Can somebody explain Bell's Inequality to me?

Question

I've just read 'In Search of Schrodinger's Cat' by John Gribbin. In it, he talks of Bell's Inequality and why it solved the final thought experiment posited by Einstein.

He didn't go into it in too much detail and I would be very interested to examine the structure and results of the experiment in order to satisfy my curiosity.

Can anyone explain it to me or any of the other analogous experiments or give me a link that I can use?

TY

Answer 1

Bell's Inequality addresses the question of hidden variables in quantum mechanics. To understand what is going on, it is important to understand how things are viewed in QM.

First, in QM there is something called the wave function. All things that are observable about a particle can be calculated in terms of this wave function. However, you only get probabilities for different measurements, not (usually) a definite result. For example, photons can have two spins along a given axis, usually called spin up and spin down. The problem is that a free photon will not have a definite spin. It will have a probability for measuring either spin up or spin down.

Now, the hope was that this situation is just an expression of our ignorance of 'what is really going on'. Einstein, in particular, thought that a single photon actually has a definite spin, but we just don't know it. The actual spin, in his mind, is determined by 'hidden variables' that have definite values but that we don't know.

Let's be clear about the differences between the two views. Suppose we have an photon beam that is half 'up' and half 'down' spins. In QM, each photon would be in a mixed state: each would be 'half up and half down'. In hidden variable theories, each photon would be either 'up' or 'down', but we would have half of each and we wouldn't know which are which.

Now, it is possible to create two photon beams from the decay of certain particles. In these two beams, the photons will have opposite spins. If the spin of a photon along one beam is up, that along the other beam will be down. Furthermore, the individual beams will be 'half up and half down' in the QM sense.

The crucial thing for Bell's theorem is that we can measure spin along more than one axis. If we measure along the same axis, the photons will have opposite spins. If we measure along slightly different axes, the photons will be mixtures of 'up' and 'down' along that axis. By counting, we can correlate how many 'up' and 'down' we get along different axes when we see an 'up' along the first axis.

Now, QM gives very definite prediction about what we will see in these correlations. Hidden variable theories have more leeway, it seems, until you realize that is a photon has a *definite* spin, the spins along other axes have to not change too quickly as you change the axis. Bell's inequality describes how fast the correlations can change as the axis is changed under the assumption that there really is a definite spin to each photon.

Now, it turns out that thr QM prediction violates Bell's inequality. Also, in actual experiments that have been done, the QM prediction has been verified and Bell's inequality has been shown not to work in the real world. What does this mean? Ultimately, that there are no hidden variables. At the quantum level, things really are inherently probabilistic. Those photons do not have definite spins while in flight. It is only when measurements occur that we get definite spins out.

Answer 2

An infinite class of inequalities formulated by Bell (1964) which seemed to be a physically reasonable condition of locality imposing restrictions on the maximum correlations of the measurements of a pair of spin 1/2 particles formed in the singlet state and moving freely in opposite directions, as considered in the Einstein-Podolsky-Rosen paradox.

Bell's inequalities can be tested in a laboratory experiment (under certain assumptions) because the statistical predictions of quantum mechanics are incompatible with any local hidden variables theory apparently satisfying only the natural assumptions of "locality," as shown by the predictions of Bell's inequality. However, at present there are no "clean" experiments unambiguously verifying the inequalities.

As in the situation explored in the EPR paradox, Bell considered an experiment in which a source produces pairs of correlated particles. For example, a pair of particles with correlated spins is created; one particle is sent to Alice and the other to Bob. On each trial, each observer independently chooses between various detector settings and then performs an independent measurement on the particle. (Note: although the correlated property used here is the particles' spin, it could alternatively be any correlated "quantum state" that encodes exactly one quantum bit.)

Same axis: pair 1 pair 2 pair 3 pair 4 ...n
Alice, 0°: + - - + ...
Bob, 180°: + - - + ...
Correlation: ( +1 +1 +1 +1 ...)/n = +1
(100% identical)
Orthogonal axes: pair 1 pair 2 pair 3 pair 4 ...n
Alice, 0°: + - + - ...
Bob, 90°: - - + + ...
Correlation: ( -1 +1 +1 -1 ...)/n = 0.0
(50% identical)
When Alice and Bob measure the spin of the particles along the same axis (but in opposite directions), they get identical results 100% of the time. But when Bob measures at orthogonal (right) angles to Alice's measurements, they get identical results only 50% of the time. In terms of mathematics, the two measurements have a correlation of 1, or perfect correlation when read the same way; when read at right angles, they have a correlation of 0; no correlation. (A correlation of -1 would indicate getting opposite results for each measurement.)

So far, the results can be explained by positing local hidden variables — each pair of particles may have been sent out with instructions on how to behave when measured in the two axes (either '+' or '-' for each axis). Clearly, if the source only sends out particles whose instructions are identical for each axis, then when Alice and Bob measure on the same axis, they are bound to get identical results, either (+,+) or (-,-); but (if all four possible pairings of + and - instructions are generated equally) when they measure on perpendicular axes they will see zero correlation.

Now, consider that Alice or Bob can rotate their apparatus relative to each other by any amount at any time before measuring the particles, even after the particles leave the source. If local hidden variables determine the outcome of such measurements, they must encode at the time of leaving the source a result for every possible eventual direction of measurement, not just for the results in one particular axis.

Bob begins this experiment with his apparatus rotated by 45 degrees. We call Alice's axes a and a', and Bob's rotated axes b and b'. Alice and Bob then record the directions they measured the particles in, and the results they got. At the end, they will compare and tally up their results, scoring +1 for each time they got the same result and -1 for an opposite result - except that if Alice measured in a and Bob measured in b', they will score +1 for an opposite result and -1 for the same result.

Using that scoring system, any possible combination of hidden variables would produce an expected average score of at most +0.5. (For example, see table at the right, where the most correlated values of the hidden variables have an average correlation of +0.5, i.e. 75% identical. The unusual "scoring system" ensures that maximum average expected correlation is +0.5 for any possible system that relies on local hidden variables.
Bell's Theorem shows that if the particles behave as predicted by quantum mechanics, Alice and Bob can score higher than the classical hidden variable prediction of +0.5 correlation; if the apparatuses are rotated at 45° to each other, quantum mechanics predicts that the expected average score is 0.71.