Bell's theorem

The Bell's inequality is a bound on mean values ​​of measured values, which was given in 1964 by John Bell. The inequality holds in all physical theories that are realistic and locally, and where you can regardless of the system to be measured randomly select, " this or that " to measure. These restrictions imply simplified that it is theory such as Newtonian mechanics or the Maxwellian electrodynamics is a " classic ".

Accordingly, it is not a " relatively unimportant mathematical quirk ", but a relation of fundamental importance for the comparison of quantum mechanics and classical physics.

Precise:

Quantum mechanics is not a realistic and local theory. The calculated mean values ​​in quantum mechanics violate Bell's inequality. Therefore, the quantum mechanics can not be completed by adding hidden variables to a real and local theory (as opposed to a collection of Albert Einstein ).

In entangled photon pairs the violation of Bell 's inequality has been measured. Their polarization properties are consistent with quantum mechanics and are not compatible with the assumption of reality and locality.

This means that are not yet defined all readings prior to the measurement or that the readings are not locally depend on distant, unpredictable decisions or that you can not choose arbitrarily, " this or that " to measure.

Derivation

We consider polarization measurements on pairs of photons that are emitted from a source in opposite directions and are measured separately in two places.

Polarizing filter to polarize photons in a plane perpendicular to the direction of propagation direction let photons that are polarized in the direction, unhindered by and absorb with security photons whose polarization direction is vertical. Here, a unit vector in the plane perpendicular to the propagation direction of the photon.

Turning the filter in its plane, we obtain a filter that polarizes in a rotated direction. The probability that a photon polarized in the direction, freely passing through a filter that is polarized in the direction according to the law of Malus

With the rest being absorbed probability. Here is the of and included angle.

The polarization of the investigated photon pairs is due to their origin not independent but intertwined: If the directions of the polarizing filter at both measurement locations match, then the one photon absorbed if and only if also the other photon is absorbed.

For polarization measurements of photons assumed the reality of adoption, in any case, stand firm for all directions, whether the photon will be absorbed, even though in each case the polarization filter can be measured only in one direction.

In the measurements, it is assumed that one can choose the direction of polarization of the two filters will. Which direction of the polarizing filter one chooses not depends on the photon pair.

Location subject to the photon pair, that the direction of a polarizing filter does not affect it if the other photon is absorbed. This presents you, thereby ensuring that the directions are only chosen randomly so late that one, even with the speed of light signals can know nothing of this choice in other measurement.

We consider a series of repeated measurements on photon pairs and number them consecutively with. If at the th measurement of the polarized filter in a direction and the photon goes through, we write this result in as, is it absorbed, we set. With we denote the outcome that would result in an attempt number if we would measure the polarization in the direction of the first photon of the pair, and with the corresponding result if we were to measure the polarization in the direction of the second photon of the pair. Correspond to or, depending on whether the second photon of the pair in the test with the number by the second polarized in the direction filter comes or not.

As the results, and only the values ​​1 or -1 may have the inequalities hold in all cases

For either, then both sides are equal to 0, or it holds, then the right hand side has the value 2 and the left side is 2 or -2.

Since in the same direction both filters the one photon is then absorbed exactly, even if the other photon is absorbed, apply to all cases

Used in the inequalities, it follows

The mean of the products of the measurement results is the sum of the products, divided by the number of trials,

Is obtained according to the averages of the measurement results and.

Adding up the inequalities and inform you the result by the number of trials, we obtain the Bell's inequality for averages of products of polarization values

In the derivation of Bell's inequality, it was assumed that in each trial in the three directions are fixed, the results of polarization measurements and, although it can be actually measured in only one direction. We have also assumed that the results do not depend on a particle is measured in the direction at the other particles that it is therefore not depending on the direction of the second filter is results or the results of the first photon. Finally, we assume that the average over all imaginary experimental results with the average over all trials actually exported matches and that no property of the photon pair affects the random choice of measurement directions.

