Logical difficulty from combining counterfactuals in the GHZ-Bell theorems

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.


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. It has a similarity to Bell's theorem in that it considers a mathematical relation among predicted results of alternative measurements that are not all performed. The alternatives consist of procedures that if performed together are non-commuting, and whose results then depend on their order of execution. Thus, if all the measurements were actually performed, their explicit non-commutation would have to be taken into account in predicting measurement outcomes.
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.
However, examples given in Section 2.2 reveal that counterfactual reasoning commonly fails in the classical domain if it neglects the non-commutation of component procedures. It is then shown that parallel reasoning leads to the paradoxical results of the GHZ and Bell theorems in its neglect of non-commutation. The conclusion is that the discrepancy between quantum mechanical eigenvalues and calculations using counterfactuals of non-commuting 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. In Section 3, the accepted interpretation of the GHZ theorem is described following the treatment by Mermin [2], Home [3], Afriat and Selleri [4], and Greenberger [5], but in a manner showing the roll of counterfactual reasoning. In Section 4, a similar inconsistency due to use of counterfactuals in the Bell theorem is outlined. In this case, the logical inconsistency is manifested by violation of the Bell inequality, an algebraic expression that must be satisfied by cross-correlations of any data sets whatsoever.

Definition
In general, the term counterfactual refers to the predicted result of an unperformed act or consequence of a condition that is not true. In the present paper, the definition is further narrowed to distinguish it from various alternatives [6]. If one considers two procedures A and B that do not commute, the result of carrying out a sequence of the two depends on whether A or B is performed first. However, one may consider each procedure in isolation from the other in an exclusive-OR sense. The predicted measurement results of such procedures, that if performed together require non-commutation to be taken into account, are herein termed counterfactuals.
(Since measurement outcomes for commuting procedures have simultaneous existence in quantum mechanics, they are of little concern here.)

Flaws in classical counterfactual reasoning
It has been stated in the context of "no-go" theorems for hidden variables that counterfactual reasoning is used frequently in the classical world without any problem: it is logically trustworthy. The author proposes that this belief is in error as will now be shown by classical counter-examples.
We first take note of characteristics of classical non-commuting operations using a semifacetious example given several years ago by Leon Cohen in a lecture at the Naval Research Laboratory: putting on shoes and socks. Consider this example from the point of view of counterfactuals. One may put on shoes alone, or socks alone in an exclusive-OR sense, and these acts have perfectly well defined meanings. However, one cannot consider these acts in a logical-AND sense unless non-commutation is taken into account. In that case, putting on socks and then shoes gives a different result from putting on shoes and then socks. Thus, converting the logical-OR case to an AND case in the sense of simultaneous existence, or conversion to commutation without conditionality, makes no physical or logical sense.
Classical operations are commonly non-commutative. Consider [7] the rotation of rigid bodies in three-dimensional space. A rotation of + about the x-axis followed by + about the y-axis produces a different final orientation than if these rotations are carried out in reverse order.
Navigation on the surface of the earth is non-commutative: traveling 100 miles north followed conclusion [8] stating that (counterfactual) results of non-commutative operations "cannot be combined to form a meaningful quantum description" in the consistent histories interpretation of quantum mechanics, and that their joint use is meaningless.

THE GHZ THEOREM
The Pauli spin operators ! x , ! y , and ! z are used to define three-particle operators A 1 , A 2 , A 3 : where the superscripts indicate the particle to which the operator is applied. In the theorem, A 1 , A 2 , and A 3 are ultimately applied to an entangled state of three spin 1/2 particles and each corresponds to a measurement of the product of their spins. Using the anti-commutation properties of the spin operators, and the fact that operators on different particles commute, it is found that A 1 , A 2 , and A 3 commute. For example, to show that A 1 and A 2 commute, multiply Using the anti-commutation property of Equation (2), The other commutations may be demonstrated similarly.
One may now consider the product operator A 1 A 2 A 3 : Since operations on different particles commute, this may be written as long as the right to left sequence of operators on each particle is unchanged from Equation (5).
Relation (6) may be simplified by using Equation (2), particularly the anti-commutation relation, to obtain It is emphasized that the minus sign in Equation (7a) that results from multiplying A 1 A 2 A 3 is due to the non-commutation of operations on particle 2. Finally from Equation (2) it follows similarly to the examples just given, that A 1 , A 2 , A 3 , and A 4 commute for any ! .
The GHZ theorem depends on the above state-independent properties of the A i and further consequences that follow from the fact that they have a common entangled eigenstate In Equation (8), ! i and ! i for particles i = 1, 2, 3 designate the eigenkets of ! z with eigenvalues +1 and -1, respectively. From the well known relations [9]: and the definitions of A 1 , A 2 , and A 3 , one obtains Then from Equation (7), which follows from the spin anti-commutation relations, action of A 4 on ! yields Since the A i 's commute and have a common eigenstate ! , they are simultaneously measurable on three particles in state ! . Thus, for example, the same value of A 1 occurs at each of its occurrences in the sequence A 1 A 2 A 1 . However, a measurement of A i must be made in such a way that only the product of the spin values and not their individual values are revealed [5].
Otherwise, a state produced by measuring an individual A i would collapse ! to one yielding a specific spin eigenvalue for each particle, and this state would not be an eigenstate of any other A i . Thus, Equations (10) and (11) would no longer hold.
A measurement of A 1 that reveals individual spins would collapse Equation (13) to one of these terms. Suppose it is the first for which all spins are positive. Now consider measurement of ! y 3 : the resulting measured spin product is no longer necessarily +1. On the other hand, if A 2 is measured first, its spin product will be the predicted +1, but a following measurement of A 1 may now produce a spin product other than +1. The situation is the same for any pair of A i 's if actual spins are revealed. The spin products corresponding to measurement of any one A i satisfy Equation (10) or (11), but measurement of a second A j , j ! i , no longer necessarily satisfies these relations. Thus, spin-revealing measurements of different A i do not commute.
The usual argument of the GHZ theorem that states that Equations (10) and (11) ± where values for each of the two occurrences of m y i , i = 1, 2, 3 are equal as deduced above.
However, quantum mechanics gives a different result for the combined operation of A 1 , A 2 , and A 3 on ! due to non-commutation of operations on particle 2: On the assumption that the use of combined counterfactuals in the proof of "no-go" theorems

