Of the several objections to quantum mechanics spearheaded by Einstein (who disliked the theory for being probabilistic), the EPR paradox was the subtlest and most successful. It was not fully resolved until 1964, when John Stewart Bell derived the Bell inequalities, which showed that there are observable differences between quantum mechanics and alternative hidden variable theories.
The EPR paradox draws attention to a phenomenon predicted by quantum mechanics known as quantum entanglement, in which measurements on spatially separated quantum systems can instantaneously influence one another. As a result, quantum mechanics violates a principle formulated by Einstein, known as the principle of locality[?] or local realism, which states that changes performed on one physical system should have no immediate effect on another spatially separated system.
The principle of locality is persuasive, because it seems at first sight to be a natural outgrowth of the theory of special relativity. According to relativity, information can never be transmitted faster than the speed of light, or causality would be violated. Any theory which violates causality would be deeply unsatisfying, and probably internally inconsistent. However, a detailed analysis of the EPR scenario shows that quantum mechanics violates locality without violating causality, because no information can be transmitted using quantum entanglement.
Nevertheless, the principle of locality appeals powerfully to physical intuition, and Einstein, Podolsky and Rosen were unwilling to abandon it. They suggested that quantum mechanics is not a complete theory, just an (admittedly successful) statistical approximation to some yetundiscovered description of nature. Several such descriptions of quantum mechanics, known as "local hidden variable theories", were proposed. These deterministically assign definite values to all the physical quantities at all times, and explicitly preserve the principle of locality.
In 1964, 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 local hidden variable theories are untenable and that the principle of locality does not hold. Therefore, 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 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 zcomponent of the spins that they receive. According to quantum mechanics, each of Alice's measurements will 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 twoelectron 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 zcomponent 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).)
It is possible to explain this phenomenon without resorting to quantum mechanics. Suppose our electronproducing 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 zcomponent 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 zcomponent 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. 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
Similarly, if Alice measures spin in a direction and Bob measures in c direction, the probability that both obtain +1/2 is
Finally, if Alice measures spin in c direction and Bob measures in b direction, the probability that both obtain the value +1/2 is
The probabilities N are always nonnegative, and therefore:
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.
Suppose a, b, and c lie on the xz 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:
σ_{y} 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:
But this inequality is violated for θ = π/8:
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.
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 nonlocal 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 manyworlds 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.
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.
Beginning in the early 1970s, 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 spacelike separated observations (that is to say, there is no time for even a light signal to propagate from one observation event to the other.)
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