Entanglement and teleportation

[seratend]

I'm quite familiar with Bell's analysis. One day the thought
struck me that Bell's Theorem isn't really dealing with
what is happening in the experiments. It isn't dealing
with the actual observational context. We're not analyzing
a variable (Lambda), we're analyzing a constant (the
entanglement). So, of course, the correct correlation
function can't be generated via individual contexts
wrt Lambda.

What do you want to proove? Bell theorem assumes non contextual random variables.
Contextual random variables (as well as contextual sample space) are known to be able to reflect quantum probabilites, there are plenty of documents concerning this subject in arxiv (for example quant-ph/0301027). There is nothing new about that.

Seratend.

 

[vanesch]

You don't need Lambda anyway to get Bell's Theorem.

This is correct, but as Sherlock talked about a semiclassical model, I was presuming Maxwellian light ; then the only degree of freedom left is Lambda of course. But you can feel free to include into lambda all hidden or non-hidden parameters that come from the source.

All you need to believe is that the wave function had a definite value for ANY possible polarizer setting at one of the detectors independent of the setting at the other. That is at the base of your presumption no matter how you try to describe it. Specifically, that results for polarizer settings A, B AND C could all exist simultaneously. If you say there are only A and B, you are describing the QM view.

Indeed, the actual factorisation P(A,B ; L) = P(A ; L) x P(B ; L) is not even necessary (but helpful when we DO have a model that gives us P(A ; L) such as is the case in the semiclassical treatment). The only thing it assures, as you point out, is that, given a statistical distribution p(L), P(A,B), P(A,C) and P(B,C) have simultaneous meaning within a Kolmogorov probability universe ; or said differently, that they can be derived from a hypothetical P(A,B,C) even if that P(A,B,C) is not, even in principle, ever measurable.
It is in fact sufficient to postulate the existence of a P(A,B,C) ; but that would run into the objection of being non-physical given that we cannot measure it ; even though this is a weaker hypothesis than the Bell factorization hypothesis. But from the Bell factorization hypothesis we can easily construct the "forbidden" P(A,B,C) by just integrating
P(A ; L) x P(B ; L) x P(C ; L) over p(L)
Indeed, summing P(A,B,C) over C and ~C gives us then P(A,B) ; summing over A and ~A gives us P(B,C) and summing over B and ~B gives us P(A,C) ; simply by bringing the sum inside the integral over p(L).
As such, the (stronger) factorization hypothesis gives us a natural framework to define a hypothetical P(A,B,C) ; and once such a P(A,B,C) exists from which we can derive the P(A,B), P(A,C) and P(B,C), Bell's inequalities are satisfied (and hence incompatible with QM).

cheers,
Patrick

 

[Sherlock]

I have a specific objection: your idea that the incident light produces a "presumed" function Theta which has a) hidden variables, but no Lambda; and b) the correlations still change according to distant settings prepared while the photons are in flight.

Theta is the angular difference between the polarizers associated
with a given pair. I don't know where you got the idea that the
incident light has anything to do with Theta.

And, of course the value of the correlation function changes if
you change Theta while the emitted light associated with
a pair of photons is still incident on one or both of the polarizers.
But, there is always one and only one Theta associated with
a given coincidence interval.

You don't need Lambda anyway to get Bell's Theorem. All you need to believe is that the wave function had a definite value for ANY possible polarizer setting at one of the detectors independent of the setting at the other. That is at the base of your presumption no matter how you try to describe it. Specifically, that results for polarizer settings A, B AND C could all exist simultaneously. If you say there are only A and B, you are describing the QM view.

Vanesch has called for you to present some details of your Theta. The burden is now on you to present something other than a few words if you want to make a convincing argument. It may make "sense" to you, but it doesn't make sense to me. You may as well just say that you assume you are right, and are leaving the details of your argument to someone else.

No offense please, but I think maybe you haven't read what
I've written. Otherwise you wouldn't be asking for details of Theta.

 

[Sherlock]

Bell theorem assumes non contextual random variables.

That's exactly why Bell's theorem is irrelevant wrt questions of
'locality' and 'realism'.

Contextual random variables (as well as contextual sample space) are known to be able to reflect quantum probabilites, there are plenty of documents concerning this subject in arxiv (for example quant-ph/0301027). There is nothing new about that.
Seratend.

In the observational context (that's described by cos^2 Theta),
Theta isn't analyzing a variable, random or otherwise.
Theta is analyzing the entanglement, which is assumed to
be produced via emission, and to not vary from pair to pair.

I'll check out the reference you provided. Thanks.

 

[Sherlock]

... as Sherlock talked about a semiclassical model, I was presuming Maxwellian light ; then the only degree of freedom left is Lambda of course. But you can feel free to include into lambda all hidden or non-hidden parameters that come from the source.

I'm not assuming anything about the details of the incident light.
Except that it's entangled at emission, which means that whatever
is incident on the polarizers is the same at both ends.

