Logical difficulty from combining counterfactuals in the GHZ-Bell theorems

TL;DR

This paper critically examines the logical consistency of combining counterfactuals in GHZ-Bell theorems, suggesting that their conclusions about local hidden variables may be invalid due to non-commutative measurement issues.

Contribution

It introduces a new analysis of non-commutative measurement effects, challenging the traditional interpretations of GHZ and Bell theorems regarding local hidden variables.

Findings

Counterfactual results of non-commuting measurements are logically inconsistent with actual measurement sequences.

The traditional conclusions about local hidden variables from GHZ and Bell theorems are questioned.

Non-commutative aspects undermine the logical foundation of these quantum theorems.

Abstract

In eliminating the fair sampling assumption, the Greenberger, Horne, Zeilinger (GHZ) theorem is believed to confirm 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. In the present paper, the non-commutative aspects of these unperformed measurements are critically examined. Classical examples and the logic of the GHZ construction are analyzed to demonstrate that combined counterfactual results of non-commuting operations are in general logically inconsistent with performed measurement sequences whose results depend on non-commutation. The Bell theorem is also revisited in the light of this result. It is concluded that negative conclusions regarding local hidden variables do not follow from the GHZ and Bell theorems as historically reasoned.