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Bell's theorem

Table of contents
1 Bell's inequality
2 Derivation of the inequality
3 Hidden variables
4 Comparison with Quantum Mechanics
5 Implications of Violation of Bell's Inequality
6 Related Thought Experiments
7 Experimental Confirmation
8 See also
9 References

Bell's inequality

The history and physical implications of this derivation is discussed on EPR page.

Briefly: based on certain assumptions about the the microscopic world, which include

  1. 'locality'
  2. 'realism'
  3. 'joint measurability'

and may be other, less obvious assumptions,

a mathematical relation (namely an inequality) is derived concerning outcome of some measurements of microsocopic particles. Experiments violate that relation. Conclusion is often that those assumptions, and in particular realism and locality, are not compatible, they cannot both be true in any consistent theory.

The following is a simplified description of the EPR scenario, developed by Bohm and Wigner.

We follow the approach in Sakurai (1994).

Derivation of the inequality

Pick three arbitrary directions a, b, and c in which Alice and Bob can measure the spins of each electron they receive. We assume three hidden variables on each electron, for the three direction spins. We furthermore assume that these hidden variables are assigned to each electron pair in a consistent way at the time they are emitted from the source, and don't change afterwards. We do not assume anything about the probabilities of the various hidden variable values.

Alice and Bob are two spatially separated observers. Between them is an apparatus that continuously produces pairs of electrons. One electron in each pair is sent towards Alice, and the other towards Bob. The setup is shown below:

The electron pairs are specially prepared so that if both observers measure the spin of their electron along the same axis, then they will always get opposite results. For example, suppose Alice and Bob both measure the z-component of the spins that they receive. According to quantum mechanics, each of Alice's measurements will give either the value +1/2 or -1/2, with equal probability. For each result of +1/2 obtained by Alice, Bob's result

will inevitably be -1/2, and vice versa.

Mathematically, the state of each two-electron composite system can be described by the state vector

.

Each ket is labelled by the direction in which the electron spin points. The above state is known as a spin singlet. The z-component of the spin corresponds to the operator (1/2)σz, where σz is the third Pauli matrix. (The quantum mechanics of spin is discussed in the article spin (physics).)

Hidden variables

It is possible to explain this phenomenon without resorting to quantum mechanics. Suppose our electron-producing apparatus assigns a parameter, known as a hidden variable, to each electron. It labels one electron "spin +1/2", and

the other "spin -1/2". The choice of which of the two electrons to send to Alice is decided by some classical random process. Thus, whenever Alice measures the z-component spin and finds that it is +1/2, Bob will measure -1/2, simply because that is the label assigned to his electron. This reproduces the effects of quantum mechanics, while preserving the locality principle.

The appeal of the hidden variables explanation dims if we notice that Alice and Bob are not restricted to measuring the z-component of the spin. Instead, they can measure the component along any arbitrary direction, and the result of each measurement is always either +1/2 or -1/2. Therefore, each electron must have an infinite number of hidden variables, one for each measurement that could possibly be performed.

This is ugly, but not in itself fatal. However, Bell showed that by choosing just three directions in which to perform measurements, Alice and Bob can differentiate hidden variables from quantum mechanics. 

b c   a b c  freq
+ + +   - - -   N1
+ + -   - - +   N2
+ - +   - + -   N3
+ - -   - + +   N4
- + +   + - -   N5
- + -   + - +   N6
- - +   + + -   N7
- - -   + + +   N8
Each row describes one type of electron pair, with their respective hidden variable values and their probabilites N. Suppose Alice measures the spin in the a direction and Bob measures it in the b direction. Denote the probability that Alice obtains +1/2 and Bob obtains +1/2 by

P(a+,b+) = N3 + N4

Similarly, if Alice measures spin in a direction and Bob measures in c direction, the probability that both obtain +1/2 is

P(a+,c+) = N2 + N4

Finally, if Alice measures spin in c direction and Bob measures in b direction, the probability that both obtain the value +1/2 is

P(c+,b+) = N3 + N7

The probabilities N are always non-negative, and therefore:

N3 + N4 < N3 + N4 + N2 + N7
This gives
P(a+,b+) < P(a+,c+) + P(c+,b+)

which is known as a Bell inequality. It must be satisfied by any hidden variable theory obeying our very broad locality assumptions. We will now show that the predictions of quantum mechanics violate this inequality.