Indeed, the actual factorisation P(A,B ; L) = P(A ; L) x P(B ; L) is not even necessary (but helpful when we DO have a model that gives us P(A ; L) such as is the case in the semiclassical treatment). The only thing it assures, as you point out, is that, given a statistical distribution p(L), P(A,B), P(A,C) and P(B,C) have simultaneous meaning within a Kolmogorov probability universe ; or said differently, that they can be derived from a hypothetical P(A,B,C) even if that P(A,B,C) is not, even in principle, ever measurable.
It is in fact sufficient to postulate the existence of a P(A,B,C) ; but that would run into the objection of being non-physical given that we cannot measure it ; even though this is a weaker hypothesis than the Bell factorization hypothesis. But from the Bell factorization hypothesis we can easily construct the "forbidden" P(A,B,C) by just integrating
P(A ; L) x P(B ; L) x P(C ; L) over p(L)
Indeed, summing P(A,B,C) over C and ~C gives us then P(A,B) ; summing over A and ~A gives us P(B,C) and summing over B and ~B gives us P(A,C) ; simply by bringing the sum inside the integral over p(L).
As such, the (stronger) factorization hypothesis gives us a natural framework to define a hypothetical P(A,B,C) ; and once such a P(A,B,C) exists from which we can derive the P(A,B), P(A,C) and P(B,C), Bell's inequalities are satisfied (and hence incompatible with QM).

As you show, if you omit Lambda from the formulation, then
you get a 'nonphysical' Bell's theorem. This is the generality of
the relationship that the theorem describes. It's just arithmetic.

However, including Lambda makes it irrelevant wrt the observational
context that we're considering in the sense that it is supposedly telling
us that nature is nonlocal and that hidden variables don't exist. But, if
Lambda isn't a factor in determining rate of coincidental detection,
(which Bell and you and many others have shown)
then Bell's theorem isn't telling us that nature is nonlocal and
hidden variables don't exist. What it does give, via the degree of
violation of Bell inequalties, is sort of a rough quantitative measure
of the strength of the entanglement.

 

[DrChinese]

1. Theta is the angular difference between the polarizers associated with a given pair. I don't know where you got the idea that the
incident light has anything to do with Theta.

2. And, of course the value of the correlation function changes if
you change Theta while the emitted light associated with
a pair of photons is still incident on one or both of the polarizers.
But, there is always one and only one Theta associated with
a given coincidence interval.

1. You said:

The correlations are due to the analysis of a global property
(the entanglement of the light incident on the polarizers) that
is revealed in the context of joint polarizer settings (Theta),
but not in the context of individual measurement.

2. Your position, as stated here, is the QM position. It is not the local realistic position you are trying to defend. It is QM that states that Theta is fundamental in the sense that the act of observation creates the reality. Local realism states that there are hidden local variables.

No offense, but your words are those of someone who ignores Bell completely. The point is simple: you postulate that local reality is maintained but refuse to accept any physical meaning to that statement. Bell provides such meaning and it is generally accepted. We all agree that theta is an observable in the sense you describe. But QM says that there is no underlying reality to individual elements of reality outside of the context of a measurement, an idea you appear to deny.

 

[vanesch]

It's just arithmetic.

Yes, but it is the arithmetic used in all of science, especially medicine and sociology. It is the amazing fact that QM does NOT follow this arithmetic that is profoundly surprising, and I don't see how you can miss that point !

You are correct of course to state that quantum theory, as an epistemological stochastic theory, can just as well posit correlations as individual probabilities, without their being a) a direct causal influence or b) common cause: probabilities fall out of the sky, and just as well joint probabilities. There is no such thing as causal relation, common cause or whatever. Just probabilities. But that only goes as far as one attaches only an epistemological value to quantum predictions.
If one tries to give an ontological meaning to the quantum formalism, or even to another theory that has equivalent statistical predictions, and one assumes that, ONCE ALL COMMON CAUSES ARE TAKEN INTO ACCOUNT, PROBABILITIES AT SPACE-LIKE SEPARATED EVENTS MUST BE STATISTICALLY INDEPENDENT, (these are the assumptions of Bell's theorem) then this doesn't work. And the above assumption is part of all of classical statistical physics, so no semiclassical model can produce the probabilities generated by entangled quantum states.

However, including Lambda makes it irrelevant wrt the observational
context that we're considering in the sense that it is supposedly telling
us that nature is nonlocal and that hidden variables don't exist.

Lambda, as polarization direction of classical light, is normally what should locally describe completely what happens to the light (the intensity getting through the polarizer) ; and the above assumption, namely the random clicking of the detector as a function of the transmitted intensity, should then be a statistically independent variable of any OTHER such detection process. Only the intensity counts to give us the probability of clicking per unit of time, and this should be an independent Poisson process of any other clicking.
This is clearly NOT what quantum theory predicts.

Let me try to give you one more analogy. Assume that balls of different colors, namely blue, red or green, can be put into a bag, and transported to Alice and Bob. It is such that each time a red ball is put in Alice's bag, then also a red ball is put into Bob's bag and so on. So they are always identical colors (that's the "common cause"). Now assume that on Alice's side, someone takes the ball and throws a dice, with outcome Da, which can be 1,2,3,4,5 or 6.
Alice chooses the order of the 3 colors, let us say, red - green - blue.
Now, if the ball is red (no matter what is Da), or if the ball is green and Da>3 or if the ball is blue and Da>4, then he "clicks" to Alice ; the same thing is done at Bob's side after Bob has choosen his order of colors.
Clearly, in the case that Alice and Bob choose the same order of colors, red -green - blue, P(Alice click) = p(red) + 1/2 p(green) + 1/3 p(blue), the same for Bob, and the probability for P(Alice AND Bob click) = p(red) + 1/4 p(green) + 1/9 p(blue). So we don't have that
P(Alice AND Bob click) = P(Alice click) x P(Bob click)

However, if we take into account the common cause (the color of the ball) then we do have:
for red: P(Alice click ; red) = 1 ; P (Bob click ; red) = 1 ; P(Alice+Bob click ; red) = 1

for green: 1/2 ; 1/2 and 1/4 respectively

for blue: 1/3, 1/3 and 1/9 respectively.

