# Efficient High-Dimensional Quantum Key Distribution with Hybrid Encoding
## 4 Security Analysis
According to the results, secret key rate of QKD using \(d\)-dimensional quantum states against collective attack is evaluated as shown in Equation (8):  
\[r=\log_{2}d+2Q\log_{2}\left(\frac{Q}{d-1}\right)+\left(1-Q\right)\log_{2}\left( 1-Q\right). \tag{8}\]  
The unit of the secret key rate is (bits/sifted signal). \(Q\) is state error rate obtained from Equation (9):  
\[Q=\sum_{i\
eq j}\left\langle l_{i},p_{j}\right|\rho\left|l_{i},p_{j}\right\rangle, \tag{9}\]where \(\rho\) is the density matrix of the state shared by [PERSON] and [PERSON], and \(i,j\in\{0,1,2,...,d-1\}\). In the ideal case, no error and no Eve, since the distributed state is the state described in Equation (7), the error rate becomes trivial, \(Q=0\). Now, we investigate an improvement of a secret key rate of \(d\)-QKD with hybrid encoding compared with the original DDI-QKD. Secret key rates per sifted signal, \(r\), of \(d\)-QKD with hybrid encoding are plotted in Figure 4. Figure 4a shows the secret key rate of the original DDI-QKD (black dotted line), \(3d\)- (red dashed line), \(4d\)- (blue dot-dashed line), and \(5d\)-QKD with hybrid encoding (orange solid line) in the ideal situation. QKD with hybrid encoding using higher dimensional quantum states has a higher secret key rate than the original DDI-QKD at same error rate, since a quantum system in high-dimension can carry more information per single quanta and qudit has enhanced robustness against an optimal cloning and eavesdropping. In Figure 4b, we simulate secret key rates of \(d\)-QKD with hybrid encoding and the original DDI-QKD with a change of the realistic experimental factors, transmission loss \(\eta\) and dark count rate of single photon detectors. When a photon propagates through an optical fiber or atmosphere, there is transmission loss. So transmission efficiency is approximately proportional to the distance between [PERSON] and [PERSON] that QKD is able to be achieved. For a single photon detector, since it is very sensitive in order to detect a very weak pulse, a single photon, it is possible to be clicked by background noise even if there is no received photon. The event is called dark count. If there is no [PERSON], the probability of the detector click corresponding to the state \(|\Phi_{0}\rangle\) when [PERSON] encodes \(x\) and [PERSON] encodes \(y\) in a single photon is able to be described as follows:  
\[p(x,x) =\frac{1}{d}(1-\eta)(1-\
u)^{(d^{2}-1)}+\eta\
u(1-\
u)^{(d^{2}-1)} \tag{10}\] \[p(x,y) =\eta\
u(1-\
u)^{(d^{2}-1)} \tag{11}\]  
where \(x,y\in\{0,1,2,...,d-1\}\), \(x\
eq y\), \(d\) is the dimension of quantum states used in \(d\)-QKD with hybrid encoding, and \(\
u\) is the dark count rate per pulse. The first term in Equation (10) denotes the case when the single photon arrives at a detector and it triggers off the detector, while there is no dark count in the other detectors. The second term in Equation (10) denotes that the single photon detector is clicked due to the dark count when the single photon is lost in channel and the other detectors are not clicked. In the ideal case, no Eve and no state error, \(p(x,y)\) should be zero since the state cannot be projected on \(|\Phi_{0}\rangle\). The only case that the detector is clicked is that the single photon is lost and the detector is  
Figure 4: The secret key rate of the original detector-device-independent QKD (DDI-QKD) (black dotted line), \(3d\)- (red dashed line), \(4d\)- (blue dot-dashed line), and \(5d\)-QKD with hybrid encoding (orange solid line). (**a**) Plot of the secret key rate \(r\) (bits/sifted pulse) vs. state error rate \(Q\); (**b**) Plot of the secret key rate \(r\) (bits/sifted pulse) vs. transmission loss \(\eta\) (dB). Dark count rate of single photon detectors is assumed as \(10^{-5}\) per pulse. clicked due to the dark count. The error rate in this situation is evaluated from the equation described as follows:  
\[Q=\frac{\sum_{i\
eq j}p(i,j)}{\sum_{x,y=0}^{d-1}p(x,y)}, \tag{12}\]  
where \(i,j\in\{0,1,2,...,d-1\}\).