Einstein–Podolsky–Rosen paradox

The EPR paradox is a thought experiment which show that quantum mechanics leads to very counter-intuitive paradoxical consequences. It is named after Einstein, Podolsky, and Rosen, who published the idea in 1935. It is also referred to as the EPRB paradox after Bohm, who improved the formulation of the thought experiment.

Of the several objections to then current interpretation of the quantum mechanics spearheaded by Einstein, the EPR paradox was the subtlest and most successful. EPR paradox was not resolved or explained in a way which agrees with classical intuition to this day. It brought a new clarity and permanent shift in thinking about 'what is reality' and what is a 'state of a physical system'. Lets first review the history briefly:

Before 1936 the generaly accepted view was that a particle, such as an electron HAS a position and a momnetum but 'we just cannot KNOW both' at the same time. This view is present in a typical textbook explanation of the Heisenberg Uncertainty Principle . In such explanation, the 'more exactly we measured the position', the 'more we disturbed the particle' and it's momentum became that much more unknown. The numerical measure of uncertanainties satisifies the Heisenbergs Principle, but the interpretation is no longer accepted.

That shift was caused the EPR thought experiment, which has shown how to measure the property of particle, such as a position, WITHOUT disturbing it. 
 In todays terminology, we would say, they did the determination by measuring state of a distant but entagled particle. According to Quantum Mechanics, the state of our particle, will instantly change even though we did not disturbed it in any local way. Now that is a paradox - since it conflicts with our classical intuition - with principle of locality. This is one of basic principles of the thory of relativity.

Let's, with hindsight, point out that the very concept of 'entaglement' which
today has been verified experimentaly and looking for applications, also
conflicts with our intuitio the same way.  We could argue that EPR paper 
'discovered' entaglement, but one escape route had to be closed, escape

route which our intuition will use the escape the quantum weirdness,

of 'indeterminate state' concept, which has no classical analoyg. That escape

route was the possibility of 'hidden parameters'

It was Bell who closed that escape route. The setup of the EPR

experiment and Bells  reasoning is described given bellow in a mathematical
form. Before we proceed to that, we must mention two Bohm's contributions   and we also explain conceptual meaning of the hidden parameter using a parable:

Bohm substituted measurement of spin coordinates for measurement of momentum and position. Classical analogy of spin of a photon is polarisation of light, which is quite familiar. However mathematical description of this property in QM is complex as you will see below. The experiment measuring spin is however easier then the the original EPR setup.

We now describe the concept of EPR using word 'red' and 'green' for 'spin up' and 'spin down':

Imagine that white particle splits in two, one green one red. (Here the color (spin) is conserved and red+green=white).

One flies left, one right, and we do not know which one is which.

When Alice, on the left, will notice (measure) that her is red she will instantly ant surely know that Bob's measurement on the right, far far away, will be green.

 So! You will say. That is no paradox here. The one which went left

was always red, the one which went right was always green. There is no need for any 'instant sync at distance'

That is indeed a sensible and intuitive explanation of the experimental result, and we call 'a hidden parameter' intrperetation.

Why hidden? Because when you look at the mathematical quantity, which according to QM describes the 'state' of that particle, it does not have that color there. It has a possibility of red, and possibility of green. These possibilities or 'potentia' for one component of spin (an angle of polariser) are complementary to such potentia for another component (another angle of polariser). Because they are complementary, just like postion and momentum, they cannot be both be determined at the same time. QM says they do not both exist. Potentia is converted to pure state, red when we measure it. Instantly, the other, entagled particle, has her potentia to jump to green. To avod that weirdness, hidden parameter theory says, it was there, it was red for x-component and red for y-component, (violating the Heisenberg's principle) and we just were not able to see it. It was hidden.

Our intuition tend to believe they must be there, because otherwise we would have to admit the 'ugly action at distance' which Einstein disliked. Bohm disliked it too and so he constructed a quite interesting hidden parameter theory which did agree with experiment.

Bell disliked the 'ugly action at distance' aka non-locality as well. (Actually, I do not know of anyone who likes them). However, there was an early mathematical proof by Von Neumann that said that what Bohm hoped he had, a local realistic theory, which would give same results as QM is impossible.

Bell discovered two things, One, that von Neumann porof was wrong. Two, that Bohm's theory was non-local. Actually, he corrected the error and generalised von Neumanns proof to whole a class of theories.

In 1964, when John Stewart Bell derived the Bell inequalities, which showed that whole class of theories, known as hidden variable theories is either in conflict with experiments or non-local.

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 yet-undiscovered 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 his 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.

A Simple Model

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:

EPR thought experiment

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

\left|\psi \right\rangle ={1 \over {\sqrt {2}}}\left(\left|z+\right\rangle _{A}\left|z-\right\rangle _{B}-\left|z-\right\rangle _{A}\left|z+\right\rangle _{B}\right)

.

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.

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 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
+ + +   - - -   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.

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 σ_y) |c-⟩ |²

= 1/2 ( | ⟨c+| cos θ |c-⟩ |² + | ⟨c+| i sin θ |c+⟩ |² )

= 1/2 sin² θ

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:

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 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 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:

Quantum teleportation

References

A. Einstein, B. Podolsky, and N. Rosen: Can quantum-mechanical description of physical reality be considered complete? Physical review 47, 777 (1935).
Bell, J.S.: On the Einstein-Poldolsky-Rosen paradox. Physics 1, pp. 195-200 (1965)
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
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