The basic Leggett inequalities don't contradict the quantum theory, neither the classical physics

TL;DR

This paper demonstrates that the basic Leggett inequalities do not inherently contradict quantum theory or classical physics when derived without specific hidden-variable assumptions, highlighting the importance of consistent probability distributions.

Contribution

The paper derives the basic Leggett inequalities in a fully general way, showing they are compatible with quantum mechanics if the same probability distribution is used throughout.

Findings

Basic Leggett inequalities do not contradict quantum theory when derived generally.

Contradictions arise only when different probability distributions are used for different averages.

The inequalities are compatible with both quantum and classical physics under consistent probability assumptions.

Abstract

The basic Leggett inequalities, i.e. those inequalities in which the particular assumptions of Leggett's hidden-variable model (e.g. Malus law) were not yet introduced, are usually derived using hidden-variable distributions of probabilities (although in some cases completely general, positive probabilities would lead to the same result). This fact creates sometimes the illusion that these basic inequalities are a belonging of the hidden-variable theories and are bound to contradict the quantum theory. In the present text the basic Leggett inequalities are derived in the most general way, i.e. no assumption is made that the distribution of probabilities would result from some wave function, or from some set of classical variables. The consequence is that as long as one and the same probability distribution is used in the calculus of all the averages appearing in the basic Leggett inequalities, no contradiction may occur. These inequalities may be violated only when different averages are calculated with different distributions, for example, some of them calculated with the quantum formalism and the others with some distribution of classical parameters.