Logical Difficulty from Combining Counterfactuals in the GHZ-Bell Theorems

Louis Sica

doi:10.4236/am.2013.410a3012

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.

Highlights

The Greenberger, Horn, Zeilinger (GHZ) theorem [1] has achieved a status similar to that of Bell’s theorem in its acceptance as a proof that local hidden variables are impossible in quantum mechanics
The use of counterfactuals in no local hidden variables theorems relies on the assumption that counterfactual reasoning is intrinsically sound classically, but not quantum mechanically
The conclusion is that the discrepancy between quantum mechanical eigenvalues and calculations using counterfactuals of noncommuting procedures can no longer be taken as proof that local hidden variables are inconsistent with quantum mechanical observations

Summary

Introduction

The Greenberger, Horn, Zeilinger (GHZ) theorem [1] has achieved a status similar to that of Bell’s theorem in its acceptance as a proof that local hidden variables are impossible in quantum mechanics. The use of counterfactuals in no local hidden variables theorems relies on the assumption that counterfactual reasoning is intrinsically sound classically, but not quantum mechanically. The conclusion is that the discrepancy between quantum mechanical eigenvalues and calculations using counterfactuals of noncommuting procedures can no longer be taken as proof that local hidden variables are inconsistent with quantum mechanical observations. A definition of counterfactuals and examples showing inconsistencies in their classical use are given in Sections 2.1 and 2.2. The logical inconsistency is manifested by violation of the Bell inequality, an algebraic expression that must be universally satisfied by cross-correlations of any data sets whatsoever

Definition

Flaws in Classical Counterfactual Reasoning

The GHZ Theorem

A2 A3 A4

Bell’s Theorem

Conclusions