John Bell
All things being equal
Abstract
Current Bell tests are riddled with uncertainties, but a definitive resolution can be achieved through a local test utilizing modern
software techniques. The Toffoli gate, also known as the CCNOT gate, is a universal reversible logic gate that operates on three qubits.
It flips the third qubit (negates its value) only when the first two qubits are both 1.
Sometimes referred to as the C-C-gate, where ‘C’ stands for ‘controlled,’ the Toffoli gate's function is straightforward.
It has three inputs, typically labeled A, B, and C, and three outputs labeled P, Q, and R. Outputs P and Q are simply A and B, respectively,
while R behaves like a conditional inverter. Specifically, R equals the logical NOT of C if both A and B are true; otherwise, R remains equal to C.
Introduction
John Stewart Bell was a renowned physicist from Northern Ireland, best known for Bell's theorem.
This theorem is a significant contribution to quantum mechanics, particularly concerning quantum entanglement and hidden-variable theories.
Bell's theorem demonstrates that no local hidden-variable theory can reproduce all the predictions of quantum mechanics, which has profound
implications for our understanding of the nature of reality.
John Bell's theorem, often referred to as Bell's inequality, addresses the question of whether the predictions of quantum mechanics can be reproduced
by any local hidden variable theory. Bell's theorem states that no local hidden variable theory can reproduce all the predictions of quantum mechanics.
This implies that the results of certain measurements on entangled particles cannot be explained by any theory that relies on hidden variables that
do not influence each other instantly over a distance. This phenomenon supports the non-local nature of quantum mechanics and the idea that the universe
is fundamentally probabilistic.
The greatest problems in Gedankenexperiment is the precise definition of the experiment itself. This arises from the fact that such a thought experiment
is often designed to prove rather than disprove a particular hypothesis. In the Bell test scenario this is particularly prevalent. The definition of a
Bell test is as follows:
Basic Bell Test
The simplest modern Bell's inequality test is termed the the CHSH inequality test named after its authors' initials. This hypothetical experiment
is stated as follows;
A Bell test involves two parties Alice and Bob who are located at widely separated locations. A pair of entangled particles
is transmitted, one to Alice and the other to Bob. When Alice receives her particle, she chooses to perform one of two possible measurements.
The measurements are denoted by \(A_0\) and \(A_1\). Both \(A_0\) and \(A_1\) are "binary measurements": the result of \(A_0\) is
either "+1 (Spin Up) or −1 (Spin Down)" and likewise for \(A_1\). When Bob receives his particle, he chooses to make one of two measurements,
\(B_0\) and \(B_1\), which are also both "binary"
It is important to describe exactly what the following truth table represents. It literally shows a binary option which in mathematical terms is a one or zero
and not +1 and -1. The table depicts every possible combination of selections that can be made by Alice and Bob and whether they coincide or not.
Simply put, if a "1" or "0" appears in both columns, both Alice and Bob made the same selection, otherwise they didn't.
| Event | \(A_0\) | \(B_0\) | Result |
|---|
| 1 | 0 | 0 | \(\checkmark\) |
| 2 | 0 | 1 | \(\times\) |
| 3 | 0 | 2 | \(\times\) |
| 4 | 1 | 0 | \(\times\) |
| 5 | 1 | 1 | \(\checkmark\) |
| 6 | 1 | 2 | \(\times\) |
| 7 | 2 | 0 | \(\times\) |
| 8 | 2 | 1 | \(\times\) |
| 9 | 2 | 2 | \(\checkmark\) |
Where:
\(\checkmark\) When The choices correlate.
\(\times \) When there is no correlation
This creates the situation where the computer can actually calculate the results, using the standard Bell test equation as follows:
$$ \left | E(a,b) + E(a,b') + E(a',b) - E(a',b') \right | \leq 2 $$
Where: \(E(a,b), E(a,b'), E(a',b), E(a',b')\) are the expected values (or correlations) of measurements on pairs of particles. \(a\) and \(a'\) represent
measurements made by Alice, \(b\) and \(b'\) being measurements made by Bob.
The Claim
The claim that Bell and his supporters "hang their hat on" occurs when there is an additional selection of an entangled electron with a spin of \(45^\circ\).
