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Negativity and contextuality as criteria for classicallity in discrete phase-space and other quasi-probability representations of quantum theory


In recent years several quasi-probability representations of finite dimensional quantum mechanics have been proposed as analogs of the phase space representation of continuous quantum systems. I will describe a formalism based on the theory of frames which allows us to characterize the set of possible quasi-probability representations for finite dimensional quantum systems that satisfy two reasonable conditions. This formalism leads to a direct proof that any such representation (that reproduces the quantum statistics) is non-classical in the sense that either the states or the measurements must be modeled by negative valued functions. This condition turns out to be equivalent to a proof of contextuality. This formalism may lead to a new method for assessing the degree of non-classicality of a given quantum information task or process.