Bell inequality

The Paradox Defined

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.

Of the several objections to the then current interpretation of the quantum mechanics spearheaded by Einstein, the EPR paradox was the subtlest and most successful. The EPR paradox has not been 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'. First, let's briefly review the history:

How the EPR Paradox Affects Our Understanding of Particles

The shift was caused by the EPR thought experiment, which has shown how to measure the property of a particle, such as a position, without disturbing it. In today's terminology, we would say that they did the determination by measuring the state of a distant but entangled particle. According to quantum mechanics, the state of our particle will instantly change even though we did not disturb it in any local way. It is called a paradox, since it conflicts with our classical intuition —specifically,with the principle of locality, which is also one of the basic principles of the theory of relativity.

The very concept of 'entanglement' also conflicts with our intuition the same way. One possibility is that quantum mechanics is just wrong. However, experiments have shown that entanglement does occur, and in fact quantum entanglement has practical applications in the field of quantum cryptography and quantum computation. In quantum cryptography, an entangled signal is sent down a communications channel making it impossible to intercept and rebroadcast that signal without leaving a trace. In quantum computation, entangled states allow simulatanous computations to occur in one step.

We could argue that the EPR paper 'discovered' entanglement. The concept, also called 'nonlocal behaviour' and 'quantum weirdness' has no classical analogy. It is the fact that QM treats two particles, which interacted in the past (and so became entangled) and then separated spatially (i.e., 'flew apart'), as one object. When one such particle one is changed, the other will change too. Instantly. That behaviour, which Einstein called 'spooky action at distance' Einstein considered unacceptable. Before it was accepted as real and inevitable by most physicists one escape route had to be closed, namely the possible existence of 'hidden parameters'.

It was Bell who closed that escape route. The setup of the EPR experiment and Bell's theorem are described in separate pages. Here we proceed historically and first describe two Bohm's contributions and then explain the conceptual meaning of the hidden parameter using a parable of color.

Further Explanation Using Color

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

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

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

"So!", you may say: "there is no paradox here!". The one which went left was always red, the one which went right was always green. Alice just did not know which was the case, until she did her measurement. There is no need for any 'instant sync at distance', no need for spooky action."

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

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 both be determined at the same time. QM says they do not both exist. Potentia is converted to pure state, red or green, when we measure it. Instantly, the other, entagled particle, has her potentia to jump to green or red. To avoid 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 'spooky action at distance' which Einstein disliked. Bohm disliked it too and so he constructed a quite interesting hidden parameter theory which did agree with the experiment. Bell disliked the 'action at distance' aka non-locality as well. (Actually, I do not know of anyone who likes it). 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 investigated and discovered two things:

1. that von Neumann's proof was wrong.
2. that Bohm's theory was non-local.

Eventually, he corrected the Von Neumann's error and generalised von Neumann's proof to a whole class of theories.

And so, in 1964, John Stewart Bell did show that whole class of theories, known as hidden variable theories is either non-local or the have to satisify Bell inequality. Quantum mechanics predicts that inequality is not satisified. To make sure, additional experiments were made to confirm that predicted action at distance is indeed instant (or at least faster than light).

Modern Perspectives on the EPR Paradox

However, the book is not closed yet on this issue. The QM experiments are different from experiments on macroscopic scale, which is directly accessible to our senses. In QM we obtain count of clicks n a Geiger_counter or spots on a photographic plates and those results have to be interpreted by some abstract reasoning. There are assumptions explicitely made or hidden and effects (such as quantum_efficiency which may be just artefacts of todays measuring devices or a fundamental limits not fully accounted for by today's theory. For this reason the topic remains active and some people are still looking for .

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
• Original papers in pdf format
• Recent doubts about conclusions
• Lasting Contribution of the EPR paper

Referenced By