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.
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:
Before 1936, the generally accepted view was that a particle, such as an electron, has a position and a momentum 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 an explanation, the 'more exactly we measure the position', the 'more we disturb the particle' and its momentum becomes that much less certain. The numerical measure of uncertainty satisfies Heisenberg's principle, but this (local realistic) interpretation is no longer accepted in professional circles (it still lives in popular books).
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', which has been verified experimentally and is today being considered for practical applications, also conflicts with our intuition the same way. 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 spatialy (meaning '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.
Bohm substituted measurement of spin coordinates for measurement of momentum and position. The classical analogy of spin of a photon is polarization of light, which is quite familiar. However, the mathematical description of this property in QM is complex as you will see below. The experiment measuring spin is, however, easier than the the original EPR setup.
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 be 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 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:
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 then light).
Today most physicists agree that local hidden variable theories are untenable and that the principle of locality does not hold. Therefore, the EPR paradox would only be a paradox because our physical intuition does not correspond to physical reality.
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 [ways to escape the logic of the Bell's theoren].