| Frame theory offers a lens through which to view a large portion of quantum information theory, providing an organizational principle to those topics in its purview. In this thesis, I cut a trail from foundational questions to practical applications, from the origin of the quantum probability rule to quantum cryptography, by way of a standard quantum measurement helpful in quantum tomography and representation of quantum theory. Before embarking, preparations are undertaken by outlining the relevant aspects of frame theory, particularly the characterization of generalized orthonormal bases in terms of physical quantum measurements, as well as several aesthetically appealing families of measurements, each possessing a high degree of symmetry.; Much more than just elegant, though, these quantum measurements are found to be useful in many aspects of quantum information theory. I first consider the foundational question of justifying the quantum probability rule, showing that putting a probability valuation on generalized quantum measurements leads directly to the Born rule. Moreover, for qubits, the case neglected in the traditional formulation of Gleason's theorem, a symmetric three-outcome measurement called the trine is sufficient to impel the desired form. Keeping with foundational questions, I then turn to the problem of establishing a symmetric measurement capable of effortlessly rendering quantum theory in terms of classical probability theory. Numerical results provide an almost utterly convincing amount of evidence for this, justifying the subsequent study of its use in quantum tomography and detailed account of the properties of the reduction to probabilistic terms.; Saving perhaps the most exciting topic for last, I make use of these aesthetic ensembles in the applied field of quantum cryptography. A large class of streamlined key distribution protocols may be cut from the cloth of these ensembles, and their symmetry affords them improved tolerance to eavesdropping over the traditionally-studied schemes. Because the ability to put quantum key distribution protocols into practice is essentially right around the corner, I conclude by examining the prospects for implementing the new protocols in free space and their ability to boost the operating signal intensity, currently a major obstacle in the development of practical schemes. |
