Transformations on quantum states form a basic building block of every quantum information system. From photonic polarization to two-level atoms, complete sets of quantum gates for a variety of qubit systems are well known. For multi-level quantum systems beyond qubits, the situation is more challenging. The orbital angular momentum modes of photons comprise one such high-dimensional system for which generation and measurement techniques are well-studied. However, arbitrary transformations for such quantum states are not known. Here we experimentally demonstrate a four-dimensional generalization of the Pauli X-gate and all of its integer powers on single photons carrying orbital angular momentum. Together with the well-known Z-gate, this forms the first complete set of high-dimensional quantum gates implemented experimentally. The concept of the X-gate is based on independent access to quantum states with different parities and can thus be easily generalized to other photonic degrees-of-freedom, as well as to other quantum systems such as ions and superconducting circuits.Introduction -High-dimensional quantum states have recently attracted increasing attention in both fundamental and applied research in quantum mechanics [1][2][3][4][5]. The possibility of encoding vast amounts of information on a single photon makes them particularly interesting for large-alphabet quantum communication protocols [6][7][8][9], as well as for investigating fundamental questions concerning local realism or quantum contextuality [10,11]. The temporal and spatial structure of a photon provides a natural multi-mode state space in which to encode quantum information. The orbital angular momentum (OAM) modes of light [12] comprise one such basis of spatial modes that has emerged as a popular choice for experiments on high-dimensional quantum information [13]. While techniques for the generation and measurement of photonic quDits carrying OAM are well known [14][15][16], efficient methods for their control and transformation remain a challenge. No general recipe is known so far, and experimentally feasible techniques are known only for special cases.Here we experimentally demonstrate a fourdimensional X-gate and all of its integer powers with the orbital angular momentum modes of single photons. The four-dimensional X-gate is a generalization of the two-dimensional σ x Pauli transformation and acts as a cyclic ladder operator on a four-dimensional Hilbert space. The cyclic transformation required for this gate was designed through the use of the computer algorithm MELVIN [17] and recently demonstrated with classical states of light [18]. The Z-gate for OAM quDits (the generalization of the two-dimensional σ z Pauli transformation) introduces a mode-dependent phase,

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Recent studies demonstrated that the optical channels encoded by Orbital Angular Momentum (OAM) are capable candidates for improving the next generation of communication systems. OAM states can enhance the capacity and security of high-dimensional communication channels in both classical and quantum regimes based on optical fibre and free space. Hence, fast and precise control of the beams encoded by OAM can provide their commercial applications in the compatible communication networks. Integrated optical devices are good miniaturized options to perform this issue. This paper proposes a numerically verified integrated high-frequency electro-optical modulator for manipulation of the guided modes encoded in both OAM and polarization states. The proposed modulator is designed as an electro-optically active Lithium Niobate (LN) core photonic wire with silica as its cladding in a LN on Insulator (LNOI) configuration. It consists of two successive parts; a phase shifter to reverse the rotation handedness of the input OAM state and a polarization converter to change the horizontally polarized OAM state to the vertically polarized one. It is shown that all four possible output polarization-OAM encoded states can be achieved with only 6 V and 7 V applied voltages to the electrodes in the two parts of the modulator.In the past decades, interests in exploiting Orbital Angular Momentum (OAM) as a new degree of freedom for encoding the information in optical communication channels have been enhanced 1 . Utilizing this technique in classical communication based on both optical fibre 2 and free space 3-7 , and also in quantum communication 8-10 demonstrated a promising increased data transmission rate for the future networks. In addition, secure transmission is one of the concerns of communication systems. Besides the classical cryptography techniques, Quantum Cryptography (QC) improved the security between authorized partners connected by a quantum channel. Quantum Key Distribution (QKD) protocols are used intensively for approaches of QC. As an example, BB84 QKD protocol employs four states belonging to two conjugate bases (such as horizontal, vertical, left circular and right circular polarization states) for data encoding 11,12 . By combining polarization with OAM, generation of a rotation invariant qubit is possible. In this case, QKD is proved to be independent of alignment 13,14 . This toolbox can be a good choice for free-space optical and quantum communication with moving objects, such as satellites or flying platforms. The rotation invariant qubit in QKD has been operated experimentally over a distance of 210 m 15 . Moreover, the possibility of using OAM together with polarization for encoding the high-dimensional QKD states improves noise resistance and increase the content of information carried by each photon 16 . Carrier modes with phase term of exp(ilφ) have OAM corresponding to lℏ; per photon. In this term l (l = 0, ±1, ±2, …) is the topological charge of the mode and its sign determines the rotation handedness...

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