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.
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.
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.
Introduction Since its inception, fifty years ago, Bell's theorem has had a long history not only of experimental tests but also of theoretical developments. Studying pairs of correlated quantummechanical particles separated in space, in a composite “entangled” state, Bell [1] showed that the joint ascription of hidden variables and locality to the system led to an inequality that is violated by the predictions of quantum mechanics. Fifteen years later, experiments confirmed the predictions of quantum mechanics, ruling out a large class of local realist theories. One of the most meaningful theoretical developments that followed Bell's work was the Greenberger–Horne–Zeilinger (GHZ) theorem, also known as Bell's theorem without inequalities. In 1989, the American physicists Daniel Greenberger and Michael Horne, who had been working on Bell's theorem since the late 1960s, together with the Austrian physicist Anton Zeilinger, introduced a novelty into the testing of entanglement, extending Bell's theorem in a different and interesting direction. According to Franck Laloë [2], For many years, everyone thought that Bell had basically exhausted the subject by considering all really interesting situations, and that two-spin systems provided the most spectacular quantum violations of local realism. It therefore came as a surprise to many when in 1989 Greenberger, Horne, and Zeilinger (GHZ) showed that systems containing more than two correlated particles may actually exhibit even more dramatic violations of local realism. The trio analyzed the Einstein, Podolsky, and Rosen 1935 argument once again and were able to write what are now called Greenberger–Horne–Zeilinger (GHZ) entangled states, involving three or four correlated spin-1/2 particles, leading to conflicts between local realist theories and quantum mechanics. However, unlike Bell's theorem, the conflict now was not of a statistical nature, as was the case with Bell's theorem, insofar as measurements on a single GHZ state could lead to conflicting predictions with local realistic models [3–5]. In this paper we present the history of the creation of this theorem and analyze its scope.
The Greenberger-Horne-Zeilinger (GHZ) version of the Einstein-Podolsky-Rosen (EPR) paradox is widely regarded as a conclusive logical argument that rules out the possibility of describing quantum phenomena within the framework of any model of local hidden variables in which the observers are free to choose their own experimental settings. In this paper, we show, however, that the GHZ argument implicitly relies on a subtle, though crucial, assumption that is not required by fundamental physical principles, and which had gone unnoticed. Namely, we note that the argument implicitly assumes the existence of an absolute angular frame of reference with respect to which the polarization properties of the hypothetical hidden configurations of the entangled particles, as well as the orientations of the measurement devices that test them, can be defined. We further note that such an absolute frame of reference may not exist if the hidden configurations of the entangled particles spontaneously break the gauge rotational symmetry. Indeed, by skipping the disputed requirement we are able to build a model of local hidden variables for the GHZ state, which complies with the 'free-will' hypothesis and reproduces the quantum mechanical predictions. Our model, thus, completes-in the EPR sense-the quantum description of the GHZ state.
Most of the standard proofs of the Bell theorem are based on the Kolmogorov axioms of probability theory. We show that these proofs contain mathematical steps that cannot be reconciled with the Kolmogorov axioms. Specifically we demonstrate that these proofs ignore the conclusion of a theorem of Vorob'ev on the consistency of joint distributions. As a consequence Bell's theorem stated in its full generality remains unproven, in particular, for extended parameter spaces that are still objective local and that include instrument parameters that are correlated by both time and instrument settings. Although the Bell theorem correctly rules out certain small classes of hidden variables, for these extended parameter spaces the standard proofs come to a halt. The Greenberger-Horne-Zeilinger (GHZ) approach is based on similar fallacious arguments. For this case we are able to present an objective local computer experiment that simulates the experimental test of GHZ performed by Pan, Bouwmeester, Daniell, Weinfurter and Zeilinger and that directly contradicts their claim that Einstein-local elements of reality can neither explain the results of quantum mechanical theory nor their experimental results.
The Greenberger-Horne-Zeilinger~(GHZ) version of the Einstein-Podolsky-Rosen~(EPR) paradox is widely regarded as a conclusive logical argument that rules out the possibility of describing quantum phenomena within the framework of a local and realistic model of hidden variables in which the observers are free to choose their own experimental settings. In this paper we show, however, that the GHZ argument implicitly relies on an additional crucial assumption, which is not required by fundamental physical principles and had gone unnoticed. Namely, we note that the argument implicitly assumes the existence of an absolute angular frame of reference with respect to which the polarization properties of the hypothetical hidden configurations of the entangled particles as well as the orientation of the measurement apparatus that test the system can be defined. We further note that such an absolute frame of reference would not exist if the hidden configurations of the entangled particles spontaneously break the gauge rotational symmetry. Indeed, by skipping this unnecessary additional assumption we are able to build an explicitly local and realistic model of hidden variables for the GHZ state, which complies with the 'free-will' hypothesis and reproduces the quantum mechanical predictions, and thus completes the description of the system in the EPR sense.
Greenberger-Horne-Zeilinger (GHZ) states are characterized by their transformation properties under a continuous symmetry group, and N-body operators that transform covariantly exhibit a wealth of GHZ contradictions. We show that local or noncontextual hidden variables cannot duplicate the predicted measurement outcomes for covariant transformations, and we extract specific GHZ contradictions from discrete subgroups, with no restrictions on particle number N or dimension d except for the general requirement that $N\ensuremath{\ge}3$ for GHZ states. However, the specific contradictions fall into three regimes distinguished by increasing demands on the number of measurement operators required for the proofs. The first regime consists of proofs found recently by Ryu et al. [Phys. Rev. A 88, 042101 (2013)], the first operator-based theorems for all odd dimensions d, covering many (but not all) particle numbers N for each d. We introduce alternative methods of proof that define second and third regimes and produce theorems that fill all remaining gaps down to N=3 for every d. The common origin of all such GHZ contradictions is that the GHZ states and measurement operators transform according to different representations of the symmetry group, which has an intuitive physical interpretation.