What is the difference between the EPR paradox and Bell's inequalities? I am a newbie here, hope I will be able to get accustomed on this forum.
I am trying to understand what quantum entanglement is. Obviously, for this it is very useful to understand Bell's theorem. What is the difference between the EPR paradox and Bell's inequalities?
If I am correct, with EPR, we are talking about the momentum and coordinate of entangled particles, while with Bell theorem we are talking either about polarization, or about the spin. Did I understand correctly that Bell's inequalities are measured for both spin and polarization?
The main question is why EPR is only a thought experiment, while violation of Bell's inequalities has been verified experimentally.
Suppose we have particles A and B, resulting from the decay of particle C. By measuring the momentum of particle A, we can recalculate the momentum of particle B through the law of conservation of momentum. Next, we measure the coordinate of the particle B. According to the uncertainty relation, we cannot know exactly the B momentum and coordinate at the same time. Hence, the B coordinate will be measured inaccurately. Why can't we check this experimentally? In the experiment, first measure momentum of A, then coordinate of B, and the experiment will confirm that the coordinate B has inaccurate values.
 A: What is the difference between the EPR paradox and Bell's inequalities?
The EPR argument starts by defining the criterion for designating an `element of reality' as

`If, without in any way disturbing a system, we can predict with
certainty (i.e. with probability equal to unity) the value of a
physical quantity, then there exists an element of physical reality
corresponding to this physical quantity.'

The EPR paper argued that because you can measure the property of one particle (not disturbing the other) and know the other immediately because they are entangled, the disposition of the second particle must exist before the measurement as you have not measured that particle, but you know something about it. EPR argued that if the predictions of quantum mechanics are valid without any non-locality (spooky action at a distance), then each particle's reaction to the configuration of the experiment (the measurement) must be predetermined. This, EPR argued, gives existence to the disposition of the particles before their measurement. Given the reality criterion, EPR concluded that quantum mechanics could not be a physically complete description of reality. This Bohr rejected, he argued that their disposition does not exist before the measurement. Bohr argued that the two particles comprise a single system, and measuring one particle is the same as measuring both, a non-local argument.
Bell showed that there are limits on the strength of correlations between the particles given different settings of the experiment. If we assume that the outcome of the experiment is predetermined, then there are certain limits to the inequality, this limit is called Bell's inequality.
The difference is that the EPR paper is an argument against non-locality, and Bell's inequalities are a way of testing this.
Did I understand correctly that Bell's inequalities are measured for both spin and polarization?
This is how Bell reformulated the argument, it is experimentally easier to measure spin or the polarisation of light than it is to measure a particle's momentum. It is, however, the same argument, just with different physical quantities.
The main question is why EPR is only a thought experiment, while violation of Bell's inequalities has been verified experimentally?
As mentioned above, it is easier to measure the polarisation of light. Testing Bell's inequalities is a way of testing the argument put forward in the EPR paper.
A: EPR describes two particles, A and B, which are correlated (Not Entangled) and then move away in mirrored trajectories. The uncertainty principle says it’s impossible to measure both the momentum and the position of a particle exactly. But it is possible to measure just the position of particle A. Then with the exact position of particle A known, and if particle B was truly correlated then the exact position of particle B can be known. EPR proves that particle B simultaneously has a position that is real and a momentum that is real.
EPR sets up a way to measure the momentum or position of B by knowing the measurement of particle A, without particle B being physically disturbed. EPR sets up a paradox that questions quantum mechanics predictions that both values cannot be known, but EPR does seem to show that there must be predetermined values. The EPR paper says: "We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete." Anyone who believes this should be interested in finding the elements of reality that are missing in quantum mechanics.
“Interact” or “Entanglement” are words that came after EPR. These are words that confuse the situation. The EPR thought experiment was obviously describing two particles that are correlated. Not only correlated but perfectly correlated in speed, trajectories, polarization, timing and linear dependency. And if we are talking perfect correlation there may be other factors to include.
Bell’s inequalities set up a mathematical construct that tries to limit the outcomes of these two perfectly correlated particles. I say tries because he only included the first three and ignored timing or linear dependency.
Everyone talks about duality but when it comes down to it, particles are never seriously considered. Waves, waves, waves are all we hear about and discussing photons with real physical properties is usually a big no no. Bell's theorem/inequality states that any physical theory that incorporates local realism cannot reproduce all the predictions of quantum mechanical theory.
For the sake of discussion, I assume that “predictions of quantum mechanics” means Malus Law or cos2theta. After all, most articles on this subject come with an overlay diagram of linear and non-linear slopes depicting classical and QM predictions. These articles make the argument that a physical model cannot reproduce the results of Malus Law.
What if real objects, large enough to see, could be physically correlated in a way that they do reproduce Quantum prediction that match Malus Law? Below I have set up such a situation (Not a theory) where the results do match.
Similar to the original EPR experiment where two particles are prepared there is another interesting experiment that tests the predictions of quantum mechanics. This experiment uses multiple polarizers where particles are sent through the first polarizer and then measured against the second one. The second polarizer can be rotated to different angles and the results do match quantum mechanics or Malus Law.
In order to prove (contrary to Bell’s inequality) that particles can be physically correlated to match Malus Law, I’ll go to an extreme of choosing large ordinary objects. Of course, testing one is not enough and the average results of testing thousands, at multiple set points will be required.
I could have chosen one of a hundred different objects but to make a point and to be specific, the objects I chose will be throwing knives. Each one 12-inch-long, one inch tall and 1/8” thick.
Their correlation involves a few things such as: (1) Every knife travels at the same speed and reaches a tester at the same time. (2) As they travel toward the testers, they rotate vertically end over end at the same rpm. (3) The tester/analyzer is a wall with a one-inch wide slot. The wall can be rotated to different set points ranging from vertical to horizontal.
When the slot is set vertical, all the knives pass through but when the slot is set horizontal, no knife can pass through. If you rotate the slot five degrees from vertical, most knives will still make it through but now there’s a slim chance the rotating knife could make contact with one of the slots edges. With the slot set vertically, all knives make it through and at five degrees it’s obvious to see the odds have been slightly reduced.
When you rotate the slot to 25 degrees it becomes much harder for a knife to pass. You can see that if the knife is rotated just right as it reaches the slot, it will make it through. As a matter of fact, if you take the time to truly visualize this you will see that a number of new things come into play. The knifes rotation, which is analogues to frequency, plays a big part, especially in relation to the knife’s proximity to the slot’s edges. In other words, if the timing and rotation are not just right, there is a higher chance the knife will hit one of the edges.
If you rotate the slot to 85 degrees from vertical (not quite horizontal) most likely a knife will not pass through but there is a very slim chance that if it’s pointed at the slot as it gets there, it will. The probability is low but still possible.
After throwing thousands of knives at various set points ranging from vertical to horizontal you accumulate the results. The results will show that the number of knives passing through is directly proportionate to the set point angle. More interesting is that the proportional results are NOT LINEAR. Instead, you’ll find that the results do match Malus Law, cos2theta and the predictions of quantum mechanics.
This experiment can be done and proves that adding just one more element of reality (In this case a very real and obvious element) that the results can always be explained physically without any uncertainty.
