Bell's theorem

Bell's inequality

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

Note that the inequality derived here is not the original discovered by John Bell, nor has it been used in any of the actual Bell test experiments. In the latter, the tests used are based on the modified inequalites devised by Clauser et al., designed for use in the real situation in which not all particles are detected (see CHSH or CH74 tests ). The CHSH inequality and related ones are currently the ones in favour, though they are not applied in a way that the originators would have approved. Unless detection efficiencies are very high they are subject to loopholes and are in danger of giving misleading results. Violation does not necessarily mean that no "local hidden variable" theory is possible.

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

1. locality
2. realism
3. joint measurability

and other technical assumptions a mathematical relation (namely an inequality) is derived concerning the outcome of some measurements of microsocopic particles. Experiments are currently generally accepted as having violated that relation. The conclusion is often that those assumptions, and in particular realism and locality, are not compatible, they cannot both be true in any theory based on quantum mechanics.

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:

(This is just a thought-experiment, remember. Real experiments on pairs of electrons are not feasible and most "Bell test experiments" have instead been based on either the polarisation direction or the frequency and phase of light, assumed to be in "photons".) 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)σ_z, where σ_z 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.

 a 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. (Note, however, that these assumptions include one that Bell took for granted but which is not true in optical Bell tests, namely that every single particle was detected. The failure of this assumption results in the best known "loophole" that enables hidden variables to survive apparent violations of the test.) We will now show that the predictions of quantum mechanics violate this inequality.

Comparison with quantum mechanics

Suppose a, b, and c lie nearly on the x-z plane (in fact, they need to be linear independent and therefore can't lie exactly on a plane), and c lies on the z-axis bisecting a and b with angle θ. We can calculate each of the probabilities with the help of the rotation operator. Consider P(c+, b+), which in Quantum Mechanics is equal to the squared scalar product between the above mentioned initial state |ψ〉 = 1/√2 ( |c+, c-〉 - |c-, c+〉 ) and the final state |c+, b+〉:

where we used the special definition of the scalar product

if the tensor product

is involved. In addition, we used 〈c_i |c_j〉 = δ_ij (the |z_i〉 are a orthonormal base of eigenstates of the spin operator S_z with eigenvalues ±hbar/2), the expansion of e^x as a power series, unitarity of σ_y, and that σ_y |c+〉 = i |c-〉. This last equation can be obtained from the following standard QM results:

σ_y is the second Pauli matrix, which generates the rotation operator D(y, θ) = exp(- i θ/2 σ_y). The other two probabilities can be obtained with similar calculations. Bell's inequality then becomes:

But this inequality is violated for:

If Alice and Bob actually perform the experiment exactly as described above using three axes that are separated by angles within the above interval 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, many "Bell test" experiments have been carried, almost all giving violations of the inequality tested. The latter, though, is not in practice the one derived above but usually the CHSH or related inequality. This has on occasion been violated by tens of standard deviations, but the presence of loopholes means the possibility of a "local hidden variable" explanation has not finally been ruled out.

In 1982 Alain Aspect and his team conducted an experiment in which the detector settings were switched during the flight of the photons, so that there was no time for even a light signal to propagate from one observation event to the other. In 1998 Weihs, Jennewein, et al. at the University of Innsbruck further demonstrated the violation for space-like separated observations.

See also

• Quantum teleportation
• EPR
• BellTestLoopholes

References

• Bell, John S.: On the Einstein-Podolsky-Rosen paradox, Physics 1, 195 (1964), reproduced as Ch. 2, pp 14-21, of J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, (Cambridge University Press 1987).
• J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Proposed experiment to test local hidden-variable theories, Physical Review Letters 23, 880-884 (1969)
• Hardy, L.: Nonlocality for 2 particles without inequalities for almost all entangled states. Physical Review Letters 71: (11) pp. 1665-1668 (1993)
• Sakurai, J.J.: Modern Quantum Mechanics. Addison-Wesley, USA 1994, pp. 174-187, 223-232
• Quantum entanglement and Bell's Theorem
• A. Aspect et al.: Experimental Test of Bell's Inequalities Using Time-Varying Analyzers , Physical Review Letters 49, 1804 (1982)
• Weihs, Gregor et al., Violation of Bell’s inequality under strict Einstein locality conditions , Physical Review Letters 81, 5039 (1998)
• A. Aspect: Bell's inequality test: more ideal than ever. Nature, vol 398, 18 March 1999. http://www-ece.rice.edu/~kono/ELEC565/Aspect_Nature.pdf