BELL'S THEOREM
The present analysis would not be complete without a review of Bell's theorem, and the contribution of the use of counterfactuals of noncommuting operations to its flaws. Previously identified logical problems in the theorem will only be outlined here since their analysis has been given in [10][11][12][13][14].
It is easy to show that the inequality that Bell derived is universally satisfied by crosscorrelations of any three or four (as appropriate) data sets consisting of ±1' s [10,11]. This fact depends only on the assumed existence of the data sets, and is independent of any other property such as their origin in random, deterministic, local, or nonlocal processes. Bell did not realize that his inequality resulted from the use of cross-correlation alone in its development. He attributed the result to consciously chosen assumptions: all measurements are represented by a function of random variables (counterfactually) defined at all instrument settings, and locality.
He then assumed that the random function he postulated resulted in a second order stationary process [15] in that all correlations could be represented by the same co-sinusoidal function of coordinate differences. Spin measurement operations on a given particle at different instrument settings were mathematically treated as commutative [14].
However, for more than two measurements on a side in Bell experiments, actual performable measurements are noncommutative according to quantum mechanics. Bell mistakenly indicated in his book that this consideration could be ignored by using noncommuting counterfactuals interchangeably with measurements [16] to produce counterfactual experimental data. Of course, since the Bell inequality results from the mere fact of cross-correlation, it is satisfied by the crosscorrelations of data sets of commutative second order stationary processes as assumed by Bell and it is thus derivable upon the assumption of such processes, even though it holds generally.
In the quantum mechanical two particle experiments to which this inequality has been applied, consideration of more than one measurement per particle implies that noncommutation must be taken into account. The author has previously identified two experiments yielding more than one measurement per side that could produce data for cross-correlation under this condition. One would use an additional apparatus in tandem on either side of the usual Bell experiment operating in a retrodictive mode [10]. The second would use separate experiments from which correlations conditional on the usual outcomes could be computed [12,13]. Either of these would yield a third correlation that is functionally different from those obtained in standard Bell experiments such that the three would satisfy the Bell inequality as required by basic mathematics.
In practice, the data from Bell experiments have not been cross-correlated, and each pair of correlations is derived from an independent experimental run. If the underlying process were second-order stationary as assumed by Bell, there would only be one correlation functional form to determine, and ensemble averaged cross-correlations would yield the same function as that measured in independent runs. The Bell inequality would be satisfied by this correlation function as measured in independent runs up to small statistical fluctuations. The violation of the Bell inequality by experimentally confirmed cosine correlation functions proves that the underlying process is not statistically stationary, contrary to what is widely assumed. This is consistent with the non-commutative process described by quantum mechanics that predicts different correlation functional forms between some variables. The logical inconsistencies in the usual interpretation of the Bell theorem begin with combining counterfactuals of noncommutative processes with real data, and are manifested in the violation of the Bell inequality, an inequality that must be universally satisfied by the cross-correlations of any data sets whatsoever consisting of ±1' s .

CONCLUSION
If a theory calculates results of actually performable experiments, the results must logically depend on taking non-commuting operations properly into account. The no-hidden-variables theorems historically contrast quantum results based on non-commutation with classical results based on its neglect. The narrative accompanying these theorems is that neglect of noncommutation of counterfactuals is logically sound in the classical domain, so it is appropriate to attribute the peculiarly inconsistent results that follow to non-locality, or the non-existence of hidden variables or pre-existing values for measurements. But if, as has been shown, the use of counterfactual reasoning in no-hidden-variables theorems is flawed both quantum mechanically and classically, the usual paradoxical choices emerging from these theorems no longer have logical motivation. That said, lack of validity of no-hidden-variables theorems does not, in and of itself, imply that local hidden variables exist.
The content of this paper was presented in [17]. The present paper discusses the central idea in greater detail than does the longer [18], while the latter includes additional variations not dealt with here.

ACKNOWLEDGEMENTS
I would like to thank Mike Steiner of Inspire Institute for useful critical comments on the manuscript, and Armen Gulian and Joe Foreman of the quantum group at Chapman University Burtonsville, MD for many useful discussions relating to the presentation of the material.