Logical Difficulty from Combining Counterfactuals in the GHZ-Bell 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 noncommutative 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 noncommuting operations are in general logically inconsistent with performed measurement sequences whose results depend on noncommutation. 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.

Bell Correlations without Entanglement: A Local Wave Model Using Gaussian-Poisson Statistics and Single Count-Pair Selection

“…The possibility of an alternative to the entanglement description has been thought to be ruled out by violation of Bell's inequalities. However, the Bell theorem is fundamentally flawed, as shown in [1] and references therein, so it may no longer be used ab initio to deny efforts to construct alternative descriptions of these phenomena.…”

“…Based on the geometry of parametric down converter sources for Bell correlation experiments (see Figure 1) [3], two complex amplitudes 1 U  and 2 U  are introduced corresponding to superpositions of orthogonally polarized light amplitudes in regions denoted as Beams 1 and 2 of the figure. 1 U  and 2 U  are superpositions of random waves:…”

“…The action of polarization analyzers placed in each of Beams 1 and 2 is now described. These transmit and reflect components of 1…”

“…It will be shown below that the covariances result from modification of the statistics to represent selection of single event pairs in a detector time-constant (as implemented in experiments), rather than arbitrary numbers of event pairs as implied by the statistics thus far. The appropriate modifications are obtained by subtracting their common mean value 0 I from each of the intensity components of 1 S and 2 S : I I I I I I I I I I I I I I I…”

“…If valid, this view would rule out the possibility of local descriptions of these correlations. However, it is shown in [1] and references therein that conclusions based on the Bell theorem as historically reasoned may not be logically drawn. Thus, it becomes appropriate to investigate possible alternatives to quantum descriptions that are widely believed to imply nonlocality.…”