The reason is that, after having taken all COMMON CAUSES into account, the fundamentally stochastic elements (the dice at each place) are supposed to be statistically independent. This is the fundamental hypothesis in Bell's theorem, and in fact in all statistical analysis ever done to find causal (or common origin) relations.

if
Lambda isn't a factor in determining rate of coincidental detection,
(which Bell and you and many others have shown)
then Bell's theorem isn't telling us that nature is nonlocal and
hidden variables don't exist. What it does give, via the degree of
violation of Bell inequalties, is sort of a rough quantitative measure
of the strength of the entanglement.

But I tried to illustrate that Lambda IS such a factor, in the case the source is polarised: in the case that the source (lambda) is perpendicular to both a and b (theta = 0) the rate of coincidental detection is ZERO !
While it isn't when the source is parallel to both a and b.

cheers,
Patrick.

 

[Sherlock]

... a priori P(A,B) is the probability of detector A and detector B clicking, in an arbitrary interval of, say, 10 ns.
P(A) is the probability of detector A clicking in an arbitrary interval of 10 ns and P(B) is the probability of detector B clicking in an arbitrary interval of 10 ns.

Wrt P(A,B), Theta is analyzing a global constant
which is not revealed (ie., is not relevant to P(A) or
P(B)) in the individual context.

And, wrt P(A) or P(B) the individual polarizers are analyzing
a variable which is not relevant to P(A,B), which variable if you globalize it to account for P(A,B) gives an incorrect description.

When the light is polarized, and is perpendicular to a and b (theta = 0), then it is ALWAYS An and Bn that trigger, never A or B. So this leads to P(A,B) = 0. When the light is polarized and parallel to a and b, then it is always A and B that trigger, never An or Bn, and we have P(A,B) = 1.

You're confusing the issue. Lambda is irrelevant.

If An and Bn trigger in the same interval, then that is a
coincidental detection at An and Bn.

Coincidental detection at An and Bn is the
same as coincidental nondetection at A and B.

We're dealing with the probability of coincidental detection,
which means getting the same detection attributes at A and
B for a given interval.
The probability of coincidental detection when Theta = 0
is 1.

The incident light can be polarized any way you like. As
long as the two polarizers are analyzing the same thing,
and are aligned, then you get coincidental detection.

Indeed, there is no way to get that function using lambda, and that's exactly the content of Bell's theorem !

And that tells you ... what? :)

There are at least three possible physical reasons why Lambda
doesn't account for P(A,B).

(1) nature is nonlocal

(2) hidden variables don't exist

(3) Lambda isn't what Theta is measuring

I've simply chosen (3) as my starting point
in attempting to understand some of this stuff. It seems
like the most parsimonious assumption to make at this time.
Formulations of P(A,B) using Lambda simply don't describe
the 'reality' of the observational context.

All of your analysis, and Bell's analysis, is consistent
with (3) -- which option is also consistent
with standard physics.

The second option (2) is inconsistent with individual
results.

The first option (1) is just speculation, the
foundation of which is a misinterpretation of the
meaning of Bell's work and the qm formulation.

 

[Sherlock]

Your position, as stated here, is the QM position. It is not the local realistic position you are trying to defend.

In a sense that's right. I'm understanding the QM prediction
by seeing that the entanglement, not Lambda, is the relevant
emission-produced global parameter of the incident light.

And this is only revealed in the observational context defined
by Theta.

Everything is quite locally explained in this view, and
there is a global parameter of the incident light which
exists prior to polarization at the polarizers and detection
at the detectors.

So, you could call this a local realist interpretation of
the qm formalism. At least for the types of experiments
we're considering.

It is QM that states that Theta is fundamental in the sense that the act of observation creates the reality. Local realism states that there are hidden local variables.

Theta isn't fundamental. It's just crossed linear polarizers.
The entanglement is fundamental. Local realism says that
the entanglement (remember that it's assumed to be constant,
never varying from pair to pair like Lambda does) is produced
at emission.

No offense, but your words are those of someone who ignores Bell completely. The point is simple: you postulate that local reality is maintained but refuse to accept any physical meaning to that statement.

No offense taken. :) But, if read my posts carefully, then you'll
see how local reality is maintained in the view I've been
presenting, and why Bell's analysis is irrelevant to the question
of local reality.

Bell provides such meaning and it is generally accepted. We all agree that theta is an observable in the sense you describe.

Bell provided a formulation in terms of Lambda, an emission-produced,
*variable* global parameter of the incident light. He showed that
this formulation is incompatible with qm predictions and experiments
have shown that his formulation is incompatible with reality.

Now, one can just stop there and say that Bell has shown that
there's no local reality. And, that's what it seems that most
people do. But, that is a wrong conclusion because there still
exists the possibility that Bell's formulation is simply irrelevant
to the observational context -- which is the view I currently take.

I've explained that the correlations aren't caused by a *variable* global parameter of the incident light, but rather are produced by a *constant*
global parameter of the incident light when viewed in the context
of the variable Theta (the angular difference between the polarizers).

But QM says that there is no underlying reality to individual elements of reality outside of the context of a measurement, an idea you appear to deny.

Yes, I would deny the idea that there's no reality underlying
measurement results. Quantum theory was developed
with the idea that a qualitative understanding of the reality
underlying measurement results was not possible. It's not
denying the existence of an underlying reality. That would
be silly. Obviously the instruments are measuring *something*. :)

 

[DrChinese]

In a sense that's right. I'm understanding the QM prediction by seeing that the entanglement, not Lambda, is the relevant
emission-produced global parameter of the incident light.

