The spatial state of non-interacting photons
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High-dimensional quantum systems are becoming an increasingly important area of study. Due to their ability to encode more information than a two-dimensional system, high-dimensional systems are useful in many applications, from quantum communication to quantum computing. In particular, spatial states of light, such as orbital angular momentum and spatial position, are inherently high-dimensional by nature and lend themselves well to manipulation and measurement. As light is commonly used in communication applications, spatial states could extend the information capacity of quantum communication and make it easier to detect eavesdroppers in the system. This thesis comprises four experiments in which the spatial state of photons is manipulated and measured. The ﬁrst experiment describes a ﬁlter for two dimensional anti-symmetric spatial states. We use a pair of photons entangled in multiple orbital angular momentum states in order to test the ﬁlter. We are able to manipulate which two-dimensional subspaces are in symmetric states and which are in anti-symmetric states, and as such we are able to ﬁlter out particular subspaces, eﬀectively engineering high-dimensional states via Hong-Ou-Mandel interference. In the second experiment, we use the anti-symmetric state ﬁlter in a four-photon system. We begin with two pairs of photons, with entanglement within the pairs but not between the pairs. Combining one photon from each pair in our anti-symmetric state ﬁlter, we create entanglement between the other two photons, achieving entanglement swapping. Additionally, due to the two-dimensional nature of the ﬁlter, we transcribe entanglement into several two-dimensional subspaces in the process. In the third experiment, we investigate the quantum teleportation that occurs as a side eﬀect of the entanglement swapping. We demonstrate teleportation of several two-dimensional OAM states, and we describe the result of attempted high dimensional teleportation. In the fourth and ﬁnal experiment, we turn our attention from the OAM of light to the spatial position of light. Using our four-photon system and anti-symmetric state ﬁlter, we demonstrate ghost imaging between photons that have never interacted. This is enabled by taking advantage of the correlations produced when entanglement swapping occurs in the ﬁlter.