Intersubjective consistency asks when different observers of a physical system, reasoning from a shared theory, will agree on their probability assessments. Classical probability theory provides a benchmark in the form of the Agreement Theorem (Aumann, 1976). We study the quantum mechanics (QM) analog to the classical Agreement Theorem in a finite-dimensional tripartite setting in which two observers obtain information via local projective measurements. We represent their mutual awareness by common certainty, which is formally an infinite hierarchy of certainty operators concerning the probability of an event of interest. We first prove that if the observers' measurements commute with one another and with the property of interest, common certainty forces their probability assessments to coincide, thereby recovering a QM analog to the classical Agreement Theorem. We then construct a one-parameter family of qutrit-qubit-qubit models in which one observer's measurement does not commute with the property and common certainty of disagreement occurs -- a distinctively QM phenomenon. The mechanism is coherence between branches distinguished by one but not the other observer's measurement. When measurement outcomes are stored in a classical register, the resulting dephasing removes this coherence and restores agreement. Finally, we prove that QM does not permit the extreme 0-1 configuration in which Alice is certain of a property and is also certain that Bob is certain of its negation. These results aim to turn debate in QM about observer-dependent facts into precise mathematical conditions for when intersubjective agreement or disagreement in QM can be sustained.
It is a scientific truism nowadays that quantum mechanical predictions are of statistical or rather of probabilistic nature: a quantum theory assigns a probability distribution to each measurable quantity (observable) of a physical system prepared according to a given preparation prescription (state). Such sets of probability distributions represent the most general quantitative predictions of physical theories. The aim of this study is to determine all theoretical frameworks which yield such a kind of output taking the probabilistic nature of quantitative physical predictions as initial hypothesis. Astonishingly, this project succeeds with a minimal number of additional assumptions, which arise quite naturally in the course of the analysis, and which essentially amount to regularity conditions. The result is that there exist only two classes of physical theories providing quantitative probabilistic predictions: classical theories where observables are represented by real functions over a phase space and states are represented by probability measures upon the phase space, and quantum theories where observables are represented by selfadjoint operators acting on a complex Hilbert space and states are represented by state operators. On empirical grounds, classical theories cannot provide the all-encompassing, fundamental frame for the mathematical description of physical systems. Quantum theories remain, and there is no more complete formal frame for quantitative physical theories. As an additional result, it is shown that there exist physical systems which possess a quantum mechanical but no classical description; a prominent example is the spin system of two spin-1/2 particles. The classical hidden variables model, which J. Bell relates to this system in order to prove his theorem, cannot be a true description of it because no such classical model exists, and Bell's theorem loses its foundation.
In this paper, we prove a theorem that imposes that certain coupled physical systems (particles or fields) have non-commuting observables even without postulating canonical commutation relation.
More precisely, we prove a theorem that ensures that it is forbidden for certain coupled physical systems to exist in a physical world as classical systems (i.e. systems describable by classical mechanics); we will call the theorem on Non-Classicality.
We will see in the present paper and in next papers, that the Theorem on Non-Classicality has important consequences among which the implication of a Quantum Mechanics more fundamental than Standard Quantum Mechanics.