It is suggested that the result of a measurement when calculated using the cosine of the angle does not coincide with a linear measurement and
the difference becomes apparent at \(45^\circ\). As it is said a picture is worth a thousand words, the graphic below perfectly displays this erroneous claim.
This "creative" graph shows the plot of the quantum mechanical prediction in blue and what is generally termed the "local" realistic model shown in red.
The y-axis shows the percentage of correlations versus the difference in spin in degrees between the two entangled electrons on the x-axis.
One might wonder why the local model is represented by a linear graph. To address this, the values at \(0^\circ\) and \(1^\circ\) can be calculated using
the cosine function. While the cosine of \(0^\circ\) is indeed 1 the cosine of \(1^\circ\) is 0.5403, which is certainly not binary.
This raises the question of why use the labels of +1 and -1 and refer to them as binary. In essence, if the calculation of "measurement"
values were consistent, the values would not be linear and there would be no disagreement, and the results would be identical.
When considering multiple angles of spin, an important question can be asked, how did Bob actually receive an electron with a spin of \(45^\circ\) when
Alice received a Spin UP or Spin Down? The only way this is possible is that the apparatus emitting the electron was rotated by \(45^\circ\) with respect
to Alice after receiving her electron. Even worse, unless it was agreed prior, Bob would be unaware of the apparatus's rotation. If the experiment and
apparatus emitting the electron were both consistent, Bob whether aware of the fact or not, must have received an electron with the opposite spin and once
more there would be no inconsistency in measurement.
The final question is how to predict definitively that there is no inconsistency. This is possibly the simplest of all, instead of comparing apples to
oranges, binary and decimal, labels against real values, any three angles can be used which are represented as \(\text{A}\), \(\text{B}\) and \(\text{C}\).
From the table of possible combinations the probabilities can be produced where \(r_{\text{AB}}\), \(r_{\text{BC}}\) and \(r_{\text{AC}}\) are the sample
correlations;
| Possible Combinations |
Probability Experiment |
| \(\text{A}\) |
\(\text{B}\) |
\(\text{C}\) |
\(r_{\text{AB}}\) |
\(r_{\text{BC}}\) |
\(r_{\text{AC}}\) |
| 1 |
1 |
1 |
\(\checkmark\) |
\(\checkmark\) |
\(\checkmark\) |
| 1 |
1 |
0 |
\(\checkmark\) |
\( \times \) |
\( \times \) |
| 1 |
0 |
1 |
\( \times \) |
\( \times \) |
\(\checkmark\) |
| 1 |
0 |
0 |
\( \times \) |
\(\checkmark\) |
\( \times \) |
| 0 |
1 |
1 |
\( \times \) |
\(\checkmark\) |
\( \times \) |
| 0 |
1 |
0 |
\( \times \) |
\( \times \) |
\(\checkmark\) |
| 0 |
0 |
1 |
\(\checkmark\) |
\( \times \) |
\( \times \) |
| 0 |
0 |
1 |
\(\checkmark\) |
\(\checkmark\) |
\(\checkmark\) |
Where in the Probability experiment:
\(\checkmark\) is when The choices correlate and \(\times \) when they dont.
The Pink rows are the only results considered in experiments.
Count of correlations \(\checkmark\) obtained (green rows) is 6.
Count of correlations \(\checkmark\) obtained regardless of non-correlations (pink rows) is 6.
The number of times the experiment was run is 24.
Conclusion
The requirements of the theory raise immediate suspicion which are highlited above in bold. That is the emphasis on "binary" results ranging
between "1 and -1". From the outset, this is inaccurate, +1 and -1 are simply convenient labels and are no more numeric than Spin Up and Spin Down.
Furthermore, probabilities are always positive and range from 0 to 1, as do binary selections, where 0 represents an impossible event (with no chance
of occurring) and 1 represents a certain event (guaranteed to occur). Therefore, binary results do not span from -1 to 1. The results of any measurement
must be decimal which is evidenced by the experimental method of obtaining the spin of the electron at \(45^\circ\) using the cosine function.
Cosines produce decimal results not binary, this fact is critical to understand the error made in every one of the Bell test experiments.
Finally, it is obvious it is the "selections" themselves that are binary not the "measurements" of the electron spin.