Alternative schemes for measurement-device-independent quantum key distribution

Practical schemes for measurement-device-independent quantum key distribution using phase and path or time encoding are presented. In addition to immunity to existing loopholes in detection systems, our setup employs simple encoding and decoding modules without relying on polarization maintenance or optical switches. Moreover, by employing a modified sifting technique to handle the dead-time limitations in single-photon detectors, our scheme can be run with only two single-photon detectors. With a phase-post-selection technique, a decoy-state variant of our scheme is also proposed, whose key generation rate scales linearly with the channel transmittance.

Figure 1

A schematic diagram for the path-phase-encoding MDI-QKD scheme. Here, BS stands for 50 : 50 beam splitter and PM stands for phase modulator. Alice and Bob each encodes their qubits by introducing a relative phase shift between their reference and signal beams. The phase shifts are applied to the signal modes using PMs, chosen from the set $\{0,\pi /2,\pi ,3\pi /2\}$. 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. Provided that they use the same phase 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 complement bits.

Figure 2

A schematic diagram of the time-multiplexed MDI-QKD protocol. Alice and Bob each encodes their qubits onto relative phases of two optical modes separated in time, $a_r$, $a_s$, $b_r$ and $b_s$, respectively. The partial BSM, using a 50 : 50 BS, is performed in the relay owned by a possibly untrusted party.

Figure 3

QBER for the MDI-QKD scheme in Fig. 1 with coherent-state sources, conditioned on partial knowledge of the overall phase. The overall QBER, calculated by Eq. (B15), represents the case where no phase information is shared. Curves labeled “W/2(3)-bit comm” represent cases where Alice and Bob postselect states coming from the same phase region, out of $N=4\left(8\right)$ phase bands in Eq. (10). The conditional QBER is calculated numerically using Eq. (B20). No background noise or misalignment is assumed.

Figure 4

Key rate comparison for single-photon and decoy-state MDI-QKD schemes. The setup parameters are listed in Table . The solid line indicates the key rate for the $X$–$Y$-basis-encoding scheme with decoy states plus three-bit communication for overall phase postselection, shown in Eq. (12). The dashed line shows the performance of the original MDI-QKD with $X$–$Z$-basis encoding. The related formulas for simulation can be found in Appendices 6 and 7. The $\mu$'s are optimized for the two decoy-state curves.

Figure 5

Optimal average photon number of coherent state for the $X$–$Y$-basis-encoding scheme (solid line) and the original MDI-QKD scheme (dashed line, $X$–$Z$ encoding) with decoy states. The setup parameters are listed in Table and the related formulas can be found in Appendix 7.