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Bell's Theorem and Negative Probabilities
By David R. Schneider
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__Introduction__

*Author's note: This article is based on Bell's Theorem (1). I have reformulated the presentation to make it a little easier to see that "Negative Probabilties" are a seemingly paradoxical consequence of his work. The Bell Inequalities can be presented in many forms, and most are essentially equivalent. I do not know if this particular presentation format or derivation has been used by others, I can only assume it has. I follow conventional interpretation of both QM (quantum mechanics) and Bell. I assume the reader already has basic familiarity with Bell test setups such as Aspect(2). For a more rigorous proof, look to Bell, Aspect and others.*

*Figure 1:John S. Bell (1928-1990), author of Bell's Theorem. We don't need to see the pocket protector to know this guy is a major geek.*

Some have posed the question,

(Hidden variables means that there are microscopic properties of fundamental particles that we are unable to observe directly by means of testing, perhaps due to technological limitations that might not exist at some future time. Since we can't observe them, they are "hidden" now - but perhaps if we knew more about them then that might explain the otherwise mysterious behavior of particle spin.)

Bell saw it differently. He took EPR at face value, as many physicists did, and concluded that there were no hidden variables. A strange position, to be sure, but not contradicted by the facts. Building from this position, he then went further: he showed that hidden variables would lead to a disagreement with the predictions of QM in certain specific cases (see Figure 2 below). Please recall: a single counter-example is sufficient to disprove any theory, and this forms the basis of our argument. So Bell's Theorem is based on EPR, and demonstrates that the following 3 things cannot all be true (i.e. at least one must be false):

i) The experimental predictions of Quantum Mechanics are correct with respect to spin predictions (a testable hypothesis)

ii) Hidden variables exist (particle attributes really exist independently of observation - this is akin to saying that the moon is there when you are not looking at it); and

iii) Locality holds (a measurement at one place does not affect a measurement result at another, more or less in accordance with Einstein's Special Theory of Relativity which makes the speed of light a universal "speed limit").

QM predicts that certain classical scenarios, if they existed, would have negative likelihood of occurance (in defiance of common sense). Any so-called local realistic theory - in which ii) and iii) above are assumed to be true - will make predictions for values of these scenarios which are significantly different than the QM predicted values. QM does not acknowledge the existence of these scenarios, often called hidden variables (HV), so it does not have a problem with this consequence of Bell's Theorem. (I.e. there are no negative probabilities in QM itself.) We will ignore the iii) case here, as if you accept that locality fails anyway then there is no particular conflict between i) and ii). In other words, we assume that Einstein's Special Relativity holds in the sense that causes cannot propagate to the future faster than c (the speed of light). Again, our objective is to see the effect of the "hidden variable" or "Realistic" assumption and how that specifically leads to results that defy our intuitive common sense.

In the entangled photon scenarios, the Realistic view - which maps to assumption ii) above - states that the photon polarization is determinate as of the point in time that the photons' existence begins. The attribute we measure is considered "determinate" because its value is determined *before* the act of measurement. To repeat, this is a questionable assumption as we shall see below. Even though the entangled photons can only be measured at 2 different angles before they are disturbed, the Realistic view states that they could potentially have been measured at other angles as well. Thus, the Realistic view is that the existence of the photon polarization is independent of the act of measurement (and is a result of the state of the hidden variables). On the other hand, QM (Heisenberg Uncertainty Principle) says that the photon spin (polarization) exists only in the context of a measurement, and the the act of observation is somehow fundamental to the measurement results. Here is the paradox that is a partial result of Bell's Theorem:

Time(created) < Time(measured)

The Left is set at angle

Basic diagram of a Bell test experiment with single channel polarizers; in single channel tests we refer to polarization as + or - with a + meaning the photon was detected. The polarizer angle can be varied and the test re-run. The + and - results from each side are compared and the correlations are counted. The correlation percentage is then computed. The +/- pattern is completely random for either detector. However, for entangled photon pairs, a pattern only emerges once the detector results are compared. Creating entangled pairs is very difficult, and requires special apparatus.

