Contrasting classical and quantum theory in the context of quasi-probability
Ferrie, Christopher Scott
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Several finite dimensional quasi-probability representations of quantum states have been proposed to study various problems in quantum information theory and quantum foundations. These representations are often defined only on restricted dimensions and their physical significance in contexts such as drawing quantum-classical comparisons is limited by the non-uniqueness of the particular representation. In this thesis it is shown how the mathematical theory of frames provides a unified formalism which accommodates all known quasi-probability representations of finite dimensional quantum systems. It is also shown that any quasi-probability representation is equivalent to a frame representation and it is proven that any such representation of quantum mechanics must exhibit either negativity or a deformed probability calculus. Along the way, the connection between negativity and two other famous notions of non-classicality, namely contextuality and nonlocality, is clarified.