Bell's theorem/Inequality

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Bell's theorem/Introduction         Bell's theorem/Probability         Bell's theorem/Inequality

Anthropromorphism and skepticism[edit]

A working (stipulative) definition of local hidden variable theory involves the anthropomorphism of elementary particles, and the adoption of that Wikipedia calls scientific skepticism, which views humans as local entities who react only to stimulus that reaches them by the ordinary senses.[1]

In other words, we pretend photons are people and show that what they do is downright "spooky".

Case 1: "Local" and "remote" filters held 90° apart[edit]

The "remote" (other) photon can be used to measure on hidden variable of the local photon.

It has been experimentally verified that if the two polarizers (local and remote) are oriented 90° apart, either both entangled photons pass or both are blocked. Moreover, the photons must obey this rule even if one of the filters is suddenly removed (as in case c). It is generally assumed that photons are unable[2] to communicate after parting company, since such communication would require information to exceed the speed of light. The photons must either pre-arrange their states at creation, or posses precognition skills, because the orientation of the observing polarizers is subject to change. [3]

The image shown to the right is closely related to the famous EPR paradox.

Case 2: Three "local" filters held randomly at 60° increments[edit]

Bell's theorem simple photon geometry.svg

In the previous section, it was argued that local photon must arrive at the measuring device with responses to all three polarization angles, X,Y,Z, already decided. With three variables that can take on two possible values, we have eight possibilities:

i 1 2 3 4 5 6 7 8
Xi 1 1 1 1 -1 -1 -1 -1
Yi 1 1 -1 -1 1 1 -1 -1
Zi 1 -1 1 -1 1 -1 1 -1
Δ γ β α α β γ Δ

The eight possibilities have been arranged into four pairs (α,β,γ,Δ). We presume that one of these eight states is "chosen" by the entangled photons when they were created by the 2-photon decay. Each of these eight possibilites might have equal or unequal probability of occurring. For example, it is conceivable that probability of a Δ event (X=Y=Z) is zero (but never negative[4])

Bells theorem (modified for our special case)[edit]

Statement of Bell's theorem[edit]

Bell proved that if X, Y, and Z are the photon measurements variables as defined above (to be ±1), then

where P(X=Y) is the probability that X = Y. This is not the form of Bell's theorem put forth in his original paper, but is mathematically equivalent to it.[5] Although we are restricting our discussion to the symmetric (60-60-60 degree) orientation of polarization measurements, this equation is valid for any configuration.[6]

Proof[edit]

The sum of all proabilities equals unity:

(1)     

provided the N posibilities, Pi represent distinct events.[7] If the probabilities do not represent distinct, or mutually exclusive events, then it is possible for:

(2)     

Example: Flip a coin and assume it will most likely land on the table. If it falls off the table, assume that it will land on the floor where you will certainly see it. There is a 50% chance of heads, 50% chance of tails, and a 10% chance that it will fall onto the floor. This sums to 110% because falling on the floor does not exclude heads or tails.
Bell's theorem Venn diagram[8]

To prove Bell's inequality, we begin by noting that since {X,Y,Z} can have only one of two possible values, at least two must equal each other:

(3)            is     impossible.

But we can group all observations of a pair of entangled photons as shown in the Venn diagram. Let the α, β, γ, and Δ denote the four mutually exclusive possibilities. Now, the probabilities sum to unity because one and only one of these events occurs every time a successful observation is made of the two photons:

(4)     

There are two ways that can occur (either a or a event). Hence,

(5)     

Likewise, P(Y=Z) = P(α) + P(Δ) and P(Z=X) = P(β) + P(Δ). If we add all three, each Δ event gets counted three times:

(6)     


This leads to one version of Bell's famous inequality:

(7)     


The same result may be obtained without a Venn diagram if symbolic logic is used. See:

Proof that the calculated (and observed) probabilities violate Bell's theorem[edit]

Here we use results previously obtained to show that both experiment, as well as the theory of quantum mechanics violate Bell's theorem. In other words, what we observe in the laboratory is impossible, or more precisely, inconsistent with the assumptions used to calculate the inequality shown above at (7).