And this is only revealed in the observational context defined
by Theta.

Everything is quite locally explained in this view, and
there is a global parameter of the incident light which
exists prior to polarization at the polarizers and detection
at the detectors.

So, you could call this a local realist interpretation of
the qm formalism. At least for the types of experiments
we're considering.

...

Yes, I would deny the idea that there's no reality underlying
measurement results. Quantum theory was developed
with the idea that a qualitative understanding of the reality
underlying measurement results was not possible. It's not
denying the existence of an underlying reality. That would
be silly. Obviously the instruments are measuring *something*. :)

OK, I am trying to follow your reasoning - so maybe the above gets us closer.

1. What, then, is a "global parameter of the incident light" ? If it is the entangled wavefunction, then this is a description that exactly matches QM. To use your words, this exists prior to the photons arriving at the polarizers.

2. Clearly, you get different results when you use different Thetas. But since Theta is determined AFTER emission and while the photons are separated, how is this a local effect?

3. And finally: why are you convinced that Bell does not apply, if in fact you say that local hidden variables exist? You say that "Bell's formulation is simply irrelevant to the observational context" because the global parameter is "constant" rather than "variable" (?). But that wouldn't make any difference to Bell's formulation anyway.

 

[vanesch]

Wrt P(A,B), Theta is analyzing a global constant
which is not revealed (ie., is not relevant to P(A) or
P(B)) in the individual context.

I don't know what these words are supposed to mean. Do you mean that the outcomes at A and B, when "looking at the global constant" are somehow different than those same outcomes, when we "look at them in the individual context" ?

And, wrt P(A) or P(B) the individual polarizers are analyzing
a variable which is not relevant to P(A,B), which variable if you globalize it to account for P(A,B) gives an incorrect description.

But from P(A,B) and P(A,~B) you can calculate P(A) of course: that's basic probability theory.

If An and Bn trigger in the same interval, then that is a
coincidental detection at An and Bn.

Coincidental detection at An and Bn is the
same as coincidental nondetection at A and B.

I'm talking about the probability P(A,B): the probability of joint detection in directions a and b. Whether or not there has been a joint detection at An and Bn is irrelevant, because it doesn't contribute to P(A,B). So again, what is your proposal for P(A,B) (the joint probability of a detection at A and B (and not An or Bn)) ; you claim this to be independent of the incident polarization, which means I can change it the way I want (random distribution, or a fixed, chosen direction). I'm telling you that in one case we have P(A,B) = 1 (high rate of coincident clicks between A and B) and in another case we have P(A,B) = 0 (rate of coincident clicks 0), for the same theta. So P(A,B) cannot be independent of that polarization direction, no ?

We're dealing with the probability of coincidental detection,
which means getting the same detection attributes at A and
B for a given interval.
The probability of coincidental detection when Theta = 0
is 1.

It is clearly 0 in the case that incident light is perpendicular to both a and b because no light gets through on both sides ! Again, P(A,B) is the counting rate for A and B to click together. If A and B never click, their coincident counting rate is 0, no ?

cheers,
Patrick.

 

[Sherlock]

I don't know what these words are supposed to mean. Do you mean that the outcomes at A and B, when "looking at the global constant" are somehow different than those same outcomes, when we "look at them in the individual context" ?

Theta analyzes the *relationship* between the light incident
on the polarizers regardless of the specific polarization of
the incident light.

An individual polarizer analyzes the specific polarization of
the incident light.

The outcomes at A are random.
The outcomes at B are random.
The combined outcomes at A and B are random
unless you correlate them wrt Theta.
Why? Because Theta is analyzing
the entanglement, which is something
different than what either polarizer by
itself is analyzing.

I'm talking about the probability P(A,B): the probability of joint detection in directions a and b. Whether or not there has been a joint detection at An and Bn is irrelevant, because it doesn't contribute to P(A,B). So again, what is your proposal for P(A,B) (the joint probability of a detection at A and B (and not An or Bn))

You're right, it's .5 cos^2 Theta.

; you claim this to be independent of the incident polarization, which means I can change it the way I want (random distribution, or a fixed, chosen direction). I'm telling you that in one case we have P(A,B) = 1 (high rate of coincident clicks between A and B) and in another case we have P(A,B) = 0 (rate of coincident clicks 0), for the same theta. So P(A,B) cannot be independent of that polarization direction, no ?

The coincidence rate at A and B with Theta = 0 is .5
no matter what the incident, common polarization is.

It is clearly 0 in the case that incident light is perpendicular to both a and b because no light gets through on both sides !

We don't know what the incident polarization is. Presumably,
during any run, with Theta = any angle, the incident polarization is
cycling through the entire range. So, the rate of coincidental
detection would seem to be independent of the direction of
the incident polarization.

With Theta = 0 then the coincidental detection rate should
be the same as the individual rate which is .5, and
I think that's about what's gotten experimentally.

 

[Sherlock]

OK, I am trying to follow your reasoning - so maybe the above gets us closer.

1. What, then, is a "global parameter of the incident light" ? If it is the entangled wavefunction, then this is a description that exactly matches QM. To use your words, this exists prior to the photons arriving at the polarizers.

The global parameter of the incident light is that the light incident
on polarizer_a is the same as the light incident on polarizer_b
during a given coincidence interval.
This includes being in phase, which is important wrt how the
amplitudes of the incident waves are (jointly) altered by the
polarizers, and therefore how the amplitudes of the waves
transmitted by the polarizers during a coincidence interval
are related for that interval.

This global parameter is a constant. It's what all variations
of Lambda have in common. Can we just call it Lambda?
I don't know.

