Questions about Bell's Theorem

[San K]

from http://en.wikipedia.org/wiki/Quantum_entanglement"

A simple limit of Bell's inequality has the virtue of being completely intuitive. If the result of three different statistical coin-flips A, B, and C have the property that:
A and B are the same (both heads or both tails) 99% of the time
B and C are the same 99% of the time
then A and C are the same at least 98% of the time. The number of mismatches between A and B (1/100) plus the number of mismatches between B and C (1/100) are together the maximum possible number of mismatches between A and C.

so far so good. makes sense. the maximum number of mismatches (between A and C) can only be (1/100) + (1/100) = 2%

yet in Quantum Entanglement (QE) the number of mismatches is, say, = 4%

however if there were hidden variables (in QE) the mismatch rate would be only 2% as calculated above. thus the hidden variable hypothesis weakens.

In quantum mechanics, by letting A, B, and C be the values of the spin of two entangled particles measured relative to some axis at 0 degrees, θ degrees, and 2θ degrees

This is a chance for folks like me to understand it better. i thought the spin value is plus 1/2 or minus 1/2.

how is another value possible?

Let's assume x-axis. We measure spin at 0 degrees, 40 degrees and 80 degrees say...we can only get -1/2 or +1/2 how are other values possible? what does the above mean?

Choosing the angle so that ε = 0.1, A and B are 99% correlated, B and C are 99% correlated and A and C are only 96% correlated.

between entangled photons: is the correlation not the same in all directions? does only one direction, generally, have 100% correlation?

what would be the actual (relative) value of the angle where A and B would be 99% correlated?

however can we entangle photons such that say along both x-axis and y-axis (separately) the correlation (between entangled photons) is 100%?

Imagine that two entangled particles in a spin singlet are shot out to two distant locations, and the spins of both are measured in the direction A. The spins are 100% correlated (actually, anti-correlated but for this argument that is equivalent). The same is true if both spins are measured in directions B or C. It is safe to conclude that any hidden variables that determine the A,B, and C measurements in the two particles are 100% correlated and can be used interchangeably.
If A is measured on one particle and B on the other, the correlation between them is 99%

.

Why does the correlation drop from 100% to 99%?

so say at some angle (and we call it 0 degrees) along some axis there is 100% correlation but as we change the angle the correlation drops? why? anything to do with Malus law?

also here we have considered only one axis. if the correlation between two photons along, say x-axis, was 100%, what correlation can we expect along y and z axis?

If B is measured on one and C on the other, the correlation is 99%. This allows us to conclude that the hidden variables determining A and B are 99% correlated and B and C are 99% correlated. But if A is measured in one particle and C in the other, the results are only 96% correlated, which is a contradiction. The intuitive formulation is due to David Mermin, while the small-angle limit is emphasized in Bell's original article.

this is conceptually understandable from the statistical coin-flips example above.

 

[xts]

What text do you cite?

" This is a chance for folks like me to understand it better. i thought the spin value is plus 1/2 or minus 1/2. How is another value possible? "
if we speak about electrons (other fermions) you are right. If about photons: +1/-1. But that doesn't matter for the problem. A,B,C just denote the results of spin measurments if the axis of detector (polarizator, Stern-Gerlach magnet...) is such and such. A,B,C may take only two values {1,-1} for photons, {1/2, -1/2} for electrons.

Let's assume x-axis. We measure spin at 0 degrees, 40 degrees and 80 degrees say...we can only get -1/2 or +1/2 how are other values possible? what does the above mean?" is the correlation not the same in all directions? does only one direction, generally, have 100% correlation? "
We put one detector at some axis, and other rotated by some angle. If they are parallel - output is fully correlated. If perpendicular - there is no correlation.

" what would be the actual (relative) value of the angle where A and B would be 99% correlated? "
[itex]corr=\cos^2(\theta)\, \rightarrow\, \theta\approx 6^{\circ}[/itex]

" however can we entangle photons such that say along both x-axis and y-axis (separately) the correlation (between entangled photons) is 100%? "
Sure. For entangled photons correlation for every possible axis is 100% if both detectors are in parallel." Why does the correlation drop from 100% to 99%? ... anything to do with Malus law? "
Exactly! That is Malus law." if the correlation between two photons along, say x-axis, was 100%, what correlation can we expect along y and z axis? "
Two would be enough, you can't measure polarisation parallel to the beam
Of course, correlation along any axis is 100% if both detectors are in parallel.

 

[San K]

What text do you cite?

" This is a chance for folks like me to understand it better. i thought the spin value is plus 1/2 or minus 1/2. How is another value possible? "
if we speak about electrons (other fermions) you are right. If about photons: +1/-1. But that doesn't matter for the problem. A,B,C just denote the results of spin measurments if the axis of detector (polarizator, Stern-Gerlach magnet...) is such and such. A,B,C may take only two values {1,-1} for photons, {1/2, -1/2} for electrons.

Let's assume x-axis. We measure spin at 0 degrees, 40 degrees and 80 degrees say...we can only get -1/2 or +1/2 how are other values possible? what does the above mean?" is the correlation not the same in all directions? does only one direction, generally, have 100% correlation? "
We put one detector at some axis, and other rotated by some angle. If they are parallel - output is fully correlated. If perpendicular - there is no correlation.

" what would be the actual (relative) value of the angle where A and B would be 99% correlated? "
[itex]corr=\cos^2(\theta)\, \rightarrow\, \theta\approx 6^{\circ}[/itex]

" however can we entangle photons such that say along both x-axis and y-axis (separately) the correlation (between entangled photons) is 100%? "
Sure. For entangled photons correlation for every possible axis is 100% if both detectors are in parallel." Why does the correlation drop from 100% to 99%? ... anything to do with Malus law? "
Exactly! That is Malus law." if the correlation between two photons along, say x-axis, was 100%, what correlation can we expect along y and z axis? "
Two would be enough, you can't measure polarisation parallel to the beam
Of course, correlation along any axis is 100% if both detectors are in parallel.

thanks xts. the link is from Wikipedia and i have inserted it into the original post now. i think you have answered 90% or more of my queries.

let me digest it and I will come back if i have anymore.

Sure. For entangled photons correlation for every possible axis is 100% if both detectors are in parallel.

i don't get this fully. the detectors are parallel to each other or to the polarization direction of the photons?

since the photons are separated, how do we figure out the x-axis of photon A corresponds to what axis of photon B?

i think i got it --- we can use the polarization direction as the base direction and thus the photons are now "direction synchronized" for experimental purpose/calculationson a separate note:

since the photons are perfectly correlated (spin wise in same direction).

Isn't placing detectors on each of the photons at an angle is (in a sense), equivalent to

putting just one detector on just any one of the photons...at the same angles to the polarization/chosen direction?

 

[xts]

"the detectors are parallel to each other or to the polarization direction of the photons?"
If the detectors are parallel to each other the correlation is 100%.

"since the photons are separated, how do we figure out the x-axis of photon A corresponds to what axis of photon B?"
We measure that with our detectors. If both detectors are parallel (let's say vertical) than if both detectors click we know both photons were vertical."since the photons are perfectly correlated ..."
Sorry - I can't get what you ask about...