Negative probabilities: What they are and what they are for

An observation space $\mathcal S$ is a family of probability distributions $\langle P_i: i\in I \rangle$ sharing a common sample space $Ω$ in a consistent way. A \emph{grounding} for $\mathcal S$ is a signed probability distribution $\mathcal P$ on $Ω$ yielding the correct marginal distribution $P_i$ for every $i$. A wide variety of quantum scenarios can be formalized as observation spaces. We describe all groundings for a number of quantum observation spaces. Our main technical result is a rigorous proof that Wigner's distribution is the unique signed probability distribution yielding the correct marginal distributions for position and momentum and all their linear combinations.

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