Bell’s inequalities

A simple example sufEce.s to show that the quantum mechanical prediction does not, in general, satisfy Hell s inequality. Say, for example, that all tliree unit vectors lie on the same plane, Z(a,/9) = 60deg, Z(/3,7) = 60deg, and Z(a,7) = 120deg. Substituting thes e values into Bell s inequality yields the nonsensical result that i<0. 

Bell s result, made all the more remarkable for the very few assumptions he makes to derive it, rather dramatically asserts that cither EPR s three premises are wrong or quantum mechanics is incorrect. However, recent experiments by A.spect, et.al. ([aspect82a], [aspect82b]). On and Mandel [01188], and others have shown, virtually conclusively, that nature satisfies the quantum mechanical prediction (equation 12.54) and not Bell s inequality (equation 12.55), thus strongly denying the possibility of local hidden variables. We are thus left with what is arguably one of the deepest mysteries in the foundations of physics the existence of a profoundly nonclassical correlation between spatially-far separated systems, or nonscparability. 

Contemporary developments include John Bell s [39] discovery of his famous inequality that is predicated on the assumptions of both locality and realism. Bell s inequality is violated by quantum mechanics, and consequently, it is frequently argued, one cannot accept quantum mechanics, realism, and locality. Experiments on correlated particles appear to demonstrate that the Bell... 

M. Flato, C. Piron, J. Grea, D. Sternheimer, and J. P. Vigier, Are Bell s inequalities concerning hidden variables really conclusive Helv. Phys. Acta 48(2), 219-225 (1975). 

On inserting a fourth unit vector d, the symmetrical form of Bell s inequality [41] is obtained from... 

Eq (16.9) is an instance of Bell s inequality. A more general form can be expressed... 

It is readily apparent that Bell s inequality in the form (16.7) is in disagreement with quantum mechanics. With the coincidence probabilities (16.6), the sense of the inequality is, in fact, reversed since... 

Figure 16.6 Graphical demonstration of Bell s inequality (16.10) using a three-dimensional Venn diagram. Each of eight regions represents one combination of the variables a, b and c. It is clear that P(a. b) is a subset of P(h. c) + / (c. ti).

For angles 0 = 0°, 45° and 90°, the predictions of local realism and quantum mechanics agree. For the general case, a modification of Bell s inequality appropriate for photon polarizations was derived by Clau.ser. Horne, Shimony and Holt. The CHSH relation predicts <2, where... 

The remarkable feature of this multiple entanglement is that, for any arrangement of the polarizers, the readings of any three detectors will determine the fourth with 100% certainty. Unlike the experiments testing Bell s inequality, the predictions for a GHZ state specify a definite outcome rather than a statistical distribution. 

Bell s inequality provides a clear-cut test of local reality vs. quantum mechanics. The unambiguous answer, from a variety of experiments which we describe in the next section, is that quantum mechanics wins Thus, we can conclude that we live in a Universe which does not respect local reality. Quantum entanglement—a term introduced by Schrddinger—really happens In drawing this conclusion we are actually glossing over a number of still-unresolved hair-splitting metaphysical arguments. This remarkable result is often summarized as Bell s theorem ... 

Figure 16.9 Results of Aspect experiment. Dotted curve shows prediction of quantum mechanics (multiplied by 0.955 to correct for detection efficiency). Shaded regions show where Bell s inequality is violated. [From A. Aspect, P. Grangier, and G. Roger. Phys. Rev. Lett. 49,91 (1982)].

A further modification addresses a remote loophole for local realism in which the polarization detectors might somehow be able to signal one another at subluminal speed. Aspect designed an experiment in which mirrors switched at high speed could direct the two photons to different detectors, after they were already inflight. It was verified in this "delayed choice experiment that Bell s inequality is still violated, in what is currently considered the most conclusive test of nonlocality. 

Aspect experiment An experiment conducted by the French physicist Alain Aspect (1947- ) and his colleagues in the early 1980s to test Bell s inequality see Bell s theorem). The experiment involves producing... 

In the case that a particle with spin 0 decays into two electrons, observing the spin of one electron determines the spin of the second electron. Since this indicates that information transmits faster than light, it violates the relativistic theory Einstein-Podolsky-Rosen paradox) (Einstein et al. 1935). Bohr could not provide a counterargument to this. Later, this paradox was resolved by Bell s inequality, which limits the correlation of subsequent measurements of particles that have interacted and then separated on the local hidden-variable theory (Bell 1964), and Aspect s experiment, which proves the violation of this inequality (Aspect et al. 1982). 

Quantum mechanics tells us that, before observation is made, both spins share - with equal weight - the states 11> and ].). Before a measurement, the probability of either spin to be found in either state is 50%. However, if one performs a measurement, say, in first spin, the state of the second spin becomes determined, no matter the distance between them For many years, this non-local property of entanglement has been perhaps the most controversial and debated aspect of quantum mechanics, since Einstein, Podolsky and Rosen pointed the problem out in a historical paper published in 1935 [13]. Since the EPR paper, as it became known, many decades were necessary until the discovery of a criterion to decide whether non-locality was a physical reality or just a mathematical property of the quantum formalism. This was a main contribution of John Bell, who in 1964 presented such a criterion [14]. The so-called Bell inequality is a statistical test for quantum nonlocality. However, in 1964 there were no experimental conditions to implement such a test in a real physical system. This came about only in 1982 as a seminal work published by Aspect, Grangier and Roger [15], entitled Experimental realization of Einstein-Podolsky-Rosen-Bohm gedankenexperiment a new violation of Bell s inequalities. This paper is considered - at least for the great majority of physicists - as the work where the nonlocality, inherent to entangled states, is demonstrated to be definitely part of the physical world. 

In opposite, an entangled mixed state is a state for which no such a decomposition exists. Non-locality and Bell s inequality... 

In the year of 1964 John Bell discovered a remarkable result [14], which sets the rules for deciding experimentally if the non-locality is indeed a fact of entangled systems. The result is expressed in the form of an inequality, which establishes an upper limit for the local correlations in a two-partite system. Since then, it is known as the Bell s inequality. For two qubits, Bell s inequality says that a determined quantity S, essentially a correlation function between observables, should not overcome the value 2. However, quantum mechanics is non-local, and predicts the value S = 2 /2 2.83, for an entangled state, therefore violating the Bell s inequality. 

In the year of 1982 the Bell s inequality was tested in a famous experiment [16]. A French group led by Alain Aspect used entangled photons, produced by the decay of electrons from a excited state of °Ca, and demonstrated the non-local quantum correlations between the polarization of the two photons. They determined the value of the quantity S experimentally, and found S = 2.70 0.05, which is very close to the ideal result predicted by quantum mechanics. 

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