# Efficient High-Dimensional Quantum Key Distribution with Hybrid Encoding
## 3 Experimental Implementation  
We investigate a practical implementation of experimental elements that can construct \(d\)-QKD with hybrid encoding. [PERSON] can generate a single photon OAM state by means of a spatial light modulator (SLM) [55]. SLMs usually have a limited frame rate of around 60 Hz, for fast generation of various OAM values, a digital micromirror device(DMD) is more desirable [56]. An OAM sorter based on liquid crystal devices can also generate photonic OAM states [57; 58]. [PERSON]'s path encoding system is realizable with an optical switch over \(d\)-port, and a schematic setup is shown in Figure 2a. [PERSON]'s bar path encoding system can be changed from Figure 2b by using an optical \(d\)-port switch and a \(d\)-port tritter, whose operation on path modes is the \(d\)-dimensional discrete Fourier transformation as shown in Equation (2). With the tritter, [PERSON] can choose bar path mode by selecting an input port of the tritter rather than controlling the phase shifters in Figure 2b. After [PERSON]'s encoding, cyclic transformations of OAM modes are performed in the each port. Figure 3 shows a schematic setup of three-fold OAM cyclic transformation \((+1)\) of OAM values \(\{-1,0,1\}\). The setup consists of OAM holograms, mirrors, beam splitters and OAM beam splitters (OAM BSs). An OAM BS, composed of a Mach-Zehnder interferometer with a Dove prism in each arm, sorts individual photons based on their OAM value [59]. \(\alpha\) is defined from relative angle \(\alpha/2\) between the two Dove prisms and relative phase between photons in the two arms is given by \(\exp(il\alpha)\). The three-fold OAM cyclic transformation \((+1)\) consists of three OAM BSs whose \(\alpha\) are \(\pi\), \(\pi/2\), and \(-\pi\). The first OAM BS (\(\alpha=\pi\)) and the final OAM BS (\(\alpha=-\pi\)) change the direction of propagation of a photon whose OAM value is odd and even, respectively. The second OAM BS (\(\alpha=\pi/2\)) spatially separates photons whose OAM value is 0 and 2. Photons are separated and combined spatially by using the OAM BSs according to their OAM value. With OAM holograms on each arm, the three-fold cyclic transformation of OAM modes \(\{-1,0,1\}\) is accomplished as it is shown in Figure 3. The experimental setup of the four-fold and five-fold cyclic transformation of OAM modes were proposed and demonstrated as well [60; 61; 62; 63]. While theoretical efficiency of the four-fold cyclic transformation is 100%, fidelity of 4-dimensional Bell state transformation using the four-fold cyclic transformation setup was reported as roughly 91.5% due to reflectivity of optical elements and misalignment [61]. Subsequently, \(d\)-port single photon interference is performed by using the tritter shown in Equation (2). The tritter can be implemented with only linear optical elements which are beam splitters, mirrors, and phase shifters. After the interference, an OAM value of the single photon is measured. Direct measurements of an OAM value of a single photon have been studied recently, for instance, by using refractive optical elements that convert OAM modes into transverse momentum modes [64; 65], refractive optical elements that give spatial separation of OAM modes [66], sequential weak and strong measurements [67; 68], spectrum analysis based on the rotational Doppler effect [69], and interferogram analysis with a multipixel camera [70]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Bases** & **Bob’s Operation (\(|\Phi_{3i+j}\rangle\))** \\ \hline bases 1 (\(l_{x}\), \(p_{y}\)) & \(y\to y-i\) (mod 3) \\ \hline \multirow{3}{*}{bases 2 (\(\tilde{l}_{x}\), \(\tilde{p}_{y}\))} & \(1\leftrightarrow 2\) for \(j=0\) \\  & \(0\leftrightarrow 2\) for \(j=1\) \\  & \(0\leftrightarrow 1\) for \(j=2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: An example of [PERSON]’s operation on his encoded information when \(d=3\) and the result of the measurement is \(|\Phi_{3i+j}\rangle\).