Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations

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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.
Here we show how the mathematical theory of frames provides a unified formalism which accommodates all known quasi-probability representations of finite dimensional quantum systems.