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.
### Hidden variables

### Bell's inequality

### Related Thought Experiments

### Experimental Confirmation

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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, [John Bell]? derived the Bell inequalities, showing that quantum mechanics could be experimentally distinguished from a very broad class of local 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 electron pairs are prepared in such a way that, if both observers measure the spin of their electron along the same axis, then they will always get opposite results: Whenever Alice measures a spin of her electron of +1/2 along the z axis, Bob's spin measurement along the z axis inevitably returns -1/2, and vice versa. The state of each two-electron composite system can be described by the state vector

- |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 σ_{3}), obtaining +1/2 or -1/2 with equal probability. Bob will then measure the opposite spin on his electron.

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.

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. We can then generate the following table:

Alice Bob a b c a b c freq + + + - - - N_{1}+ + - - - + N_{2}+ - + - + - N_{3}+ - - - + + N_{4}- + + + - - N_{5}- + - + - + N_{6}- - + + + - N_{7}- - - + + + N_{8}

Each row describes one type of electron pair, with their respective hidden variable values and their probabilites N. Probabilities are always non-negative, and therefore:

- N
_{3}+ N_{4}≤ N_{3}+ N_{4}+ N_{2}+ N_{7}

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+) = N
_{3}+ N_{4}

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

- P(a+,c+) = N
_{2}+ N_{4}

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+) = N
_{3}+ N_{7}

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 locality assumptions. We will now show that the predictions of quantum mechanics violate 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(2
*i*θ) |b-> |^{2} - = 1/2 ( | <b+| cos 2θ |b-> |
^{2}+ | <b+|*i*sin 2θ |b+> |^{2}) - = 1/2 sin
^{2}2θ

- = 1/2 | <b+| D(

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

- sin
^{2}2θ ≤ 2 sin^{2}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, a strange hidden variable theory might survive by somehow violating the broad assumptions we made above. However, no satisfactory one has been invented to date.)

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.

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

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 (that is to say, there is no time for even 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)

Hardy, L. Nonlocality for 2 particles without inequalities for almost all entangled states. Physical Review Letters 71: (11) 1665-1668 (1993)

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

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