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Unitary transformations are the fundamental building blocks of gates and operations in quantum information processing allowing the complete manipulation of quantum systems in a coherent manner. In the case of photons, optical elements that can perform unitary transformations are readily available only for some degrees of freedom, e.g. wave plates for polarisation. However for high-dimensional states encoded in the transverse spatial modes of light, performing arbitrary unitary transformations remains a challenging task for both theoretical proposals and actual implementations. Following the idea of multi-plane light conversion, we show that it is possible to perform a broad variety of unitary operations when the number of phase modulation planes is comparable to the number of modes. More importantly, we experimentally implement several high-dimensional quantum gates for up to 5-dimensional states encoded in the full-field mode structure of photons. In particular, we realise cyclic and quantum Fourier transformations, known as Pauli $hat{X}$-gates and Hadamard $hat{H}$-gates, respectively, with an average visibility of more than 90%. In addition, we demonstrate near-perfect unitarity by means of quantum process tomography unveiling a process purity of 99%. Lastly, we demonstrate the benefit of the two independent spatial degrees of freedom, i.e. azimuthal and radial, and implement a two-qubit controlled-NOT quantum operation on a single photon. Thus, our demonstrations open up new paths to implement high-dimensional quantum operations, which can be applied to various tasks in quantum communication, computation and sensing schemes.
A simple and flexible scheme for high-dimensional linear quantum operations on optical transverse spatial modes is demonstrated. The quantum Fourier transformation (QFT) and quantum state tomography (QST) via symmetric informationally complete positi
An open question in quantum optics is how to manipulate and control complex quantum states in an experimentally feasible way. Here we present concepts for transformations of high-dimensional multi-photonic quantum systems. The proposals rely on two n
Quantum key distribution (QKD) promises information-theoretically secure communication, and is already on the verge of commercialization. Thus far, different QKD protocols have been proposed theoretically and implemented experimentally [1, 2]. The ne
High-dimensional entangled photons are a key resource for advanced quantum information processing. Efficient processing of high-dimensional entangled photons requires the ability to synthesize their state using general unitary transformations. The le
Spatial modes of light constitute valuable resources for a variety of quantum technologies ranging from quantum communication and quantum imaging to remote sensing. Nevertheless, their vulnerabilities to phase distortions, induced by random media, im