# Bellboy and Quantum Discussion

[From Bill, Powers (2005.04.03.0919 MDT)]

Ely Dorsey 2005.04.03.01:31EST –

](http://www.geocities.com/CapitolHill/Lobby/7049/puzzles.htm)There’s “What happened to the dollar?”, where 3 men
pay \$10 apiece for separate rooms in a hotel. The clerk gives a \$5 refund
to the bellboy to return to the 3 men. The bellboy gives each one a
dollar, and keeps \$2 for himself. “Now, each man, after receiving
the \$1 rebate, paid only \$9 for his room. That makes \$27 for all three.
The bellboy kept the other \$2, which makes \$29. What happened to the
other dollar?”

Did Dr. Bell happen to have a bellboy advising
him?

Best,.

Bill P.

[From Bruce Gregory (2005.0403.1152)]

Bill, Powers (2005.04.03.0919 MDT)

There's "What happened to the dollar?", where 3 men pay \$10 apiece for separate rooms in a hotel. The clerk gives a \$5 refund to the bellboy to return to the 3 men. The bellboy gives each one a dollar, and keeps \$2 for himself. "Now, each man, after receiving the \$1 rebate, paid only \$9 for his room. That makes \$27 for all three. The bellboy kept the other \$2, which makes \$29. What happened to the other dollar?"

Nothing. There was no other dollar. The clerk received \$25. The men paid \$27 and the Bellboy kept \$2. No non-locality here.

A true believer knows the solution before he understands the problem.

[From Bill Powers (2005.04.03.1002 MDT)]

Bruce Gregory (2005.0403.1152)

There's "What happened to the dollar?",

...

Nothing. There was no other dollar. The clerk received \$25. The men paid \$27 and the Bellboy kept \$2. No non-locality here.

If you followed up the link, you found that this was an exercise given to fifth-graders, so it's not surprising that there is an answer. However, what about the so-called violation of Bell's inequality? Is this not like proving that something is logically true, and then demonstrating that it is not true? To show a violation, you would have to write down a table of pluses and minuses corresponding to the outcomes of the experiment, apply the test of Bell's Inequality, and demonstrate that the inequality is violated for that table. But we know that for any such table, no matter how it is generated, the inequality holds -- either that, or logic itself doesn't work. Somebody, as in the Bellboy and the Missing Dollar, added the wrong numbers together if they think they proved that the inequality can be violated. It can't. It's like proving that 2 = 3. You can do it if you slip in an illegal operation that nobody catches, but that's the only way.

Or maybe Rae's example of it is flawed. I don't know. Does someone else?

Best,

Bill P.

···

A true believer knows the solution before he understands the problem.

[From Bruce Gregory (2005.0403.1340)]

Bill Powers (2005.04.03.1002 MDT)

If you followed up the link, you found that this was an exercise given to fifth-graders, so it's not surprising that there is an answer.

Or maybe Rae's example of it is flawed. I don't know. Does someone else?

There is a nice discussion of the possibilities at:

http://www.upscale.utoronto.ca/GeneralInterest/Harrison/BellsTheorem/BellsTheorem.html

A true believer knows the solution before he understands the problem.

[From Bill Powers (2005.04.03.1215 MDT)}

Bruce Gregory (2005.0403.1340) –

There is a nice discussion of
the possibilities at:

Here is an excerpt:

But consider trying to measure,
say, Number(A, not B). This is the number of electrons that are
spin-up for zero degrees, but are not spin-up for 45 degrees. Being
“not spin-up for 45 degrees” is, of course, being
spin-down for 45 degrees. But we know that by measuring the spin
of an electron at an angle of zero degrees irrevocably changes the number
of electrons which are spin-down for an orientation of 45 degrees. If we
measure at 45 degrees first, we change whether or not it is spin-up for
zero degrees. Similarly for the other two terms in this application of
the inequality. This is a consequence of the Heisenberg Uncertainty
Principle. So this inequality is not experimentally testable.

Why can’t you just go ahead and naively do the experiment and write down
the results? It doesn’t matter if you think the second result is changed
by the first measurement; just write the measurements down anyway.You can
change the order of doing the measurements, which theoretically will
change the results. Write the results into the table anyway: just do it.
Change the order at random, or don’t change it – it doesn’t matter. When
you end up you’ll have a table of the three states determined over and
over – and THE INEQUALITY WILL HOLD TRUE. That’s because once you have
the table, you will ALWAYS find that Bell’s inequality holds true. It
can’t do anything else. The difficulties cited above come from stopping
the experiment before you do it and reasoning out what the results will
be according to what you believe will happen, and as a consequence not
doing the experiment.
So proof of violations of Bell’s Inequality seem to be based on not
doing experiments. If anybody had ever just gone ahead and done the
experiments and written down the results in a table, and then analyzed
the table, Bell’s Inequality would have been shown to be true.

