We present a brief historical introduction to the topic of Bell's theorem. Next we present the surprising features of the three particle Greenberger-Horne-Zeilinger (GHZ) states. Finally we shall present a method of analysis of the GHZ correlations, which is based on a numerical approach, which is effectively equivalent to the full set of Bell inequalities for correlation functions for the given problem. The aim of our numerical approach is to answer the following question. Do additional possible local settings lead for the GHZ states to more pronounced violation of local realism (measured by the resistance of the quantum nature of the correlations with respect ``white'' noise admixtures)?
We present a brief historical introduction to the topic of Bell's theorem. Next we present the surprising features of the three particle Greenberger-Horne-Zeilinger (GHZ) states. Finally we shall present a method of analysis of the GHZ correlations, which is based on a numerical approach, which is effectively equivalent to the full set of Bell inequalities for correlation functions for the given problem. The aim of our numerical approach is to answer the following question. Do additional possible local settings lead for the GHZ states to more pronounced violation of local realism (measured by the resistance of the quantum nature of the correlations with respect ``white'' noise admixtures)?
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.
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.
The Bell theorem can take the form of several inequalities, as we have seen in §4.2. Moreover, since it was discovered, the theorem has stimulated the discovery of several other mathematical contradictions between the predictions of quantum mechanics and those of local realism. We review a few of them in this chapter: GHZ contradictions (§5.1) and their generalization (§5.2), Cabello's inequality (§5.3), and Hardy's impossibilities (§5.4). Finally, in §5.5, we discuss the notion of contextuality and introduce the BKS theorem. GHZ contradiction 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 [188, 189]. They involve a sign contradiction (100% violation) for perfect correlations, while the BCHSH inequalities are violated by about 40% (Cirelson bound) and deal with situations where the results of measurements are not completely correlated. In this section, we discuss three-particle systems, but generalizations to N particles are possible (§5.2). Derivation GHZ contradictions may occur in various systems, not necessarily involving spins. Initially, they were introduced in the context of entanglement swapping (§6.3.2) for four particles [188] or entanglement of three spinless particles [189].
The N-qubit states of the W class, for N>10, lead to more robust (against noise admixture) violations of local realism, than the Greenberger-Horne-Zeilinger (GHZ) states. These violations are most pronounced for correlations for a pair of qubits, conditioned on specific measurement results for the remaining $N\ensuremath{-}2$ qubits. The considerations provide us with a qualitative difference between the W state and GHZ state in the situation when they are separately sent via depolarizing channels. For sufficiently high amount of noise in the depolarizing channel, the GHZ states cannot produce a distillable state between two qubits, whereas the W states can still produce a distillable state in a similar situation.
We demonstrate how a new exactly soluble cavity QED model can be used to test local realism associated with three-particle entangled states, as proposed by Mermin. These states embody the correlations described by Greenberger, Horne, and Zeilinger (GHZ) and provide a stronger refutation of local hidden variable theories than is possible with Bell inequalities. It is shown that two photons distributed in the model’s four cavity modes coupled with a two-level atom can effectively produce three independent spin-1/2 particle states. We discuss preparation of the desired perfectly correlated GHZ state and a method to perform measurement on various field modes and atomic level correlations equivalent to a spin component along any direction.
Bell's theorem started with two qubits (spins 1/2). It is a 'no-go' statement on classical (local causal) models of quantum correlations. After 25 years, it turned out that for three qubits the situation is even more astonishing. General statements concerning higher dimensional systems, qutrits, etc, started to appear even later, once the picture with spin (higher than 1/2) was replaced by a broader one, allowing all possible observables. This work is a continuation of the Gdansk effort to take advantage of the fact that Bell's theorem can be put in the form of a linear programming problem, which in turn can be translated into a computer code. Our results are numerical and classify the strength of the violation of local causality by various families of three-qutrit states, as measured by the resistance to noise. This is previously uncharted territory. The results may be helpful in suggesting which three-qutrit states will be handy for applications in quantum information protocols. One of the surprises is that the W state turns out to reveal a stronger violation of local causality than the GHZ (Greenberger–Horne–Zeilinger) state. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.