# Efficient High-Dimensional Quantum Key Distribution with Hybrid Encoding
## 1 Introduction  
Quantum key distribution (QKD) is a novel scheme to distribute a symmetric secret key between two distant authorized parties, [PERSON], and [PERSON], by exploiting quantum mechanical phenomena. The protocol provides an information-theoretic security under potential attacks of a malicious eavesdropper, conventionally called [PERSON]. Since its first seminal proposal called BB84 protocol [1], many relevant or extended versions of QKD protocols have been proposed and studied based on quantum principles [2; 3; 4; 5; 6; 7; 8]. In recent QKD studies, the security defects due to device imperfections have been emerging as an important issue. It has been shown that [PERSON] can hack into the QKD system by exploiting the imperfection of devices. This is known as a side channel attack including photon number splitting (PNS) attack [9], faked-state attack [10], detector efficiency mismatch attack [11], detector blinding attack [12; 13], time-shift attack [14], and laser damage attack [15]. In this background, measurement-device-independent QKD (MDI-QKD) was proposed to overcome the problems coming from imperfections of measurement devices [16]. In MDI-QKD, the Bell state measurement (BSM) of two photons [17] is an essential task. However, the success probability of BSM with linear optics on single photons is upper bounded by 50% [18; 19]. Recent advanced schemes of BSM require multi-photon encoding of 2-dimensional quantum states, called qubits, to beat the limit with linear optics [20; 21; 22; 23; 24]. Then, detector-device-independent QKD (DDI-QKD) was proposed [25; 26; 27] to simplify the scheme of MDI-QKD, exploiting two different degrees-of-freedoms (DoFs) in a single photon and single-photon interference instead of two-photon interference. In its protocol, [PERSON] encodes her information into one DoF of a single photon and sends it to [PERSON], who encodes his information into another DoF of the single photon. The measurement result of the single photon reveals correlation of two DoFs in the single photon. The implementation of DDI-QKD requires only measurements on single photons and is thus less challenging than BSM performed on two photons. As its scheme issimilar to the process of BSM used in MDI-QKD, it was conjectured that DDI-QKD guarantees the same security level with MDI-QKD. However, it has been shown that not all the side channel attacks are protected with DDI-QKD [28; 29], and an assumption of the trusted measurement setup is necessary for ensuring its security. In another branch of QKD research, there has been significant effort to improve the secret key rate, for example, using \(d\)-dimensional quantum states, called qudits. There are several advantages to using qudits as a generalized information carrier. For example, qudits (\(d>2\)) can naturally carry more classical information than qubits. Compared to qubit operations, qudits has been shown to be more robust against quantum cloning (i.e., a possible eavesdropping) [30; 31; 32]. It has been also found that the efficiency of key distribution increases with qudits in an ideal situation [33; 34; 35; 36]. Various high-dimensional QKD protocols have been proposed such as a generalized version of BB84, a multipartite high-dimensional QKD [37], and MDI-QKDs using high-dimensional quantum states [38; 39; 40]. Moreover, QKD protocols using qudits have been implemented experimentally in various quantum system, for instance, energy-time eigenstates [41; 42; 43; 44; 45] and orbital angular momentum (OAM) mode of a single photon [46; 47; 48; 49; 50]. In this article, we propose a schematic configuration of high-dimensional QKD based on hybrid encoding over two different DoFs. We demonstrate that the secret key rate is improved with our scheme over previous 2-dimensional QKD based on two different DoFs of a single photon. We also present its implementation with current optical technologies, by exploiting the OAM mode of a single photon as a high-dimensional information carrier.