2. Clearly, you get different results when you use different Thetas.

The predictability of which implies that the variable
Theta is analyzing something that isn't changing randomly.
But, Lambda is changing randomly -- so that presents
a difficulty in how to talk about what it is that Theta
is analyzing.

Note that the rate of coincidental detection is
*only* due to changes in Theta.

But since Theta is determined AFTER emission and while the photons are separated, how is this a local effect?

One Theta (the angular difference between the settings
of two spacelike separated polarizers) is analyzing the
relationship between the light incident on the polarizers
during a given coincidence interval.

Theta is the common (global) variable cause of the predictably
variable joint results.

Individual results are random sequences.

Combine these random sequences wrt coincidence
intervals and they're still random sequences.

Correlate the combined results wrt Theta,
and wrt a given Theta you can expect
to get n coincidental detections per unit time
(with n varying as .5 cos^2 Theta).

3. And finally: why are you convinced that Bell does not apply, if in fact you say that local hidden variables exist?

I'm not necessarily *convinced* -- it's just one
approach. And, I'm only exploring the idea that
Bell doesn't apply to questions of locality and
realism.

No matter what is eventually learned, Bell's
analysis is extremely important.

I have the idea that local hidden variables exist
because it seems most parsimonious to assume
that the random variability of individual
results is coming from the emission events.
But, as you've noted, for all anyone knows, it
could be produced at/in the polarizer, or at/in the
detector, or between the emitter and the polarizer,
or between the polarizer and the detector, or ...

However, the setups and instruments are calibrated
very carefully, so the likelihood of something
other than the emission events being the cause
seems rather small.

You say that "Bell's formulation is simply irrelevant to the observational context" because the global parameter is "constant"
rather than "variable" (?).

The *relevant* global parameter (of the incident light)
for the *predictable* (that is, correlated wrt Theta)
joint results can't be varying randomly -- because
the number of coincidental detections per unit time
wrt a given Theta isn't random.

Bell's program was to see if the globalization of
an emission-produced variable parameter (that, if known,
would make individual results predictable) could duplicate
the qm corrlation function for singlet-type setups.

The problem is that predictable joint results
are due to the global variable Theta and a
constant global parameter of the light.

But that wouldn't make any difference
to Bell's formulation anyway.

It would make a difference in how experimental
violations of Bell inequalities are interpreted
(that is, it would make a difference wrt the
relevancy of Bell's theorem to questions of
local realism).

 

[vanesch]

We don't know what the incident polarization is. Presumably,
during any run, with Theta = any angle, the incident polarization is
cycling through the entire range. So, the rate of coincidental
detection would seem to be independent of the direction of
the incident polarization.

In the case of the entanglement |0>|0> + |90>|90> yes, but your reasoning should also hold when I use a source with a SPECIFIC, fixed polarization (for instance when there are polarizers at each output of the source!). Quantum mechanically, this corresponds to a pure state, |0>|0> for instance.
As you claim that P(A,B) does not depend on the specific polarization direction in the case |0>|0> + |90>|90>, it should also not depend on it in the case |0>|0>. Now that's where I present you with the problem that in the case I have |0>|0> (and imagine that a and b are both 0, with theta 0) I will have P(A,B) = 1 (namely EVERYTHING gets through, on both sides), and if I now change a and b to 90 degrees, keeping |0> |0>, NOTHING gets through, and P(A,B) = 0.
So clearly P(A,B) cannot be only a function of theta, but must depend also on the angle between the polarization direction of the source and a and b, in this case. But according to your claims, it is always the same value, no matter what the incident polarization, as long as theta is a constant.

cheers,
Patrick.

 

[DrChinese]

1. The global parameter of the incident light is that the light incident on polarizer_a is the same as the light incident on polarizer_b
during a given coincidence interval.
This includes being in phase, which is important wrt how the
amplitudes of the incident waves are (jointly) altered by the
polarizers, and therefore how the amplitudes of the waves
transmitted by the polarizers during a coincidence interval
are related for that interval.

2. This global parameter is a constant. It's what all variations
of Lambda have in common. Can we just call it Lambda?
I don't know.

The predictability of which implies that the variable
Theta is analyzing something that isn't changing randomly.
But, Lambda is changing randomly -- so that presents
a difficulty in how to talk about what it is that Theta
is analyzing.

3. Note that the rate of coincidental detection is
*only* due to changes in Theta.

One Theta (the angular difference between the settings
of two spacelike separated polarizers) is analyzing the
relationship between the light incident on the polarizers
during a given coincidence interval.

4. I'm not necessarily *convinced* -- it's just one
approach. And, I'm only exploring the idea that
Bell doesn't apply to questions of locality and
realism.

No matter what is eventually learned, Bell's
analysis is extremely important.

I have the idea that local hidden variables exist
because it seems most parsimonious to assume
that the random variability of individual
results is coming from the emission events.
But, as you've noted, for all anyone knows, it
could be produced at/in the polarizer, or at/in the
detector, or between the emitter and the polarizer,
or between the polarizer and the detector, or ...

However, the setups and instruments are calibrated
very carefully, so the likelihood of something
other than the emission events being the cause
seems rather small.

The *relevant* global parameter (of the incident light)
for the *predictable* (that is, correlated wrt Theta)
joint results can't be varying randomly -- because
the number of coincidental detections per unit time
wrt a given Theta isn't random.

Bell's program was to see if the globalization of
an emission-produced variable parameter (that, if known,
would make individual results predictable) could duplicate
the qm corrlation function for singlet-type setups.

