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Using negative probability for quantum solutions

Probabilities with a negative sign have been of great use in quantum physics.
Dice with negative numbers

In a new publication in the Journal of Physics A, Emeritus Prof. Yuri Gurevich and Prof. Andreas Blass (Mathematics) set out to explore the intriguing and seemingly absurd phenomenon of negative probabilities, encouraged by their practical use. While the term may sound strange, these probabilities with a negative sign have proven beneficial to the field of quantum physics.

Formally, these signed probabilities can be defined as special signed measures. But what do these signed probabilities mean in the real world?

“We don’t know,” Gurevich answers. “The standard frequency-based interpretation of probabilities makes no sense for negative probabilities. There are attempts in the literature to provide meaning for negative probabilities but, in our judgement, the problem is wide open.”

Instead, in “Negative probabilities: What are they for?” the authors address a more pragmatic question: how can we actually make use of negative probabilities? 

“It is not rare in science to use a concept without understanding its intrinsic meaning.”

Gurevich gives the example of early uses of complex numbers. The standard quantity-based interpretation of numbers makes no sense for imaginary numbers, and the intrinsic meaning of imaginary numbers wasn’t clear (and is debatable even today). Yet complex numbers were used very effectively to solve algebraic equations. 

A similar scenario arises from the fundamental properties of the quantum uncertainty principle, which asserts a limit to the precision with which the position and momentum of a particle can be known simultaneously. While knowing the probability distributions of these observables individually in a given quantum state, their joint probability distribution makes no physical sense – but it does make mathematical sense. In 1932, Eugene Wigner exhibited such a joint distribution, with some negative values. At the time Wigner wrote that this “must not hinder the use of it in calculations as an auxiliary function which obeys many relations we would expect from such a probability.”

It turns out that the various quantum applications of negative probabilities can be seen as examples of a certain application template. The authors’ first achievement was to make this template explicit. To achieve this, they introduced observation spaces, a family of probability distributions on a common sample space. They sought a single probability distribution, called the space’s grounding, which yields all the probability distributions as marginal distributions – and it may necessarily be negative.

They then solve the grounding problem for a number of notable observation spaces, particularly a variety of quantum scenarios that could be formalized as observation spaces. Chief among these was their rigorous proof that Wigner’s distribution is the unique signed probability distribution yielding the correct marginal distributions for the position and momentum of a particle, and all their linear combinations. Thanks to this paper, Wigner was proved to be right nearly 90 years later.