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Steklov Mathematical Institute Seminar
November 18, 2021 16:00, Moscow, online

Negative probabilities: What are they for?

Yu. Sh. Gurevich
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Abstract: Negative probabilities make no sense. It is absurd to talk about an urn containing -11 red balls or -17 green balls. Yet negative probabilities are profitably used in quantum optics and elsewhere. So what reality, or at least intuition, is behind negative probabilities? We have no idea.
Instead, we ask what negative probabilities are good for. It turns out that the disparate quantum applications of negative probabilities can be seen as examples of a certain application template. To make this template explicit, we introduce and study observation spaces. An observation space S is a family of (nonnegative) probability distributions P1; P2; ... on a common sample space. A question arises whether there is a single probability distribution P (a grounding for S) which yields all P1; P2; ... as marginal distributions. That P may be necessarily signed.
We study the grounding problem in general and solve it for some observation spaces of note. In the process we illustrate how naturally negative probabilities arise and how they can be usefully exploited even if one has no interest in them.
The talk is based on a recent paper with Andreas Blass in J. Phys. A.

Language: English

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