1. What you call a global parameter, is usually called a common wavefunction. This is simply the definition of entanglement. I guess I would agree that it has a global (non-local) element.

2. There is no Lambda. That is the explanation, and there is no difficulty.

3. This is a prediction of QM, specifically, what that value will be for any Theta. Competing theories need to yield the same predictions.

4. Bell applies, unless you provide non-standard definitions of locality and realism. Bell defines local reality as meaning there exists a definite value for a Theta other than the one actually observed. If you reject that definition, then naturally you won't follow Bell the rest of the way. However, that definition is fairly well respected with the physics community.

My conclusion is that you assume what you seek to prove. Your argument seems essentially identical to the logic of EPR, which also makes an unwarranted (but reasonable!) leap of logic in its final paragraph.

This discussion has reached the point at which I do not feel I can either add understanding or gain understanding. Thanks, and I will be sitting back and watching for a while.

 

[Sherlock]

In the case of the entanglement |0>|0> + |90>|90> yes, but your reasoning should also hold when I use a source with a SPECIFIC, fixed polarization (for instance when there are polarizers at each output of the source!). Quantum mechanically, this corresponds to a pure state, |0>|0> for instance.
As you claim that P(A,B) does not depend on the specific polarization direction in the case |0>|0> + |90>|90>, it should also not depend on it in the case |0>|0>. Now that's where I present you with the problem that in the case I have |0>|0> (and imagine that a and b are both 0, with theta 0) I will have P(A,B) = 1 (namely EVERYTHING gets through, on both sides), and if I now change a and b to 90 degrees, keeping |0> |0>, NOTHING gets through, and P(A,B) = 0.
So clearly P(A,B) cannot be only a function of theta, but must depend also on the angle between the polarization direction of the source and a and b, in this case. But according to your claims, it is always the same value, no matter what the incident polarization, as long as theta is a constant.

cheers,
Patrick.

I'm not sure what setup you're using. Are there two polarizers,
in series, on each side of the emitter?

 

[vanesch]

I'm not sure what setup you're using. Are there two polarizers,
in series, on each side of the emitter?

Alice Det <-- |POL Alice a| <-------|pol 0 deg| (source) |pol 0 deg| -----> |POL Bob b| --> Bob Det

Or you could use a source which emits already polarized light. It is hard to claim that putting in parallel polarizers will undo any "common" property... but even in that case you could use a polarized source.

 

[Sherlock]

In the case of the entanglement |0>|0> + |90>|90> yes, but your reasoning should also hold when I use a source with a SPECIFIC, fixed polarization (for instance when there are polarizers at each output of the source!). Quantum mechanically, this corresponds to a pure state, |0>|0> for instance.
As you claim that P(A,B) does not depend on the specific polarization direction in the case |0>|0> + |90>|90>, it should also not depend on it in the case |0>|0>. Now that's where I present you with the problem that in the case I have |0>|0> (and imagine that a and b are both 0, with theta 0) I will have P(A,B) = 1 (namely EVERYTHING gets through, on both sides), and if I now change a and b to 90 degrees, keeping |0> |0>, NOTHING gets through, and P(A,B) = 0.
So clearly P(A,B) cannot be only a function of theta, but must depend also on the angle between the polarization direction of the source and a and b, in this case. But according to your claims, it is always the same value, no matter what the incident polarization, as long as theta is a constant.

cheers,
Patrick.

P(A,B) is the rate at which coincidental results are recorded.
And, this rate will of course change as you vary the incident
polarization wrt the aligned polarizers.

But, this just changes the coincidental detection *visibility*
range.

The entanglement isn't affected by variations in the
incident global polarization. The polarization of the light
incident on the polarizers can be anything -- as long
as it's the same at both ends, then, with Theta = 0,
A and B will record identical detection attributes during
any given coincidence window.

 

[vanesch]

P(A,B) is the rate at which coincidental results are recorded.
And, this rate will of course change as you vary the incident
polarization wrt the aligned polarizers.

Ah, but you claimed before that this rate was something like 1/2 cos^2(theta), INDEPENDENT of the incident polarization lambda, which was an "irrelevant" quantity:

The coincidence rate at A and B with Theta = 0 is .5
no matter what the incident, common polarization is.

and now that rate of coincident results (remember that the total INCOMING flux is always the same, and if we are normalizing again on A or An, we find P(A,B) strongly varying) is OF COURSE dependent on the polarization detection.

Now if we work out quantum-mechanically what we are supposed to obtain when we have a |0>|0> state, and we test this with an <a| <b| measurement, we have the joint probability
P(A,B) = |<a|<b|0>|0>|^2 = |<a|0>|^2 |<b|0>|^2 = cos^2(a) cos^2(b)

(that's a quantummechanical prediction).

So if we now rotate this polarized source over lambda, we have the joint probability P(A,B ; lambda) = cos^2(a-lambda) cos^2(b-lambda), right ?

That's EXACTLY the same result as the semiclassical prediction I gave you a long time ago for incident light on both sides with identical polarization lambda: it is the double, and independent, application of Malus' law for each independent detector. But this time it is also the quantummechanical result for a polarized source in direction lambda. Both methods work in this case (semiclassical and pure quantum).

And now I come back to my initial reasoning: if you now assume that the "entangled" source is like this (semiclassical) source with polarization lambda, but where lambda is randomly varying from event to event, then (that's basic probability theory) the overall P(A,B) without knowing lambda, is of course the P(A,B ; lambda) weighted (integrated over) the probability distribution of lambda, p(lambda). And cylindrical symmetry then makes this p(lambda) = 1/2pi.

