Efficient measurement-device-independent quantum key distribution without vacuum sources

We present an efficient decoy-state protocol measurement-device-independent quantum key distribution without vacuum source, with the heralded single-photon source (HSPS) and global optimization. This protocol is able to give a high key rate and avoid the security problem caused by the vacuum source or extremely weak source. In the protocol, the heralded outcome pulse pairs in $X$ basis are used as decoy pairs and all outcome pairs in $Z$ basis are used as signal pairs. Each side uses three different intensities for the outcome pulses, two in $X$ basis and one in $Z$ basis. In order to efficiently optimize the key rate globally, we present an efficient method to estimate what values we will observe by using the HSPS. This efficient method greatly reduces the computation complexity from the fourfold summation to a single-fold summation in the numerical simulation and makes it operable to seek the optimization points globally.

Figure 1

Schematic of the setup for MDIQKD. BS: beam splitter; PBS: polarization beam splitter; D1–D4: single-photon detectors in the measurement station of Charlie. In preparing a pulse in $X$ basis, i.e., sources $x_A,y_A$ ($x_B,y_B$), Alice (Bob) takes postselection to her (his) $T$-mode field depending on the outcome of local detection to mode $L$ done in box AO (BO). Only the heralded $T$-mode pulses are postselected for $X$-basis pulses. In preparing a pulse in $Z$ basis, no postselection is taken to the $T$ mode.

Figure 2

Optimized key rates (per pulse pair) vs transmission distance with HSPS sources and WCS sources with devices' parameters given by row $a$ of Table 2. The total number of pulses at each side is $N_t={10}^9$ here.

Figure 3

Optimized key rates (per pulse pair) vs transmission distance with HSPS sources and WCS sources with devices' parameters given by row $b$ of Table 2. The total number of pulses at each side is $N_t={10}^{10}$ here.

Figure 4

Optimized key rates (per pulse pair) vs transmission distance with HSPS sources and WCS sources with devices' parameters given by row $b$ of Table 2. The total number of pulses at each side is $N_t={10}^{13}$ here.

Figure 5

Optimized key rates (per pulse pair) vs transmission distance using HSPS sources and WCS sources in the asymptotic and nonasymptotic regime. The devices' parameters are given by row $c$ of Table 2 except $N_t$ for the asymptotic case.

Figure 6

Optimized key rates with simple treatment to nonasymptotic effects. The devices' parameters are given by row $b$ of Table 2.