# Efficient High-Dimensional Quantum Key Distribution with Hybrid Encoding
## 4 Security Analysis
Compared to this, prepare-and-measure QKD protocols using high-dimensional systems are threatened by the first proposal of detector blinding attack [12], and the original DDI-QKD was breached by the combined attack of the detector blinding attack with various blinding power and detector efficiency mismatched attacks even with the random-detector-efficiency protocol [13]. Therefore, we can conclude that the complexity of a successful side channel attack becomes higher by exploiting the proposed protocol compared to prepare-and-measure \(d\)-QKDs and the original DDI-QKD, although the proposed protocol does not provide the detector-device-independent security. It is necessary to analyze security of \(d\)-QKD with hybrid encoding to evaluate the usefulness of the protocol. The analysis of the security is able to be made through the inspection of the equivalent protocol using the entanglement distillation process (EDP) [4; 5; 6]. The idea of the method is that, if [PERSON] and [PERSON] share the maximally entangled state, [PERSON] cannot generate correlation between her state and the shared maximally entangled state of [PERSON] and [PERSON] [77]. In the method, we can analyze the security of the proposed protocol with the amount of distributed maximally entangled states. In order to use the method, an equivalent protocol of which [PERSON] and [PERSON] share an entangled state at the end should be introduced. Note that the equivalent protocol is employed only for the security analysis, so its experimental efficiency is not significant. However, it is important that the equivalent protocol is physically realizable, since any security analysis of QKD should be valid under quantum mechanics. Therefore, we will briefly introduce possible implementations of the equivalent protocol to show that it is physically reasonable. At first, [PERSON] and [PERSON] generate the three-photon entangled state shown in Equation (6):  
\[\left|\Psi\right\rangle_{ABD}=\frac{1}{d}\sum_{m,n=0}^{d-1}\left|l_{m}\right \rangle_{A}\left|l=0,p_{n}\right\rangle_{B}\left|l_{m},p_{n}\right\rangle_{D}, \tag{6}\]  
where the subscript \(A(B)\) means [PERSON]'s ([PERSON]'s) single photon state and the subscript \(D\) means a single photon that goes to tritter and OAM measurement setup. Generation of this state is possible, in principle, by using two cascade spontaneous parametric down-conversion (SPDC) crystals, spatial discrimination elements of the OAM mode, and relabelling of the OAM and path values. For a 4-dimensional system, the generation of 4-dimensional OAM mode entangled states [78] and 4-dimensional path mode entangled states [79; 80] using SPDC crystals was demonstrated. [PERSON] and [PERSON] keep their photons, and [PERSON] measures the photon \(D\) using the measurement setup. Based on the result, [PERSON] performs the corresponding unitary operation to share the maximally entangled state shown in Equation (7) :  
\[\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}\left|l_{k}\right\rangle_{A}\otimes\left|p_{ k}\right\rangle_{B}. \tag{7}\]  
[PERSON] ([PERSON]) chooses her (his) measurement basis randomly between OAM (path) modes and bar OAM (path) modes. After the measurement, [PERSON] and [PERSON] share their measurement bases and discard if the two bases are not matched. If the two bases are matched, their measurement outcomes are always identical if there is no error and no Eve. Since the maximally entangled state is distributed to [PERSON] and [PERSON], security of the protocol becomes the same with that of a \(d\)-dimensional entanglement based QKD. Security of a QKD using \(d\)-dimensional maximally entangled states was studied against individual attacks [33] ([PERSON] monitors state separately), and against collective attacks [34; 35] ([PERSON] monitors several states jointly).