So, if you suppose that light from an entangled source is just "randomly polarized, but each time identical, semiclassical light", then you work out this weighting: P(A,B) = integral over lambda of p(lambda) x P(A,B ; lambda) and you find 1/8(2 - cos(2(a-b))). (or whatever it was that I got).

Saying things like that the polarization is "irrelevant" for P(A,B) is clearly wrong as you now see yourself, from the moment that the source is polarized.
It cannot be "irrelevant" for one case, and "of course dependent" in the other, when the analyzing technique is identical (and only the source changes, which was exactly what we were trying to analyze).

I rest my case.

cheers,
Patrick.

PS: the same simple technique gives us the right coincident rate in the case of the entangled state |psi> =1/sqrt(2) (|0>|0> + |90>|90>):

|<a|<b| psi>|^2 = 1/2 |<a|0><b|0> + <a|90><b|90>|^2 =
1/2 (cos(a) cos(b) + sin(a) sin(b))^2 = 1/2(cos(a-b))^2 = 1/2 cos^2 theta.

 

[Sherlock]

Now if we work out quantum-mechanically what we are supposed to obtain when we have a |0>|0> state, and we test this with an <a| <b| measurement, we have the joint probability
P(A,B) = |<a|<b|0>|0>|^2 = |<a|0>|^2 |<b|0>|^2 = cos^2(a) cos^2(b)

(that's a quantummechanical prediction).

So if we now rotate this polarized source over lambda, we have the joint probability P(A,B ; lambda) = cos^2(a-lambda) cos^2(b-lambda), right ?

That's EXACTLY the same result as the semiclassical prediction I gave you a long time ago for incident light on both sides with identical polarization lambda: it is the double, and independent, application of Malus' law for each independent detector. But this time it is also the quantummechanical result for a polarized source in direction lambda. Both methods work in this case (semiclassical and pure quantum).

And now I come back to my initial reasoning: if you now assume that the "entangled" source is like this (semiclassical) source with polarization lambda, but where lambda is randomly varying from event to event, then (that's basic probability theory) the overall P(A,B) without knowing lambda, is of course the P(A,B ; lambda) weighted (integrated over) the probability distribution of lambda, p(lambda). And cylindrical symmetry then makes this p(lambda) = 1/2pi.

So, if you suppose that light from an entangled source is just "randomly polarized, but each time identical, semiclassical light", then you work out this weighting: P(A,B) = integral over lambda of p(lambda) x P(A,B ; lambda) and you find 1/8(2 - cos(2(a-b))). (or whatever it was that I got).

Saying things like that the polarization is "irrelevant" for P(A,B) is clearly wrong as you now see yourself, from the moment that the source is polarized.
It cannot be "irrelevant" for one case, and "of course dependent" in the other, when the analyzing technique is identical (and only the source changes, which was exactly what we were trying to analyze).

I rest my case.

Ok it's relevant. :)

You've got a source producing entangled photon pairs
with the polarization unchanging from pair to pair.

You rotate polarizer_a, whose setting we'll
denote as p_a, to find the maximum
detection rate at A. Denote this setting as
MDR_a and the detection rate at MDR_a as mdr_a.
Denote the detection rate at A for any p_a as
dr_a. dr_a should vary (mdr_a --> 0) as
|p_a - MDR_a| varies (0 --> pi/2) as
the function,

dr_a = mdr_a(cos^2 |p_a - MDR_a|).

Using the same conventions at B,
if MDR_b = MDR_a, then if mdr_b = mdr_a,
then the rate of coincidental detection,
denoted as cd_AB should vary (mdr --> 0)
as |p_a - p_b| varies (0 --> pi/2) as
the function,

cd_AB = mdr(cos^2 |p_a - p_b|).


Now, if you rotate the aligned
polarizers so that mdr decreases
by say .125, then cd_AB will vary
(mdr-.125 --> .125mdr) as |p_a - p_b|
varies (0 --> pi/2) as the
function,

cd_AB = mdr[.125(1 + 2(cos^2 |p_a - p_b|))].

So, in order to keep the classical
notion of a definite global polarization
per emission as an explanation for Bell test
results, then it must be that the polarizer
at the end that initiates the coincidence
interval was aligned with the emission
polarization of the light incident on it.

(It seems likely to be so aligned
a few times per second out of the
millions of randomly polarized
emissions per second.)

And, since the light incident on the
other polarizer for that interval is
polarized via emission the same as
the light incident on the polarizer
at the detecting end, then you get
the qm prediction, and a physical
explanation for why the projection
works.

Do you see anything wrong with this?

 

[vanesch]

Ok it's relevant. :)

You've got a source producing entangled photon pairs
with the polarization unchanging from pair to pair.

This is the state |mdr_a> |mdr_a> if I understand you well. Technically this is not called an "entangled" state, but a product state (but that doesn't mattter for the discussion here).

You rotate polarizer_a, whose setting we'll
denote as p_a, to find the maximum
detection rate at A. Denote this setting as
MDR_a and the detection rate at MDR_a as mdr_a.
Denote the detection rate at A for any p_a as
dr_a. dr_a should vary (mdr_a --> 0) as
|p_a - MDR_a| varies (0 --> pi/2) as
the function,

dr_a = mdr_a(cos^2 |p_a - MDR_a|).

Up to here, I agree. That's also what I wrote.

Using the same conventions at B,
if MDR_b = MDR_a, then if mdr_b = mdr_a,
then the rate of coincidental detection,
denoted as cd_AB should vary (mdr --> 0)
as |p_a - p_b| varies (0 --> pi/2) as
the function,

cd_AB = mdr(cos^2 |p_a - p_b|).

