Study of Wigner's theorem on joint probabilities

J.G. Krüger

In contrast to a widespread belief, Wigner's theorem allows the construction of true joint probabilities in phase space for distributions describing the object system as well as for distributions depending on the measurement apparatus. The fundamental role of Heisenberg's uncertainty relations in Schroedinger form (including correlations) is pointed out for these two possible interpretations of joint probability distributions. Hence, in order that a multivariate normal probability distribution in phase space may correspond to a Wigner distribution of a pure or a mixed state, it is necessary and sufficient that Heisenberg's uncertainty relation in Schroedinger form should be satisfied.

Georges Chevalier

Philippe Allard Guérin, Veronika Baumann, Flavio Del Santo +1 more

Abstract The notorious Wigner’s friend thought experiment (and modifications thereof) has received renewed interest especially due to new arguments that force us to question some of the fundamental assumptions of quantum theory. In this paper, we formulate a no-go theorem for the persistent reality of Wigner’s friend’s perception, which allows us to conclude that the perceptions that the friend has of her own measurement outcomes at different times cannot “share the same reality”, if seemingly natural quantum mechanical assumptions are met. More formally, this means that, in a Wigner’s friend scenario, there is no joint probability distribution for the friend’s perceived measurement outcomes at two different times, that depends linearly on the initial state of the measured system and whose marginals reproduce the predictions of unitary quantum theory. This theorem entails that one must either (1) propose a nonlinear modification of the Born rule for two-time predictions, (2) sometimes prohibit the use of present information to predict the future—thereby reducing the predictive power of quantum theory—or (3) deny that unitary quantum mechanics makes valid single-time predictions for all observers. We briefly discuss which of the theorem’s assumptions are more likely to be dropped within various popular interpretations of quantum mechanics.

Pankaj Sharan

Jürg Rätz

Ralph Sabbagh, Olga Movilla Miangolarra, Hamid Hezari +1 more

A class of signed joint probability measures for n arbitrary quantum observables is derived and studied based on quasicharacteristic functions with symmetrized operator orderings of Margenau-Hill type. It is shown that the Wigner distribution associated with these observables can be rigorously approximated by such measures. These measures are given by affine combinations of Dirac delta distributions supported over the finite spectral range of the quantum observables and give the correct probability marginals when coarse-grained along any principal axis. We specialize to bivariate quasiprobability distributions for the spin measurements of spin- 1 / 2 particles and derive their closed-form expressions. As a side result, we point out a connection between the convergence of these particle approximations and the Mehler-Heine theorem. Finally, we interpret the supports of these quasiprobability distributions in terms of repeated thought experiments.