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Necessity of negativity in quantum theory

A unification of the set of quasiprobability representations using the mathematical theory of frames was recently developed for quantum systems with finite-dimensional Hilbert spaces, in which it was proven that such representations require negative probability in either the states or the effects. In this article we extend those results to Hilbert spaces of infinite dimension, for which the celebrated Wigner function is a special case. Hence, this article presents a unified framework for describing the set of possible quasiprobability representations of quantum theory, and a proof that the presence of negativity is a necessary feature of such representations.