Reference-frame-independent measurement-device-independent quantum key distribution using fewer states

Reference-frame-independent measurement-device-independent quantum key distribution (RFI-MDI-QKD) is immune to both detector side channel attacks and slow drifts of reference frames, and thus possesses a high level of security. In the traditional protocol of RFI-MDI-QKD, each legitimate user, Alice or Bob, has to prepare six encoding states in three orthogonal bases ($X, Y$, and $Z$), which are sent to the untrusted third party, Charlie. Then all the successful counting events are utilized for parameter estimations, which suffer from a severe finite-size effect. In this paper, we propose a scheme on RFI-MDI-QKD with fewer states and demonstrate that it can be secured and made useful by utilizing semidefinite programming. Numerical simulation results reveal the equivalence between those groups of different monitoring states from Alice and Bob. In addition, when considering practical experimental conditions and taking finite-size effects into account, our proposal can achieve a better performance than the traditional six-state RFI-MDI scheme, which provides a resource-saving and simpler state-preparation approach for practical implementations without sacrificing key rates.

Figure 1

The lower bounds $C_L$ obtained by SDP optimization vs the bit-flip rate $e$ in cases where Alice and Bob have different combinations of monitoring states. The performance of $C_L$ remains the same against any rotation of the reference frames ${\beta }_{A\left(B\right)}$.

Figure 2

Key rate vs standard fiber link when using phase-randomized WCS. Comparisons of $G_{66}, G_{46}, G_{36}$, and $G_{44}$ in the asymptotic case with different misalignments of Alice and Bob's reference frames $\mathrm{\Delta }\beta =|{\beta }_B-{\beta }_A|$. The upper solid line and dashed line denote $G_{66}$ and $G_{46}$ in $0^{\circ }$ and ${45}^{\circ }$, respectively, and the lower solid line and dashed line denote $G_{36}$ and $G_{44}$ in $0^{\circ }$ and ${45}^{\circ }$, respectively.

Figure 3

The variations of the key rates with transmission distance for different groups ($G_{66}, G_{46}, G_{44}, G_{36}$) and different rotation angles of the reference frames ($\mathrm{\Delta }\beta$), by accounting for the finite-size effects with $N={10}^{10}$.

Figure 4

(a) Comparison between the key rates of $G_{66}$ and $G_{44}$ vs the total number of photon pulses at a 70-km standard single-mode fiber distance with $\mathrm{\Delta }\beta =0^{\circ }$. (b) The relative difference in key rates of $G_{66}$ and $G_{44}$.

Figure 5

$C$'s value in different groups vs the rotation angle of the reference frame.