Comparison with Quantum Mechanics

Suppose a, b, and c lie on the x-z plane, and c bisects a and b with angle θ. We can calculate each of the probabilities with the help of the rotation operator.

Consider P(c+,b+). Alice measures the spin in the c direction, and obtains +1/2 with probability 1/2. This collapses Bob's electron to |c-⟩B. Working in the state space of Bob's electron and dropping the B subscripts, we can calculate the conditional probability that Bob then obtains +1/2 when measuring the spin in the b direction:

P(z+,b+) = 1/2 | ⟨c+|b-⟩ |2
= 1/2 | ⟨c+| D(y, θ) |c-⟩ |2
= 1/2 | ⟨c+| exp(i θ σy) |c-⟩ |2
= 1/2 ( | ⟨c+| cos θ |c-⟩ |2 + | ⟨c+| i sin θ |c+⟩ |2 )
= 1/2 sin2 θ

where

σy is the second Pauli matrix, which generates the rotation operator D(y,θ). The other two probabilities can be obtained with similar calculations. Bell's inequality then becomes:

1/2 sin2 2θ < 1/2 sin2 θ + 1/2 sin2 θ

But this inequality is violated for θ = π/8:

0.25 < 0.1464...

If Alice and Bob actually perform the experiment exactly as described above using three axes that are separated by angles of π/8 and obtain the probabilities predicted by quantum mechanics, then their results will violate Bell's inequality. This would falsify the class of local hidden variable theories which we considered.

Implications of Violation of Bell's Inequality

There are several popular responses to this situation:

The first is to simply assume that quantum mechanics is wrong. However, this can be experimentally tested and experiments have supported quantum mechanics: Alice and Bob will indeed measure the predicted probabilities.

The second is to abandon the notion of hidden variables and to argue that the wave function does not contain any information about the outcome of the measurement of the values in the particles. This corresponds to the Copenhagen interpretation of quantum mechanics.

One may also give up locality: the violation of Bell's inequality can be explained by a non-local hidden variable theory, in which the particles exchange information about their states. This is the basis of the Bohm interpretation of quantum mechanics. However, this type of interpretation is regarded as inelegant, since it requires all particles in the universe to be able to instantaneously exchange information with all other particles in the universe.

Finally, one subtle assumption of the Bell's inequality is counterfactual definiteness. In reality, one can only measure the particles once without collapsing the wave function, and yet Bell's inequality involves talking about alternative measurements that cannot be performed and assuming that these would result in well defined outcomes. But relaxing this assumption one can also resolve Bell's inequality. In the Everett many-worlds interpretation, the assumption of counterfactual definiteness is abandoned because this interpretation assumes that the universe branches into many different observers each which measures a different observation.

One active area of theoretical research is to attempt to find other hidden assumptions in Bell's inequality.

Related Thought Experiments

The CHSH inequality, developed in 1969 by Clauser, Horne, Shimony, and Holt, generalizes Bell's inequality to arbitrary observables. It is expressed in a form more suitable for performing actual experimental tests.

Bell's thought experiment is statistical: Alice and Bob must carry out several measurements to obtain P(a+,b+), and the other probabilities. In 1989, Greenberger, Horne, and Zeilinger produced an alternative to the Bell setup, known as the GHZ experiment. It uses three observers and three electrons, and is able to distinguish hidden variables from quantum mechanics in a single set of observations.

In 1993, Hardy proposed a situation where nonlocality can be inferred without using inequalities.

Experimental Confirmation

Beginning with the Kocher and Commins experiment in 1967, several experiments have been carried out to test the above results, and Bell's inequality was found to be violated, in one case by tens of [[standard deviation|standard deviations]].

Experiments generally test the CHSH generalization of Bell's inequality, and use observables other than spin (which is in practice not easy to measure.) Most use the polarization of photon pairs produced during radioactive decay. However, the basic approach is very similar to the simple model presented above.

In 1998, Weihs, Jennewein, et al. at the University of Innsbruck first demonstrated the violation for space-like separated observations (that is to say, there is no time for even a light signal to propagate from one observation event to the other.)

See also

References