Device-independent quantum key distribution based on measurement inputs

We provide an analysis of a family of device-independent quantum key distribution (QKD) protocols that has the following features. (a) The bits used for the secret key do not come from the results of the measurements on an entangled state but from the choices of settings. (b) Instead of a single security parameter (a violation of some Bell inequality) a set of them is used to estimate the level of trust in the secrecy of the key. The main advantage of these protocols is a smaller vulnerability to imperfect random number generators made possible by feature (a). We prove the security and the robustness of such protocols. We show that using our method it is possible to construct a QKD protocol which retains its security even if the source of randomness used by communicating parties is strongly biased. As a proof of principle, an explicit example of a protocol based on the Hardy's paradox is presented. Moreover, in the noiseless case, the protocol is secure in a natural way against any type of memory attack, and thus allows one to reuse the device in subsequent rounds. We also analyze the robustness of the protocol using semidefinite programming methods. Finally, we present a postprocessing method, and observe a paradoxical property that rejecting some random part of the private data can increase the key rate of the protocol.

Figure 1

Comparison of guessing probabilities of key values certified with the protocols using Hardy paradox (solid blue line) and CHSH inequality (dotted black line) in the case when the distribution of settings is biased. $\epsilon$ refers to the bias defined in Sec. 3.

Figure 2

Functions ${\mathrm{\Gamma }}_0\left(\mathbf{h}\right)$ and ${\mathrm{\Gamma }}_1\left(\mathbf{h}\right)$ for uniform and nonuniform (see Sec. 7 B 2) distribution of settings. These functions give upper bounds for values of $P(A=0|a=0,b=0)$ and $P(A=1|a=0,b=0)$ for $\mathbf{h}$ given by (16).

Figure 3

Solutions of the program (17) for cases with uniform and nonuniform (see Sec. 7 B 2) distribution of settings.

Figure 4

Values of conditional entropies of the setting of Alice given the setting of Bob, $H\left(A\right|B)$, if outcomes on both sides were equal to 0. The cases with uniform and nonuniform distribution of settings, and with and without dropping strategy, are considered. The $\eta$ parameter refers to the state given by Eq. (13). In the case with nonuniform distribution, the line referring to use of dropping strategy is slightly above the one without dropping.

Figure 5

Comparison of key rates in different scenarios.