Statistical fluctuation analysis for measurement-device-independent quantum key distribution

Measurement-device-independent quantum key distribution with a finite number of decoy states is analyzed under finite-data-size assumption. By accounting for statistical fluctuations in parameter estimation, we investigate vacuum+weak- and vacuum+two-weak-decoy-state protocols. In each case, we find proper operation regimes, where the performance of our system is comparable to the asymptotic case for which the key size and the number of decoy states approach infinity. Our results show that practical implementations of this scheme can be both secure and efficient.

Figure 1

A schematic diagram for the MDI-QKD protocol, where PBS stands for polarizing beam splitter and PM stands for phase modulator. In order to encode their bits in the $z$ basis, Alice and Bob generate phase-randomized coherent states with either $H$ or $V$ polarizations at their sources. To encode a bit in the $x$ or $y$ basis, they generate $+45$-polarized signals at their encoders. A PM will introduce a relative phase shift between their reference and signal beams. The phase shifts are chosen from the set $\{0,\pi \}$ for the $x$ basis and $\{\pi /2,3\pi /2\}$ for the $y$ basis. A partial BSM, possibly performed by an untrusted party, Eve or Charlie, on the two reference and the two signal modes would establish correlations between the raw key bits of Alice and Bob. If they both use the $z$ basis, a click on exactly one of the $r$ detectors and exactly one of the $s$ detectors would imply anticorrelated bits shared between Alice and Bob. For $x$ and $y$ bases, if they both use the same basis, a joint click on detectors $r_0$ and $s_0$ implies identical bits for Alice and Bob; so does a joint click on $r_1$ and $s_1$. A joint click on $r_0$ and $s_1$, or, $r_1$ and $s_0$ would imply anticorrelated bits [38].

Figure 2

Upper bounds on the dropped terms in Eq. (10). It can be seen that the effect of dropping higher-order terms is negligible when the number of leading terms kept, $k$, is sufficiently large.

Figure 3

Key rate versus channel transmittance using vacuum+weak-decoy-state method for MDI-QKD.

Figure 4

Key rate versus channel transmittance using vacuum+two-weak-decoy-state method for MDI-QKD.