Case | A=0 degrees | B=67.5 degrees | C=45 degrees | Predicted likelihood of occurance |

[1] | A+ | B+ | C+ | >=0 |

[2] | A+ | B+ | C- | >=0 |

[3] | A+ | B- | C+ | >=0 |

[4] | A+ | B- | C- | >=0 |

[5] | A- | B+ | C+ | >=0 |

[6] | A- | B+ | C- | >=0 |

[7] | A- | B- | C+ | >=0 |

[8] | A- | B- | C- | >=0 |

The sum of all possible outcomes above:

[1] + [2] + [3] + [4] + [5] + [6] + [7] + [8] = 100% = 1

This seems innocent enough, and simple enough as well. With a Realistic view, this is true regardless of the unknown hidden variable function that controls these individual outcome probabilities (referred to as "lambda" in Bell's paper). So it is the requirement that each outcome have an expectation value >=0 that connects to the assumption of reality per ii) above.

QM says that there are only 4 cases to consider:

Case | A=0 degrees | B=67.5 degrees | C=45 degrees | Predicted likelihood of occurance |

[QM1] | A+ | B+ | n/a | >=0 |

[QM2] | A+ | B- | n/a | >=0 |

[QM3] | A- | B+ | n/a | >=0 |

[QM4] | A- | B- | n/a | >=0 |

These 4 cases also add to 1. In QM, there is no C unless it can be measured.

But in the Realistic view, [2]>=0 and [7]>=0. Combining these, we get the non-negative prediction for the Realistic side:

X = combined probability of cases [1] + [2] + [7] + [8]

Y = combined probability of cases [2] + [4] + [5] + [7]

Z = combined probability of cases [1] + [4] + [5] + [8]

X = correlations between measurements at A and B

Y = non-correlations between measurements at A and C

Z = correlations between measurements at B and C

You can review the 8 cases in

(X + Y - Z) / 2

= (([1] + [2] + [7] + [8]) + ([2] + [4] + [5] + [7]) - ([1] + [4] + [5] + [8])) / 2

Now simplify by eliminating offsetting terms:

= ([2] + [7] + [2] + [7]) / 2

= [2] + [7]

Which means that, if

Graph of QM predictions (sine wave) against Bell's Inequality (straight line). Note that almost all angles have a difference between the QM prediction and Bell's Inequality. The example angles presented here are just one sample, we could have looked at many others too and gotten similar results. The maximum deviation is actually at A=0, B=60, C=30. It only takes one such example to invalidate the entire realistic position, however.

X is determined by the angle between A and B, a difference of 67.5 degrees

Y is determined by the angle between A and C, a difference 45 degrees

Z is determined by the angle between B and C, a difference 22.5 degrees

(X + Y - Z) / 2

Substituting values from

= (.1464 + .5000 - .8536)/2

= (-.2072)/2

= -.1036

Therefore:

Which predicted result is less than zero, in conflict with the prediction of

QM predicts an expectation value for cases [2] and [7] of -.1036, which is less than 0 and seemingly absurd. However, this is born out by actual experiments, in defiance of common sense! This result means that the seemingly reasonable assumption (the Realistic view) that we started with in

(1) J.S. Bell: "On the Einstein Podolsky Rosen paradox" Physics 1 #3, 195 (1964).

(2) A. Aspect, Dalibard, G. Roger: "Experimental test of Bell's inequalities using time-varying analyzers" Physical Review Letters 49 #25, 1804 (20 Dec 1982).

(3) A. Einstein, B. Podolsky, N. Rosen: "Can quantum-mechanical description of physical reality be considered complete?" Physical Review 41, 777 (15 May 1935).

You can view a copy of these papers in .PDF form at: EPR, Bell and Aspect: The Original References

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**(c) 2005-2007 David R. Schneider.
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