. An optimal attack without setting knowledge leads to ${\tilde{I}}_{\mathrm{BE}}$ (dotted line). Only for $Q⩽Q_0\approx 0.63$ the BBM-CHSH protocol is secure, because Alice and Bob find $I_{\mathrm{AB}}⩽{\tilde{I}}_{\mathrm{BE}}$ and they will not use their key.