Bell's theorem/Inequality

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Anthropromorphism and skepticism

 * See also WikiJournal of Science/ A card game for Bell's theorem and its loopholes

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

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&deg; apart
It has been experimentally verified that if the two polarizers (local and remote) are oriented 90&deg; 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  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.

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

Case 2: Three "local" filters held randomly at 60&deg; increments
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: The eight possibilities have been arranged into four pairs (&alpha;,&beta;,&gamma;,&Delta;). 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 &Delta; event (X=Y=Z) is zero (but never negative )

Statement of Bell's theorem
Bell proved that if X, Y, and Z are the photon measurements variables as defined above (to be &plusmn;1), then


 * $$ P(X=Y) + P(Y=Z) + P(Z=X) \ge 1$$

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. Although we are restricting our discussion to the symmetric (60-60-60 degree) orientation of polarization measurements, this equation is valid for any configuration.

Proof
The sum of all proabilities equals unity:

(1)     $$\sum_1^N P_i = 1\;,$$

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

(2)     $$\sum_1^N P_i \ge 1$$


 * 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.



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)     $$X \ne Y   \quad\&\quad  Y \ne Z  \quad\&\quad Z\ne X \quad$$     is      impossible.

But we can group all observations of a pair of entangled photons as shown in the Venn diagram. Let the &alpha;, &beta;, &gamma;, and &Delta; 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)     $$P(\alpha)+P(\beta )+P(\gamma ) + P(\Delta) = 1$$

There are two ways that $$X=Y$$ can occur (either a $$\gamma$$ or a $$\Delta$$ event). Hence,

(5)      $$P(X=Y) = P(\gamma ) + P(\Delta)$$

Likewise, P(Y=Z) = P(&alpha;) + P(&Delta;) and P(Z=X) = P(&beta;) + P(&Delta;). If we add all three, each &Delta; event gets counted three times:

(6)     $$P(Y = Z) + P(Z = X) + P(Y = Z) = P(\alpha)+P(\beta )+P(\gamma ) + 3P(\Delta) $$ $$= 1 + 2P(\Delta)\ge 1$$

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

(7)      $$P(Y = Z) + P(Z = X) + P(Y = Z) \ge 1$$

The same result may be obtained without a Venn diagram if symbolic logic is used. See:
 * Probability
 * Slide 7 of http://www.slideshare.net/gill1109/epidemiology-meets-quantum-statistics-causality-and-bells-theorem

Proof that the calculated (and observed) probabilities violate Bell's theorem
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, or de facto measurement. Using this knowledge of the polarization with orientation-X, we note that the Y filter is rotated by 60&deg;. As was explained here, if the photon passed X, the probability that it passes Y is:


 * $$P(X=Y) = \frac 1 4$$

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:


 * $$P(X=Y) = P(Y=Z) = P(Z=X)= \frac 1 4$$

Therefore:  $$P(X=Y) + P(Y=Z) + P(Z=X)= \frac 3 4$$

This contradicts the inequality defined at (7).

What went wrong?
Don't Click! Don't Blink!   Click again if you are a Dr. Who fan. 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. 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. 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.

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.

Reader's comment(s)

 * If the twin particles are far enough apart that gravitational gradients must be taken into account, then one cannot safely assume a common clock. Clocks are affected by gravity. When Einstein first pointed this out, it might have been characterized as "spooky timekeeping at a distance." The derivation of Bell's Inequality is fairly straightforward if one makes the simplifying assumption of a static hidden variable.  This is how it is often explained the first time around.  But if one then takes up the case of a time-varying hidden variable, this issue of phase-locked synchronicity becomes significant.  The derivation of Bell's Inequality still works if you assume a common clock.  But if you admit there is no such thing as a common clock for particles separated in space, then the math of Bell's derivation runs into difficulties.  Mathematically, speaking, there arises a residual non-zero "beat frequency" that doesn't vanish.  Bell managed to get the presumptive hidden variable to vanish because he (tacitly) assumed a common clock.  Since Bell's Inequality doesn't hold in our cosmos, one can interpret that as evidence that there is such a thing as "spooky timekeeping at a distance." And this phenomenon is well understood to be a feature of gravitational gradients. The twin particles do not age in perfect phase-locked synchrony, and this throws a monkey wrench into Bell's derivation.  Bell's Inequality would only hold for a cosmos where there are no gravitational gradients. ~Barry Kort 2001:470:1F07:BEE:0:0:0:A8 (discuss) 00:44, 1 August 2017 (UTC)
 * I believe you have raised a valid point. This is why Bell's theorem experiments need to be performed.  It is not unusual for a successful theory to be later shown to be an approximation to a deeper and more fundamental theory that is completely different from the original theory.  If quantum mechanics is wrong, the error likely involves gravity.  And if the "impossibility" of violating Bell's inequality is not quite so impossible, gravity might explain the issue of causality.  This might throw your proverbial monkey wrench into Bell's derivation. --Guy vandegrift (discuss • contribs) 01:41, 1 August 2017 (UTC)


 * 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.--Guy vandegrift (discuss • contribs) 01:42, 1 August 2017 (UTC)

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Footnotes and references
See also:
 * quantumtantra simple proof
 * Schneider, "Bell's Theorem with Easy Math" (See also index)
 * Stanford Encyclopedia of Philosophy:Bell's theorem
 * Alain Aspect,"Bell's theorem: the naive view of an experimentalist
 * Vandegrift, "Bell’s Theorem and Psychic Phenomena"
 * "A simple proof of Bell’s inequality", Lorenzo Maccone
 * gill1109 "Epidemiology-meets-quantum-statistics-causality-and-bells-theorem" (see slide 7)
 * "Bringing home the atomic world: Quantum mysteries for anybody", Mermin, American Journal of Physics (1981)
 * John Preskill "Course Information: Quantum Computation"    Chap. 4     Chap. 4.01
 * Two youtube vidoes: simple and advanced (MIT)

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 * Footnotes