# Efficient High-Dimensional Quantum Key Distribution with Hybrid Encoding
## 2 Schematic Description  
In this section, we describe a schematic setup of \(d\)-QKD with hybrid encoding. As an example, the schematic setup of 3-dimensional QKD (\(3d\)-QKD) with hybrid encoding is shown in Figure 1. In \(d\)-QKD with hybrid encoding, we exploit OAM mode and multipath mode of a single photon as a information carrier, since the OAM mode is known to be suitable for quantum communication as it is resilient against perturbation effects [52]. As a first step of the protocol, [PERSON] generates \(d\)-dimensional information randomly. Subsequently, [PERSON] randomly chooses a encoding basis between two mutually unbiased bases (MUBs) which are written as \(\{|I_{x}\rangle\}\) and \(\{|I_{x}\rangle\}\) where \(x\in\{0,1,2,...,d-1\}\). The relation between the two MUBs is described as the \(d\)-dimensional discrete Fourier transformation on the \(d\) OAM modes which is shown in Equation (1):  
\[|\vec{l}_{x}\rangle=\frac{1}{d}\sum_{k=0}^{d-1}\omega^{xk}\left|l_{k}\right\rangle \tag{1}\]  
where \(\omega=\exp\left(2\pi i/d\right)\). [PERSON] encodes her \(d\)-dimensional information in OAM modes of a single photon [53]. For example, when \(d=3\), [PERSON]'s classical information \(x_{3}\) would be one of the dimensional integers, \(x_{3}\in\{0,1,2\}\), and she generates a quantum state denoted as \(\left|l_{x_{3}}\right\rangle\) whose OAM value is \((x_{3}-1)\). Subsequently, [PERSON] sends the encoded photon to [PERSON], who encodes his \(d\)-dimensional information in multipath modes of the single photon. [PERSON] also uses two MUBs that are described as \(\{\left|p_{y}\right\rangle\}\) and \(\{\left|p_{y}\right\rangle\}\). \(\left|p_{y}\right\rangle\) denotes a single photon state in the optical path \(p_{y}\) where \(y\in\{0,1,2,...,d-1\}\). Similarly with [PERSON]'s bases, the relation between [PERSON]'s two bases is given as the \(d\)-dimensional discrete Fourier transformation of the \(d\) path modes. Figure 2 shows a schematic setup of [PERSON]'s encoding systems. [PERSON] randomly chooses one basis between path modes and bar path modes, which are MUBs of the path modes. If [PERSON] uses a path mode, he selects one optical path among \(\{p_{0},p_{1},p_{2}\}\) corresponding to his information. If he chooses a bar path mode, he encodes his information by selecting a phases set \(\{B1,B2\}\) in Figure 2b among \(\{1,1\}\), \(\{\omega,\omega^{2}\}\), and \(\{\omega^{2},\omega\}\). After [PERSON]'s encoding, the two qudits encoded in the single photon, which can be written as \(\left|l_{x},p_{y}\right\rangle\), go into a cyclic transformation of OAM modes. A transformed OAM value of the single photon in path \(p_{y}\) is obtained with the following rule: \(x\to x+d-y\) (mod \(d\)). Subsequently, a single photon interference is performed by recombining \(d\)-path via beam splitters that have different transmissivity. The unitary transformation on path modes operated by multi-port interferometer,  
Figure 1: A schematic setup of 3-dimensional quantum key distribution (QKD) with hybrid encoding. [PERSON] uses orbital angular momentum (OAM) modes of a single photon, and [PERSON] controls the phase of each path to encode their information in the single photon. The encoded photon enters into a 3-port interferometer. After single photon interference, a OAM value and existing path of the single photon is measured. SLM: spatial light modulator; BS1: 50:50 beam splitter; BS2: beam splitter of which transmissivity is 1/3; BS3: beam splitter of which transmissivity is 2/3; OAM CT: cyclic transformation of OAM modes. called tritter [54], is defined as the \(d\)-dimensional discrete Fourier transformation on the \(d\) path modes as shown in Equation (2):  
\[\hat{U}_{d}=\frac{1}{\sqrt{d}}