Wikipedia: EPR paradox

Showing revision 1

The EPR paradox is a [thought experiment]? devised by Einstein, Podolsky, and Rosen in 1935 to attack the theory of quantum mechanics. It is the subtlest and most successful of the several objections Einstein raised against quantum mechanics, which he disliked for its use of probability.

The EPR paradox uses quantum entanglement to demonstrate that quantum mechanics violates the principle of locality (also known as local realism.) The principle, formulated by Einstein, stated that changes performed on one physical system should have no effect on another spatially separated system. This was a persuasive argument, as it appeared to be a natural outgrowth of relativity.

In order to explain away the success of quantum mechanics at producing experimental predictions, it was suggested that quantum effects arise from unknown microscopic parameters, known as "hidden variables".

In 1964, Bell? derived the Bell inequalities, showing that quantum mechanics could be experimentally distinguished from a very broad class of hidden variable theories. Subsequent experiments took the side of quantum mechanics, and most physicists now agree that the locality principle is incorrect. Thus, the EPR paradox is only a paradox because our physical intuition does not correspond to physical reality.

The following is a simplified description of the EPR scenario, developed by Bohm? and Wigner. We follow the approach in Sakurai (1994).

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 state of each two-electron composite system is

: |Ψ> = 2^-1/2 ( |z+>_A |z->_B - |z->_A|z+>_B )

Each ket is labelled by the direction in which the electron spin points. The above state is known as a spin singlet. As discussed in the article quantum entanglement, Alice can measure the z-component of the spin, (1/2)σ_z (where σ_z is the third Pauli matrix σ₃), obtaining +1/2 or -1/2 with equal probability. Whenever Alice measures +1/2, Bob's measurement of σ_z on his electron inevitably returns -1/2. Conversely, whenever Alice measures -1/2, Bob measures +1/2.

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.

Bell's inequality

Pick three arbitrary directions a, b, and c in which Alice and Bob can measure the spins of each electron they receive. We require three hidden variables on each electron. We assume that the probability of each possible set of labels is consistent for each electron pair, but need not specify what the probabilities are. We can then generate the following table:

 Alice    Bob
 a b c   a b c  freq
 + + +   - - -   N₁
 + + -   - - +   N₂
 + - +   - + -   N₃
 + - -   - + +   N₄
 - + +   + - -   N₅
 - + -   + - +   N₆
 - - +   + + -   N₇
 - - -   + + +   N₈

The N's denote the probability of each set of labels occuring, and must be non-negative. Therefore,

: N₃ + N₄ ≤ N₃ + N₄ + N₂ + N₇

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,

: P(a+,c+) = N₂ + N₄

: P(c+,b+) = N₃ + N₇

This gives

: P(a+,b+) ≤ P(a+,c+) + P(c+,b+)

which is known as a Bell inequality, which must be satisfied by any hidden variable theory obeying our very broad assumptions. We will now show that quantum mechanics violates this inequality.

Suppose a, b, and c lie on the x-z plane, c = z, and c bisects a and b with angle θ between each. The probabilities can be calculated using the [rotation operator]?:

P(a+,b+) = 1/2 | <a+|b-> |²

: = 1/2 | <b+| D(y, 2θ) |b-> |²
: = 1/2 | <b+| exp(2i θ) |b-> |²
: = 1/2 | <b+| cos 2θ |b-> |² + | <b+| i sin 2θ |b+> |²
: = 1/2 sin² 2θ

and similarly for the other two probabilities. Bell's inequality becomes:

: sin² 2θ ≤ 2 sin² θ

Simply choose θ = π/8. Then

: 0.5 ≤ 0.2929... (???)

As we claimed, the inequality is violated. Therefore, if Alice and Bob actually perform the experiment and obtain the probabilities predicted by quantum mechanics, then their results cannot be explained by hidden variables. (Conceivably, some very strange hidden variable theory might survive somehow by violating the broad assumptions we made, but no satisfactory one has been invented to date.)

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 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.

Experimental Confirmation

Beginning in the early 1970's, 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 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 (i.e. there is no time for a light signal to propagate from one observation event to the other.)

References

Bell, J.S. On the Einstein-Poldolsky-Rosen paradox Physics 1, 195-200 (1965)

Sakurai, J.J. Modern Quantum Mechanics (Addison-Wesley, USA) 1994, pp. 174-187, 223-232

www-ece.rice.edu/~kono/ELEC565[/Aspect Nature]?.pdf - review paper by A. Aspect