That is so obvious it makes me think that something is missing from these
explanations that everybody knows but nobody bothers to mention.

Best,

Bill P.

[From Bruce gregory (2005.0403.1453)]

Bill Powers (2005.04.03.1215 MDT)

So proof of violations of Bell's Inequality seem to be based on not doing experiments. If anybody had ever just gone ahead and done the experiments and written down the results in a table, and then analyzed the table, Bell's Inequality would have been shown to be true.

From the source I cited:

"The experiments have been done. For electrons the left polarizer is set at 45 degrees and the right one at zero degrees. A beam of, say, a billion electrons is measured to determine Number(right spin-up zero degrees, left spin-up 45 degrees). The polarizers are then set at 90 degrees/45 degrees, another billion electrons are measured, then the polarizers are set at 90 degrees/zero degrees for another billion electrons.

The result of the experiment is that the inequality is violated. The first published experiment was by Clauser, Horne, Shimony and Holt in 1969 using photon pairs. The experiments have been repeated many times since."

I am not sure why you claim that the experiments have not been done

A true believer knows the solution before he understands the problem.

[From Bill Powers (2005.04.03.1308 MST)]

Bruce Gregory (2005.0403.1453) --

From the source I cited:

"The experiments have been done. For electrons the left polarizer is set at 45 degrees and the right one at zero degrees. A beam of, say, a billion electrons is measured to determine Number(right spin-up zero degrees, left spin-up 45 degrees). The polarizers are then set at 90 degrees/45 degrees, another billion electrons are measured, then the polarizers are set at 90 degrees/zero degrees for another billion electrons.

The result of the experiment is that the inequality is violated. The
first published experiment was by Clauser, Horne, Shimony and Holt in 1969 using photon pairs. The experiments have been repeated many times since."

I am not sure why you claim that the experiments have not been done.

Interesting. I would like to see the table of (A, Not B) and so on. Note that Rae says it is immaterial how you write the values into the table; for example, you could fill the first and second columns at random, and then randomly duplicate either the first or the second column in the third column. The counts of occurrances would satisfy Bell's Inequality -- or so Rae claims. From his algebra I don't see any way for it NOT to be satisfied. There is no arrangement of values in the table that will not satisfy the inequality, as long as each row contains one number twice and the other number once. So any experiment that ends up with a table of values of this kind has to satisfy the inequality, doesn't it? Maybe I'm not understanding how the inequality is constructed.

The experiment described above is a little different from Rae's description, though I can't see how it would make any difference. Ray says to fill in each row, implying that you make the first, second, and third measurements, filling in a row, then repeat the experiment as many times as you like. In the version above, the columns are filled in (all the rows) one at a time. But if the inequality holds for any way of filling in the table, it shouldn't matter how the table is filled in. I think I'm missing something.

Best,

Bill P.

[From Bill Powers (2005.04.03.1328 MST)]

Bruce Gregory (2005.0403.1453)

Look at page 37 of Rae. The table of + and - numbers can be filled in by any method at all. Rae says

"Try again if you like and see if you can find a set of triplets composed of +'s and -'s that does not obey this relation: you will not succeed, because it's impossible."

So whether this table is filled in using imagination, random numbers, or experimental results, the result has to be the same: it is impossible for the relation not to be obeyed. I now see it's not even necessary for one symbol to appear twice in one row. What am I missing?

Best,

Bill P.

[From Bruce Gregory (2005.0403.1600)]

Bill Powers (2005.04.03.1328 MST)

So whether this table is filled in using imagination, random numbers, or experimental results, the result has to be the same: it is impossible for the relation not to be obeyed. I now see it's not even necessary for one symbol to appear twice in one row. What am I missing?

The real world does not allow you fill in the table with observational results because you destroy the photon in the process of making a measurement.

From the article:

"There is another objection to the experimental tests that, at least so far, nobody has managed to get totally around. We measure a spin combination of, say, zero degrees and 45 degrees for a collection of electrons and then measure another spin combination, say 45 degrees and 90 degrees, for another collection of electrons. In our classroom example, this is sort of like measuring the number of men students whose height is not over 5' 8" in one class, and then using another class of different students to measure the number of students whose height is over 5' 8" but do not have blue eyes. The difference is that a collection of, say, a billion electrons from the source in the correlation experiments always behaves identically within small and expected statistical fluctuations with every other collection of a billion electrons from the source. Since that fact has been verified many many times for all experiments of all types, we assume it is true when we are doing these correlation experiments. This assumption is an example of inductive logic; of course we assumed the validity of logic in our derivation."

A true believer knows the solution before he understands the problem.

