# Efficient High-Dimensional Quantum Key Distribution with Hybrid Encoding
## 4 Security Analysis
The dark count rate, \(\
u\), is assumed as \(10^{-5}\) per pulse in Figure 4b. In the plot, it is shown that a secret key rate becomes higher, as the dimension of quantum states used in \(d\)-QKD with hybrid encoding increases in low transmission loss regime. When the transmission loss is high, the secret key rate decreases more rapidly as \(d\) increases. QKD with hybrid encoding using higher dimensional quantum states is more influenced by the dark count of detectors, since the number of the single photon detector used in \(d\)-QKD with hybrid encoding is lager than the original DDI-QKD. Therefore, when a single photon is lost, the error rate of QKD with hybrid encoding using higher dimensional quantum states increases rapidly compared with that of the original DDI-QKD. Now, we compare \(3d\)-QKD with hybrid encoding with MDI-QKD using 3-dimensional quantum states (\(3d\)-MDI-QKD). \(3d\)-MDI-QKD was proposed to increase a secret key rate of original MDI-QKD [38]. In its key rate analysis, it is assumed that 3-dimensional BSM used in \(3d\)-MDI-QKD includes six single photon detectors and the 3-dimensional BSM setup can discriminate only three 3-dimensional Bell states among nine ones. Figure 5 shows the secret key rate of \(3d\)-MDI-QKD (red dashed line) and \(3d\)-QKD with hybrid encoding (black solid line). Secret key rate per total pulse can be obtained from (signal sifting rate) \(\times\) (secret key rate per sifted key \(r\)). The signal sifting rate is obtained from (the probability that [PERSON] and [PERSON] used the same bases) in \(d\)-QKD with hybrid encoding, and (the probability that [PERSON] and [PERSON] used the same bases) \(\times\) (the success probability of a BSM) in MDI-QKD. Since a success probability of BSM with linear optics cannot be 100% [18; 19], MDI-QKD always has a lower secret key rate per total signal than prepare-and-measure QKD protocols and QKD with hybrid encoding. Furthermore, it was proven that a generalized BSM in a high-dimensional two-photon state cannot be implemented by means of linear optical elements [19]. The scheme using multi-photon interference with linear optics can be adopted to implement MDI-QKD using qudits [81], however, the secret key rate \(R\) of the protocol is always lower than original MDI-QKD, since the signal sifting rate of the protocol is given as \(1/(2d^{2})\). There is another scheme in which the ideal signal sifting rate can reach \(1/(2d)\) by exploiting nonlinear effects, however, because of the nonlinearity, experimental efficiency of the scheme is much lower than that of the setup with linear optical elements [82]. Also, it was shown that a secret key rate of MDI-QKD using qudits (\(d>4\)) cannot exceed that of qubit MDI-QKD at low error rate even if a signal sifting rate of a \(d\)-dimensional BSM setup reaches \(1/(2d)\)[82]. Therefore,  
Figure 5: The secret key rate of \(3d\)-measurement-device-independent QKD (MDI-QKD) (red dashed line) and \(3d\)-QKD with hybrid encoding (black solid line). Plot of the secret key rate \(R\) (bits/total pulse) vs. transmission loss \(\eta\) (dB). The secret key rate per total signal is obtained from (the secret key rate per sifted key)\(\times\)(the signal sifting rate). Details are described in maintext. Dark count rate of single photon detectors is assumed as \(10^{-5}\) per pulse. it can be claimed that QKD with hybrid encoding is more suitable to exploit qudits than MDI-QKD in its implementation, although it needs additional assumptions to guarantee the security level of MDI-QKD. Here, we compare key generation efficiency of \(d\)-QKD with hybrid encoding with that of existing \(d\)-QKDs. First, compared with entanglement-based \(d\)-QKDs [43; 44; 45], our protocol has an advantage in that generation of an entangled state is not necessary. A high-dimensional time-energy entangled state is generated from spontaneous parametric down-conversion (SPDC), and entangled state generation efficiency of SPDC is not comparable with a single photon OAM mode encoder.