- #1

- 8

- 1

## Main Question or Discussion Point

The following assessment of a Bell experiment is based on N. David Mermin's example and is intended for persons with very little understanding of mathematics and physics (myself included).

(A1) A source emits a pair of particles with some opposite pieces of information.

(A2) The pieces of information from (A1) remain constant over time.

(A3) A detector can measure the pieces of information from (A1) via three methods [X, Y and Z], but only one method at a time can be used. Each method measures a specific part of the information.

(A4) There are only two values possible for the measurements from (A3). [+ and -]

A source emits pairs of particles as per the above assumptions.

One particle is sent to a detector (Alice) and the other to another detector (Bob).

There are maximum 8 combinations of particles and 9 combinations of measurements:

From the above we get the following equivalent of Bell's inequality:

We consider the case of two entangled electrons, for which their spin is measured at three angles: X=0°, Y=120°, Z=240°.

(I) First of all I would like to state my lack of understanding on how probabilities can be meaningfully compared with actual results. Take for example the case of a dice for which we have the probability of 1/6 to get any of the faces. Does that mean that necessarily after many throws we'll get each face for about 1/6 of the number of total throws?

(II) Returning to the experiment, I think the above probability chart applies only for the case in which the types of particle pairs are measured by a complete set of measurement combinations (9). But this is something that can't be guaranteed as per how the experiment is performed, so IMO the following chart would be more appropriate:

(III) And finally, let's consider the following case (one of many) which AFAIK fully complies with the requirements of the experiment:

If the source emits pair B of particles when it just happens for Alice and Bob to do combinations 1, 3, 5, 6, 7, 8, 9 of measurements and pair C of particles when it just happens for Alice and Bob to do combinations 2 and 4 of measurements, we get a ratio of 1/3 for opposite results. Considering that this is a real possibility shouldn't only the following inequality be considered for a meaningful comparison with the actual results:

**Assumptions**(A1) A source emits a pair of particles with some opposite pieces of information.

(A2) The pieces of information from (A1) remain constant over time.

(A3) A detector can measure the pieces of information from (A1) via three methods [X, Y and Z], but only one method at a time can be used. Each method measures a specific part of the information.

(A4) There are only two values possible for the measurements from (A3). [+ and -]

**Theoretical Results**A source emits pairs of particles as per the above assumptions.

One particle is sent to a detector (Alice) and the other to another detector (Bob).

*There is no predefined arrangement for what type of particle pair the source emits (it can be the same type for the entire duration of the experiment or randomly different with each consecutive emission or any other combination) and for what methods Alice or Bob will use, just that each combination of measurements is used equally.*There are maximum 8 combinations of particles and 9 combinations of measurements:

Code:

```
Particles for Results [Alice|Bob] Opposite
Alice and Bob 1 2 3 4 5 6 7 8 9 results
X Y Z X Y Z X|X X|Y X|Z Y|X Y|Y Y|Z Z|X Z|Y Z|Z probability
A + + + - - - O O O O O O O O O 1
B + + - - - + O O I O O I I I O 5/9
C + - + - + - O I O I O I O I O 5/9
D + - - - + + O I I I O O I O O 5/9
E - + + + - - O I I I O O I O O 5/9
F - + - + - + O I O I O I O I O 5/9
G - - + + + - O O I O O I I I O 5/9
H - - - + + + O O O O O O O O O 1
```

*Probability of getting opposite ("O") results at the detectors*__>__5/9**Actual Results**We consider the case of two entangled electrons, for which their spin is measured at three angles: X=0°, Y=120°, Z=240°.

*Probability of getting opposite results at the detectors = 1/2***Conclusions**(I) First of all I would like to state my lack of understanding on how probabilities can be meaningfully compared with actual results. Take for example the case of a dice for which we have the probability of 1/6 to get any of the faces. Does that mean that necessarily after many throws we'll get each face for about 1/6 of the number of total throws?

(II) Returning to the experiment, I think the above probability chart applies only for the case in which the types of particle pairs are measured by a complete set of measurement combinations (9). But this is something that can't be guaranteed as per how the experiment is performed, so IMO the following chart would be more appropriate:

Code:

```
Particles for Results [Alice|Bob]
Alice and Bob 1 2 3 4 5 6 7 8 9
X Y Z X Y Z X|X X|Y X|Z Y|X Y|Y Y|Z Z|X Z|Y Z|Z
A + + + - - - O O O O O O O O O
B + + - - - + O O I O O I I I O
C + - + - + - O I O I O I O I O
D + - - - + + O I I I O O I O O
E - + + + - - O I I I O O I O O
F - + - + - + O I O I O I O I O
G - - + + + - O O I O O I I I O
H - - - + + + O O O O O O O O O
Opposite results 1 4/8 4/8 4/8 1 4/8 4/8 4/8 1
probability
```

*Probability of getting opposite ("O") results at the detectors*__>__4/8 = 1/2(III) And finally, let's consider the following case (one of many) which AFAIK fully complies with the requirements of the experiment:

If the source emits pair B of particles when it just happens for Alice and Bob to do combinations 1, 3, 5, 6, 7, 8, 9 of measurements and pair C of particles when it just happens for Alice and Bob to do combinations 2 and 4 of measurements, we get a ratio of 1/3 for opposite results. Considering that this is a real possibility shouldn't only the following inequality be considered for a meaningful comparison with the actual results:

*The ratio between the number of opposite results and total number of measurements*?__>__3/9 = 1/3