Re: Bellboy and Quantum
Discussion
[Martin Taylor 2005.04.03.15.12]

[From Bill Powers (2005.04.03.1215
MDT)}

Bruce Gregory (2005.0403.1340) –

There is a nice discussion of the
possibilities at:
http://www.upscale.utoronto.ca/GeneralInterest/Harrison/BellsTheorem/ BellsTheorem.html

Here is an excerpt:

But consider trying to measure, say,Number(A, not B). This is the number of electrons that are spin-up
for zero degrees, but are not spin-up for 45 degrees. Being “not
spin-up for 45 degrees” is, of course, being spin-down for
45 degrees. But we know that by measuring the spin of an electron at
an angle of zero degrees irrevocably changes the number of electrons
which are spin-down for an orientation of 45 degrees. If we measure at
45 degrees first, we change whether or not it is spin-up for zero
degrees. Similarly for the other two terms in this application of the
inequality. This is a consequence of the Heisenberg Uncertainty
Principle. So this inequality is not experimentally testable.

Why can’t you just go ahead and naively do the experiment and write
down the results? It doesn’t matter if you think the second result is
changed by the first measurement; just write the measurements down
anyway.You can change the order of doing the measurements, which
theoretically will change the results. Write the results into the
table anyway: just do it. Change the order at random, or don’t change
it – it doesn’t matter. When you end up you’ll have a table of the
three states determined over and over – and THE INEQUALITY WILL HOLD
TRUE. That’s because once you have the table, you will ALWAYS find
that Bell’s inequality holds true. It can’t do anything
else.

Except experimentally.

The difficulties cited above come from
stopping the experiment before you do it and reasoning out what the
results will be according to what you believe will happen, and as a
consequence *not doing the experiment.*So proof of violations of Bell’s Inequality seem to be based on
not doing experiments. If anybody had ever just gone ahead and done
the experiments and written down the results in a table, and then
analyzed the table, Bell’s Inequality would have been shown to be
true.

Unfortunately, many people have done the experiments, using
different kinds of entities, incuding photons, electrons, and ions.
And the inequality has been shown always to fail.

That is so obvious it makes me think that
something is missing from these explanations that everybody knows but
nobody bothers to mention.

Do you think the same is true in the quantum eraser
experiment?

says that if there are three classes of something, with possible
properties A, B, and C, the classes being Class X (those that have A
and not B), Y (those that have B and not C), and Z (those that have A
and not C), the total number of the members of Classes X and Y is at
least as great as the number of class Z. This is consistent with
classical logic. The problem is that in the actual experiment, the
“logically necessary” result doesn’t hold.

Harrison says:

···

assumptions in the proof. These are:

l Logic is
a valid way to reason.

lParameters exist whether they are measured or not. For example,
when we collected the terms Number(A, not B, not C) + Number(A, B,
not C)
to get Number(A, not C), we assumed that eithernot B or B is true for every member.

Further on, after discussing the experimental results that
violate Bell’s inequality, he says:

In the last section
we made two assumptions to derive Bell’s inequality which here
become:

l Logic is valid.

``````  l Electrons have
``````

spin in a given direction even if we do not measure it.

Now we have added a third assumption in order to beat the Uncertainty
Principle:

``````  l No information
``````

can travel faster than the speed of light.

We will state these a little more succinctly as:

1. Logic is valid.

1. There is a reality separate from its
observation

2. Locality.

You will recall the
we discussed proofs by negation. The fact that our final form of
Bell’s inequality is experimentally violated indicates that at least
one of the three assumptions we have made have been shown to be
wrong.

From there, it’s a question of which assumption is wrong–that
it’s valid to deal with the real world as if its behaviour conformed
to logic, that there exists a real world, or that objects and events
in the real world can be localized.

You may call it spooky, perhaps, but if you accept the
experimental obsrvations, I think you have to deal with the apparent
spookiness. I don’t think it’s very different from the situation when
phlogiston was the obvious non-spooky way of treating heat, and the
spooky theory of tiny invisible “atoms” banging each other
about accounted for observations that good old intuitively reasonable
phlogiston didn’t.

Martin

[From Bill Powers (2005.04.03.14129 MDT)]

So proof of violations of Bell’s
Inequality seem to be based on not doing experiments. If anybody had ever
just gone ahead and done the experiments and written down the results in
a table, and then analyzed the table, Bell’s Inequality would have been
shown to be true.

Unfortunately, many people have done the experiments, using different
kinds of entities, incuding photons, electrons, and ions. And the
inequality has been shown always to fail.

Let me backtrack. When you say the inequality has been shown to fail, has
this been done by making measurements and filling in a table like the one
shown in Rae on page 37, and then counting the occurrances of up and down
spins in the table? If so then there is something extremely wrong,
because as Rae says, and I can’t contradict him, there is NO WAY those
relationships can violate the basic inequality. That inequality will hold
for ANY arrangement of ups and downs in the table, including
experimentally determined arrangements.

