Spatial Control of Photonic Quantum States (Invited)

Significance Photons have several important degrees-of-freedom available for control belonging to the spatial domain, such as the path and orbital angular momentum (OAM) degrees-of-freedom. These available multiple spatial degrees- offreedom and the high dimensionality of each degree-of-freedom provide for us diverse spatial methods, which can achieve spatial control of photonic quantum states in multiple degrees-of- freedom, high dimensionality, and multi- photons. Specifically, the preparation of spatially entangled photon states and their applications in optical quantum information have attracted extensive attention. Both path and OAM degrees-of-freedom can theoretically construct infinite-dimensional Hilbert spaces. Therefore, path and OAM degrees- of-freedom have inherent advantages in realizing the field of highdimensional coding, which has attracted more extensive attention from researchers. The high- dimensional coding and highdimensional entanglement of the path and OAM degrees-of- freedom themselves continue to break records. The multidegree-of-freedom entangled photonic modulation schemes in which two or more degrees-of- freedom are jointly involved have likewise made progress. It is important to summarize the existing research on spatial control of photonic quantum state to promote the future development of the field. Progress The encoding of quantum information in single photons has evolved from a single degree-of-freedom to multiple degrees-of- freedom encoded together. On-demand conversion of quantum information among different degree-of-freedom through linear optical elements and such active modulation has built a series of quantum optical platforms such as weak measurements and quantum walks. Two-photon entangled states also emerge from the two-dimensional space of a single degree-of-freedom and are realized in multi-degree-of-freedom and high-dimensional systems. For two- photon spatialdomain entanglement, the original polarization-entangled photons are converted into OAM-entangled photons by SLM. Different polarizations can be loaded with different OAMs by a Sagnac interferometer, and two- photon entanglement of two photons with different OAMs has been reported. Subsequent work increases the OAM quantum number of entangled photons to 10010. In addition to the above preparation of OAM entangled photons using entanglement conversion, twophoton entanglement in Hilbert space higher than 100 dimensions has been realized by preparing a high-dimensional OAM entangled source through spontaneous parametric down- conversion and further combining it with the modulation of the radial modes of photons. At the same time, measurements of high-dimensional entangled sources develop in parallel. The successful realization of HOM interference based on multi- degree- of-freedom optical quantum states paves the way for the expansion of optical quantum technology in higher dimensions. Although the quantum modulation of the space domain of multi-photons is complex and difficult, researchers continue to make important progress in the preparation of highdimensional entangled states and optical quantum information processing. The study of space- domain modulation of optical quantum states not only allows for selecting photons generated by transitions under spontaneous down conversion in free space but also considers using a variety of microstructures for the generation and modulation of photons. These microstructured devices allow for efficient beam modulation, localized control of polarization, and a significant enhancement of the efficiency of the emitted and detected photons. In the field of quantum photo generation, the generation of down converted photon pairs with spontaneous parametrization over 100 paths has already been achieved by integrating metal lens arrays with nonlinear crystals on a two-dimensional hypersurface. This holds the promise of generating highdimensional hyperentangled and multiphoton states in an integrated and efficient manner. Conclusions and Prospects In this review, starting from the linear control of a single photon in the spatial domain, we successively describe the spatial coding and transformation of a single photon, the preparation and measurement of twophoton entangled states, as well as the preparation of multi-photon high-dimensional spatial entangled states and their applications in quantum information. We mainly focus on the control in multi- degree- of-freedom and high-dimensional quantum information transformation. In addition, we discuss the recent progress on the spatial photonic quantum states in preparation, coding, measurement, and application. Meanwhile, possible solutions to some key issues are also explored. However, the study on the spatial control of photonic quantum states is still in its infancy and flourishing. There are many