Logical inconsistency in combining counterfactual results from non-commutative operations: Deconstructing the GHZ-Bell theorems

TL;DR

This paper critically examines the logical consistency of combining counterfactual results from non-commutative operations in GHZ-Bell theorems, challenging their conclusions about local hidden variables in quantum mechanics.

Contribution

It reveals that combining counterfactuals from non-commuting measurements leads to logical inconsistencies, questioning the validity of GHZ-Bell theorems' negative conclusions.

Findings

Counterfactual results of non-commuting operations are generally inconsistent with actual measurement sequences.

The analysis demonstrates logical flaws in the assumptions underlying GHZ and Bell theorems.

Negative conclusions about local hidden variables do not necessarily follow from these theorems.

Abstract

The Greenberger, Horne, Zeilinger (GHZ) theorem is critically important to consideration of the possibility of hidden variables in quantum mechanics. Since it depends on predictions of single sets of measurements on three particles, it eliminates the sampling loophole encountered by the Bell theorem which requires a large number of observations to obtain a small number of useful joint measurements. In evading this problem, the GHZ theorem is believed to have confirmed Bell's historic conclusion that local hidden variables are inconsistent with the results of quantum mechanics. The GHZ theorem depends on predicting the results of sets of measurements of which only one may be performed, i.e., counterfactuals. In the present paper, the non-commutative aspects of these unperformed measurement sequences are critically examined. Three classical examples and two variations on the GHZ construction are analyzed to demonstrate that combined counter factual results of non-commuting operations are in general logically inconsistent with performable measurement sequences that take non-commutation into account. As a consequence, negative conclusions regarding local hidden variables do not follow from the GHZ and Bell theorems as historically reasoned.