A famous experiment by Pan et al. (Nature 403, 515 (2000)) on a three-photon Bell state is considered to prove the existence of entanglement without relying on Bell inequalities. We show that the theory this proof is based on contains errors in the calculations that are beyond repair. Indeed, the formalism to describe the internal dynamics of electrons (at rest) and photons in quantum mechanics is based on the representations of the rotation group. For electrons, who have spin 1/2, the representation is SU(2), for photons, who have spin 1 it is SO(2). Rotations are just a subject of elementary Euclidean geometry, which does not contain any physics, let alone "quantum magic". Therefore all quantum paradoxes that oppose classical mechanics and QM (when it is purely based on the algebra of SU(2)), can only be due to logical errors and a lack of understanding of the group representation theory. The three-photon Bell state is extended but not non-local and as such does not imply a "spooky action at a distance". (This version supersedes the previous version which contained an error).
A famous experiment by Pan et al. (Nature 403, 515 (2000)) on a three-photon Bell state is considered to prove the existence of entanglement without relying on Bell inequalities. We show that the theory this proof is based on contains errors in the calculations that are beyond repair. Indeed, the formalism to describe the internal dynamics of electrons (at rest) and photons in quantum mechanics is based on the representations of the rotation group. For electrons, who have spin 1/2, the representation is SU(2), for photons, who have spin 1 it is SO(2). Rotations are just a subject of elementary Euclidean geometry, which does not contain any physics, let alone "quantum magic". Therefore all quantum paradoxes that oppose classical mechanics and QM (when it is purely based on the algebra of SU(2)), can only be due to logical errors and a lack of understanding of the group representation theory. The three-photon Bell state is extended but not non-local and as such does not imply a "spooky action at a distance". (This version supersedes the previous version which contained an error).
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.
Quantum physics violates the basic assumptions of local realism. Particles can be entangled with each other in ways that allow correlations stronger than possible in classical physics. This was first realized by Einstein, Podolsky and Rosen (EPR) in 1935 and led them to conclude that quantum theory was incomplete. It took about 30 years before J. S. Bell found observable consequences of this phenomenon by showing how entangled states violate inequalities for two-particle correlations derived purely from the assumption that the world obeys local realism. Violations of Bell inequalities were first observed experimentally in the Aspect experiment, which will be discussed below. Such experiments are technically challenging and care is needed to avoid any loopholes in the experimental setup that would still permit the description of the experiments by a local realistic theory. Bell inequalities describe violations of local realism on average, meaning that each individual experimental outcome could occur in a local realistic theory. It is the statistics of these outcomes that violates local realism. In contrast, another class of quantum states, so-called Greenberger–Horne–Zeilinger (GHZ) states, which contain three particles, can be shown to violate local realism in each measurement outcome. We will also discuss this class of states and experiments by Zeilinger's group, which demonstrated this type of violation of local realism for the first time. Our aim here is not to discuss the exact relations between quantum entanglement, non-locality and realism, and their physical and philosophical implications. While this is a highly interesting topic which attracts a lot of attention from researchers, such an exploration would go beyond the scope of this book.
EPR paper contains an error. Its correction leads to a conclusion that position and momentum of a particle can be defined precisely simultaneously, EPR paradox does not exist and uncertainty relations have nothing to do with quantum mechanics. Logic of the EPR paper shows that entangled states of separated particles do not exist and therefore there are no nonlocality in quantum mechanics. Bell's inequalities are never violated, and results of experiments, proving their violation, are shown to be false. Experiments to prove absence of nonlocality are proposed where Bell's inequalities are replaced by precise prediction. Interpretation of quantum mechanics in terms of classical field theory is suggested. Censorship against this paper is demonstrated.
The Greenberger-Horne-Zeilinger (GHZ) theorem exhibits an ``all versus nothing'' contradiction between Ernstein, Podelsky, and Rosen's local realism and quantum mechanics by using a maximally entangled state of three qubits. However, it is shown in this paper that no extension of GHZ's proof is possible for any entangled state of two qubits, including Bell states. The proof is obtained by showing the fact that there are no more than three distinct elements (except for a different sign) in a set of mutually commuting nonlocal spin observables of two qubits. This also leads to a generic setup of the Kochen-Specker theorem for the two-qubit system.
A proof of Bell's theorem without inequalities valid for both inequivalent classes of three-qubit entangled states under local operations assisted by classical communication, namely Greenberger-Horne-Zeilinger (GHZ) and W, is described. This proof leads to a Bell inequality that allows more conclusive tests of Bell's theorem for three-qubit systems. Another Bell inequality involving both tri- and bipartite correlations is introduced which illustrates the different violations of local realism exhibited by the GHZ and W states.
A recent discovery by Hardy [Phys. Rev. Lett. 71, 1665 (1993)] is developed and extended. It shows that the inconsistency between quantum mechanics and Einstein-Podolsky-Rosen assumptions about locality and reality can be demonstrated for two photons or particles with spin 1/2, as in experiments that test Bell inequalities, but without inequalities, as in what Greenberger, Horne, and Zeilinger discovered for three or four particles [in Bell's Theorem, Quantum Theory, and Conceptions of the Universe, edited by M. Kafatos (Kluwer, Dordrecht, 1989), p. 74]. Here all the possibilities are found. The measurement choices can be different for the two particles. They are described directly in terms of angles for spin measurements. Hardy's result and its converse are both proved: almost any state will do (with measurement choices made to accommodate) and any choice of two different measurement possibilities for each particle will do (with a state made to accommodate). A simple example is given; it allows the calculations to be done quickly with Pauli matrices.