Quantum mechanical averages

In quantum mechanics results for the mean value of the polarization measurements

This follows from:

But it is the linear combination of the average values ​​, as found in the Bell inequality,

Include not less than 1 for all directions one chooses, for example, as between the bisector and 60 degrees,

Is obtained for the linear combination of the quantum mechanical averages

In marked contrast to Bell 's inequality for local realistic theories.

Thus, the quantum mechanical probabilities can not (as opposed to the adoption of Albert Einstein ) resulting from a more complete local theory due to the ignorance of hidden parameters that define the output of every conceivable measurement.

Experimental studies

Requirements

In order to prove the violation of Bell 's inequality convincing, the experiment must meet the following requirements:

Widerlegungsexperimente

Since the late 1960s, many experiments were conducted to prove the violation of a Bell 's inequality:

• CA Kocher and Eugene Commins (1967 ) observed correlations in pairs of photons, which are emitted by excited atoms of calcium.
• Stuart Freedman and John Clauser (1972 ) used this process to demonstrate a first violation of Bell 's inequality.
• Aspect, Dalibard and Roger (1982 ) used a different process in the calcium atom, the higher count rates and thus resulted in a more significant injury. In addition, both polarization filters were 12 meters away and the choice of measurement directions was carried out by a fast random generator only after both photons left the source.
• Weihs and colleagues ( 1998) used polarization- entangled photons that were produced by spontaneous parametric fluorescence. The polarization filters were 400 meters, so that information transmission over the direction of measurement due to the finite speed of light was not possible.
• Rowe and colleagues ( 2001), a violation of the inequality to demonstrate the basis of measurements of ions in a trap succeeded. In this case, all events could be detected (see: Requirements for the experiment ).

The result of each experiment - that Bell's inequality is violated - shows explicitly that the relevant physics of quantum phenomena involved, is not classical.

Conclusions

You can dismiss it as wrong, quantum mechanics is not easy. It is in line with the experimental findings.

One can instead Einstein's postulates, in particular the notion of hidden variables give up and accept the fact that the wave function only defines the probability of the measured values ​​, but not which measured value occurs in each individual case. This is the Copenhagen interpretation of quantum mechanics, which prevails among physicists. So construed, quantum mechanics is non- real, as opposed to the ideas of Einstein, Podolski and Rosen ( see Einstein - Podolsky-Rosen paradox) because a measurement is not easy reads a property, but finds (more precisely: we produce ), which previously was not certain. In addition, the quantum mechanics is non- local, because extending the quantum state of the photon pair over both test sites.

In its Copenhagen interpretation of quantum mechanics is therefore not sufficient Einstein's receivables to a full, real and local description of physics. This Einstein had recognized and criticized. But he was wrong to believe that quantum mechanics could be real and locally by adding hidden variables.

It is the locality to give up and stick to reality, such as in Bohmian mechanics. Bohm suggests the wave function as a non- local reference field of classical particles. Whether this interpretation leads to physical insights, is controversial among physicists.

Related

The CHSH inequality ( 1969 by Clauser, Horne, Shimony and Holt developed ) generalized Bell's inequality to arbitrary observables. It is experimentally easier to verify.

DM Greenberger, MA Horne and A. Zeilinger 1989 described an experimental setup, the GHZ experiment with three observers and three electrons to distinguish with a single set of measurements, the quantum mechanics of a quasi- classical theory with hidden variables.

Hardy examined in 1993 a situation theoretically non-locality can be shown with the.

The experiments on the violation of Bell 's inequality leave open whether (as in the Copenhagen interpretation ) in addition to the assumption of locality, the adoption of an "objective reality " must be abandoned. Leggett formulated in 2003 an inequality, which is independent of the assumption of locality and should allow to verify the assumption of objective reality. Recent experiments of Gröblacher et al. suggest that the Leggettsche inequality is violated. The interpretation of the results is, however, debatable.

Others

Published in 2001, Karl Hess and Walter Philipp mathematician essays in which they pointed to a possible loophole in Bell 's theorem. Their argument and its model has been criticized by Zeilinger and others.