To show me I am wrong, all you have to do is present a table showing the
measured spins in the three columns, and demonstrate that the numbers of
combinations of +'s and -'s do not obey Bell’s Inequality.

Perhaps the problem is that in fact it is impossible to fill in the table
that Rae describes. Maybe that’s what he’s saying that I interpreted to
mean that the experiments weren’t done. But the way he described it, it
would certainly be possible.

New point. On pages 39-40 where he describes the experiments, he seems to
be describing experiments in which only two of the three spins can be
obtained at any given time. Thus in any table of results, one of the
entries for each triplet would be blank. That’s different, of course,
from saying that all three relationships hold at the same time – the two
on the left of the inequality and the relationship between their sum and
the quantity on the right. The counts that Rae talks about couldn’t be
done if there were blanks in the table.

Just to make it clear what we’re talking about, here is the table I’ve
been talking about, with up and down spins being indicated by + and
-:

`````` Condition

1    2    3

+    +    -

-    -    +

-    +    -
``````

and so on, in any random arrangment.

The Bell Inequality is then simply

n[1+,2+] + n[2-,3+] >= n[1+,3+],

where n[1+,2+] (for example) means number of rows in which
spin is up in both condition 1 and condition 2.

Since this inequality holds for any arrangement of +'s and -'s
whatsoever, it is not false for any arrangement. Therefore if this table
is filled in with experimental determinations, whether done
simultaneously or in any sequence, the inequality must hold true. Rae’s
description of how to obtain the numbers precludes any dependence on
assumptions or experimental conditions or anything else. The inequality
is simply a mathematical fact about tables of this kind and has nothing
to do with reality or experiments. Is that a misunderstanding on my part?
If so, Rae seems to be under the same impression.

That is so obvious it makes me
think that something is missing from these explanations that everybody
knows but nobody bothers to mention.

Do you think the same is true in the quantum eraser experiment?

if there are three classes of something, with possible properties A, B,
and C, the classes being Class X (those that have A and not B), Y (those
that have B and not C), and Z (those that have A and not C), the total
number of the members of Classes X and Y is at least as great as the
number of class Z. This is consistent with classical logic. The problem
is that in the actual experiment, the “logically necessary”
result doesn’t hold.

Martin, try to answer this simple question. How are the results of the
experiments tabulated? Are they in a table showing triples of results, as
in the Rae table above? If they are, then the inequality holds. There is
simply no way it can not hold, because this is a property of the table
and has nothing to do with how the table was constructed. It is true of
all possible tables of this kind.

The only guess I can make is that such tables have never been constructed
– that the results have been presented in some other way, and never
written down in a table of triplets of + and - spins.

## Harrison says:

We have made two assumptions in the proof. These are:
lLogic is a valid way to reason.lParameters exist whether they are measured or not. For example, when
we collected the terms Number(A, not B, not C) + Number(A, B, not
C)
to get Number(A, not C), we assumed that either not
B
or B is true for every member.

Why do we have to assume anything? Aren’t these three quantities EACH the
results of experimental determinations? In other words, we measure A, B,
and C, noting down whether the spins are + or - after each determination.
That completes one row of the table. Then we repeat as many times as we
like, and finally sum up the numbers of + and - marks for each condition.
Isn’t that the way it’s done?

You may call it spooky, perhaps,
but if you accept the experimental observations, I think you have to deal
with the apparent spookiness.

Since we’re talking about perceptions here and not external reality, the
spookiness is relative to what we expect. The spookiness is in the
explanation we offer, not in nature. I’m not yet convinced that we don’t
have an internal contradiction here, which is the only kind of spookiness
we have to disallow unconditionally.

Best,

Bill P.

[From Bruce Gregory (2005.0403.1806)]

Bill Powers (2005.04.03.14129 MDT)

Why do we have to assume anything? Aren't these three quantities EACH the results of experimental determinations? In other words, we measure A, B, and C, noting down whether the spins are + or - after each determination. That completes one row of the table. Then we repeat as many times as we like, and finally sum up the numbers of + and - marks for each condition. Isn't that the way it's done?

No. The experiments are done by passing a beam of photons through a polarizer. This is not a measurement, as you point out, it is called a state preparation. As a result of this state preparation we have a collection of photons in the same state. Now we pass the photons through a second polarizer oriented at an angle to the first polarizer and count the number of photons that are transmitted. This tells us the percentage of photons that are simultaneously in both states. Now we take another beam of photons and repeat the experiment with the second polarizer at a different orientation. This tells us the percentage of photons that are in both states with the new arrangement. We cannot perform both measurements on the same photons because a photon is absorbed when we measure that it has passed through the the second polarizer.

The Rae table is a thought experiment designed to show that the statistic obtained in the real experiments are inconsistent with the assumption that the photons have a definite state of polarization prior to a measurement.

A true believer knows the solution before he understands the problem.