Phase Matching Quantum Key Distribution based on Single-Photon Entanglement

Abstract

Two time-reversal quantum key distribution (QKD) schemes are the quantum entanglement based device-independent (DI)-QKD and measurement-device-independent (MDI)-QKD. The recently proposed twin field (TF)-QKD, also known as phase-matching (PM)-QKD, has improved the key rate bound from O(η) to O[Formula: see text] with η the channel transmittance. In fact, TF-QKD is a kind of MDI-QKD but based on single-photon detection. In this paper, we propose a different PM-QKD based on single-photon entanglement, referred to as single-photon entanglement-based phase-matching (SEPM)-QKD, which can be viewed as a time-reversed version of the TF-QKD. Detection loopholes of the standard Bell test, which often occur in DI-QKD over long transmission distances, are not present in this protocol because the measurement settings and key information are the same quantity which is encoded in the local weak coherent state. We give a security proof of SEPM-QKD and demonstrate in theory that it is secure against all collective attacks and beam-splitting attacks. The simulation results show that the key rate enjoys a bound of O[Formula: see text] with respect to the transmittance. SEPM-QKD not only helps us understand TF-QKD more deeply, but also hints at a feasible approach to eliminate detection loopholes in DI-QKD for long-distance communications.

Figure 1

Schematic diagram of SEPM-QKD. An untrusted third party, Charlie, generates single-photon entanglement, by injecting a photon from a heralded single-photon source into a beam splitter. Alice and Bob generate a local weak coherent (WC) state $|\gamma e^{i\left({\phi }_{a(b)}+k_{a(b)}\pi \right)}\rangle$ with ${\phi }_{a(b)}\in \left\{-\frac{\pi }{4},0,\frac{\pi }{4},\frac{\pi }{2}\right\}$ and $k_{a(b)}\in \left\{0,1\right\}$ to test the quantum nonlocal correlation in wave space and generate the final key. ${\phi }_{a(b)}$ is a random phase used to construct Bell inequality and is also used for phase matching measurement. Random bit $k_{a(b)}$ can be regarded as the measurement setup for homodyne detection of wave-state.

Figure 2

Simulation of SEPM-QKD under intensities of local coherent light. The key rate decreases with increasing attenuation of the coherent light intensity whereas the transmission distance increases as the attenuation increases. When the average photon-number of the coherent state is far less than 1, the key rate is approximately proportional to the square of the amplitude of the coherent state according to Eq. 14. For coherent states with high intensity, the proportion of the particle-like correlation between Alice and Bob will also increase (Eq. 5). This will increase the bit error rate of the final key, so the transmission distance will be reduced. In addition, there are two more key rate curves, orange and purple dotted lines, which correspond to the fitting results without considering beam-splitting (BS) attacks. It can be found that the BS attack will have an important effect on the key rate at long transmission distance.

Figure 3

Key rate comparison between different QKD protocols. The simulation results of the other QKD are taken from refs^,. Compared with single-photon based BB84- and MDI-QKD schemes which obey the PLOB bound by Pirandola et al. (PLOB bound), SEPM-QKD has the same $\sqrt{\eta }$ dependence on transmission distance as PM-QKD and TF-QKD which obey the single-repeater bound. In BB84- and MDI-QKD protocols, the carrier of information is a single-photon, and the detection probability is proportional to the transmission coefficient $\eta$, so the key rate has a $\eta$ on the transmission distance. While for PM- and SEPM-QKD protocols, the carrier of information is a wave-photon, and the detection probability is proportional to the square root of the transmission coefficient $\sqrt{\eta }$, so the key rate has a $\sqrt{\eta }$ on the transmission distance. In the simulation of SEPM-QKD, the amplitude of coherent state is $\gamma =0.001$. Its average intensity is several orders of magnitude lower than that in other QKD protocols and BS attack is also considered in SEPM-QKD, which results in the key rate of SEPM-QKD being much lower than that of other QKD protocols.

Figure 4

Schematic diagram of BS attack. Suppose that the transmission loss of a single-photon state is captured and stored by Eve in BS attack scheme. In this attack scheme, Eve synchronizes his light source with Alice and Bob’s. After Alice and Bob publicly announce random phase values, selection of measurement bases and response results of detectors, Eve uses the same measurement method to measure the stored photon states. Eve finally infers Alice and Bob’s keys based on his measurement result $E_{A,B}$.