Negative probability

TL;DR

This paper discusses the concept of negative probability distributions, focusing on Wigner's distribution, which uniquely produces correct marginals for all linear combinations of position and momentum despite having negative values.

Contribution

It provides a simple proof of the uniqueness of Wigner's distribution and discusses related issues in the context of negative probabilities in quantum mechanics.

Findings

Wigner's distribution uniquely produces correct marginals for all linear combinations.

Negative probabilities can still yield valid marginal distributions.

The paper offers a simple proof of the distribution's uniqueness.

Abstract

This article was written for the Logic in Computer Science column in the February 2015 issue of the Bulletin of the European Association for Theoretical Computer Science. The intended audience is general computer science audience. The uncertainty principle asserts a limit to the precision with which position x and momentum p of a particle can be known simultaneously. You may know the probability distributions of x and p individually but the joint distribution makes no physical sense. Yet Wigner exhibited such a joint distribution f(x,p). There was, however, a little trouble with it: some of its values were negative. Nevertheless Wigner's discovery attracted attention and found applications. There are other joint distribution, all with negative values, which produce the correct marginal distributions of x and p. But only Wigner's distribution produces the correct marginal distributions for all linear combinations of position and momentum. We offer a simple proof of the uniqueness and discuss related issues.