First we consider X and Y; Using the known relation between the polarization of two entangled photons, it is possible to make what might be called a virtual[9], or de facto measurement. Using this knowledge of the polarization with orientation-X, we note that the Y filter is rotated by 60°. As was explained here, if the photon passed X, the probability that it passes Y is:

But the same result occurs if the photon at orientation-X is blocked, only this time polarization-Y has a 3/4 probability of passing, and therefore is also blocked with a probability of 1/4. By symmetry, the same argument applies to the other two pairs of filters:

    

Therefore:  

This contradicts the inequality defined at (7).

What went wrong?[edit]

The short answer is that nobody knows.

The consensus among experts seems to involve something akin to the anthropomorphism used in this introduction to Bell's theorem. Two measurements were made, one "local", and the other "remote". If the police interview two witnesses and they give the same answers to all questions, it is assumed that they either both witnessed the same event, or they discussed their answers before the interview.[10] If the photons were people, the experimental evidence establishes that they either "knew" in advance which combination of orientations would be used, or the first photon to be measured could "communicate" the nature of its "interrogation" to the other photon. In other words, photons seem to possess either telepathic skills or precognition. These are disturbing thoughts for anyone who self-identifies as a scientific skeptic.

Quantum mechanics precludes the possibility of simultaneously knowing or measuring certain combinations of variables. The most famous such pair is position and momentum. Apparently, since X and Y and Z cannot be simultaneously measured, they cannot simultaneously exist.[11] Hidden variable theory is any theory that simultaneously assigns numerical values X, Y, and Z. Strictly speaking, it is usually said that Bell's experiments don't establish the non-existence of hidden variable theories, but the non-existence of local hidden variable theories.[12]

To put it bluntly, Bell's theorem tells us that any successful hidden variable theory will need to attribute psychic abilities to atoms and other elementary particles. So far, no such theory has ever been developed to the point where it is widely used.[13]

Wikipedia-logo.png Search for Weeping_Angel on Wikipedia.

Weeping angels don't exist if you look at them (they cover their eyes because if two weeping angels simultaneously see each other they mutually "lock" into nonexistence). Hidden variables don't exist unless you look, and Weeping Angels don't exist if you do look. They seem to be complementary entities.

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Footnotes and references[edit]

See also:

Footnotes
  1. Because despite a large number of claims that have been debunked, there is no reliable evidence for w:telepathy or w:precognition.
  2. "unable" represents an anthropomorphism here.
  3. In fact, there are not three, but an infinite number of possible orientations for polarizers.
  4. Unless you believe this website: http://drchinese.com/David/Bell_Theorem_Negative_Probabilities.htm (which is actually quite rigorous -- "negative probability" is just a fun way for Dr. Chinese to explain the impossibility of disobeying Bell's inequality)
  5. Bell spoke in terms of correlation coefficients.
  6. The equation is valid for any configuration if all "remote" filters are all displaced by exactly 90°This 90 degree displacement permits us to ascertain the local photon's polarization by observing the remote photon's ability to pass a given filter.
  7. by "distinct possibilities" we mean that two different possibilities. We can't for example add the chances of it raining tomorrow if we are counting dice because "three with rain" and "three without rain" are not different outcomes if you care about dice.
  8. See http://skepticsplay.blogspot.com/2009/03/bells-theorem-explained.html and http://www.johnboccio.com/research/quantum/notes/paper.pdf
  9. Here, "virtual" means that no measurement is taken, but since we know the outcome of said measurement by measuring the "remote" photons polarization along that axis, we have what might be called a de facto measurement.
  10. In police work, there is a third alternative: The two witnesses made the same mistake in interpreting what they saw. Nevertheless, the same answers to a question implies that these answers were somehow predetermined before the interview took place.
  11. This is not intended to be a rigorously mathematical or logical statement.
  12. http://web.mit.edu/leozhou/www/8.06paper.pdf
  13. Jeffrey Bub wrote: "The conclusions of this paper will therefore be that there are no hidden variable theories of quantum phenomena in the usual sense, that the term 'hidden variable theory' (to describe the work of Bohm and his collaborators) is unfortunate and misleading." Bub, Jeffrey. "What is a hidden variable theory of quantum phenomena?." International Journal of Theoretical Physics 2.2 (1969): 101-123.

Bell's theorem/Introduction         Bell's theorem/Probability         Bell's theorem/Inequality