No, this is not correct (according to quantum mechanics).
cd_AB = mdr cos^2(p_a-MDR_a) cos^2(p_b - MDR_a).

You can see this easily when you calculate the in - product of the ket of the joint detection (<a|<b|) with the state of the light |mdr_a>|mdr_a>, squared, which gives you the probability of observing this (joint) state. But note that it takes on (in this case) the form of a product of the detection probabilities at A and at B respectively (I didn't put this in, it came out of the QM calculation, but it is a result of the fact that the initial state was not an entangled state but a pure product state).

cheers,
Patrick.

 

[Sherlock]

No, this is not correct (according to quantum mechanics).
cd_AB = mdr cos^2(p_a-MDR_a) cos^2(p_b - MDR_a).

We've set p_a = MDR_a and are varying p_b so,

cos^2(p_a - MDR_a) = 1, and |p_b - MDR_a| = |p_b - p_a| so,

cd_AB = mdr{cos^2 |p_b - p_a|).


So, what about thinking of the detection that
initiates a coincidence interval as being aligned
with the (global) emission polarization?

 

[vanesch]

We've set p_a = MDR_a and are varying p_b so,

cos^2(p_a - MDR_a) = 1, and |p_b - MDR_a| = |p_b - p_a| so,

cd_AB = mdr{cos^2 |p_b - p_a|).

Ah, sorry, I missed that in your previous message. Yes, then it is correct.

So, what about thinking of the detection that
initiates a coincidence interval as being aligned
with the (global) emission polarization?

Do you mean: IF I have a detection at A, then necessarily the incident light must be parallel to a ? Then the detection rate at A (with fixed incident polarization) wouldn't follow Malus' law as a function of a. Indeed, you see yourself that if a is not perfectly aligned with the incident, fixed polarization, we still get clicks at A (diminished by a factor given by Malus' law). So these clicks start coincidence intervals when the incident light is NOT aligned with the global emission polarization in this case. The global emission polarization is, say, 0 degrees (fixed) by the source, and a = 45 degrees. They are clearly not aligned, nevertheless, there is still a click rate which is half of the MDR at A.


cheers,
Patrick.

 

[Sherlock]

Do you mean: IF I have a detection at A, then necessarily the incident light must be parallel to a ? Then the detection rate at A (with fixed incident polarization) wouldn't follow Malus' law as a function of a. Indeed, you see yourself that if a is not perfectly aligned with the incident, fixed polarization, we still get clicks at A (diminished by a factor given by Malus' law). So these clicks start coincidence intervals when the incident light is NOT aligned with the global emission polarization in this case. The global emission polarization is, say, 0 degrees (fixed) by the source, and a = 45 degrees. They are clearly not aligned, nevertheless, there is still a click rate which is half of the MDR at A.

For the coincidence interval initiated by a detection at A,
the maximum photon flux, ie., the maximum detection rate, mdr_a,
is 1. So, the polarizer setting at A, (p_a), is MDR_a for that
interval.

So, assuming a definite but randomly varying global polarization
produced at emission, the math seems to be telling us that the
qm projection along the transmission axis of the polarizer
associated with a coincidence-interval-initiating detection must
be parallel to (or at least *very* closely aligned with) the
global polarization of the light incident on the polarizers
for any given coincidence interval.

 

[vanesch]

For the coincidence interval initiated by a detection at A,
the maximum photon flux, ie., the maximum detection rate, mdr_a,
is 1. So, the polarizer setting at A, (p_a), is MDR_a for that
interval.

Wait, wait... I thought that the MDR_a was obtained when we were TESTING different a values, and then found the a value of maximum intensity ! Which will of course correspond to the polarization direction of the polarized source. With the same source, we can now CHANGE a to another, set value, and we will still have a detection rate (smaller than mdr_a). So you cannot claim that each click then corresponds to an event where the source polarization is coincident with a because the source has fixed polarization, and we changed a from that polarization.

So, assuming a definite but randomly varying global polarization
produced at emission, the math seems to be telling us that the
qm projection along the transmission axis of the polarizer
associated with a coincidence-interval-initiating detection must
be parallel to (or at least *very* closely aligned with) the
global polarization of the light incident on the polarizers
for any given coincidence interval.

Yes, and that's the "magic" of course. Because classically this cannot be, and the proof of that is exactly what I said above (and in at least 5 posts before): for a source with a FIXED polarization (of which we KNOW the polarization and which is NOT randomly varying), it is clearly not true that on a click, this means that the source polarization is parallel to the detected polarization. So if it is not true for a fixed source, why should it suddenly become true for a random varying source ?

cheers,
Patrick.

 

[Sherlock]

For the coincidence interval initiated by a detection at A,
the maximum photon flux, ie., the maximum detection rate, mdr_a,
is 1. So, the polarizer setting at A, (p_a), is MDR_a for that
interval.

Wait, wait... I thought that the MDR_a was obtained when we were
TESTING different a values, and then found the a value of maximum intensity !

In one case (nonvarying source polarization) dr_a varies as you vary p_a, and in
the other (randomly varying source polarization) dr_a doesn't vary as you vary p_a.
In either case we can get a value for mdr_a and MDR_a for any interval.

Which will of course correspond to the polarization direction of the polarized source.

That seems like an ok assumption (sort of obvious even), but
it's unnecessary for the formulation I've presented.

With the same source, we can now CHANGE a to another, set value, and we will still
have a detection rate (smaller than mdr_a).

In the first case (where dr_a varies with p_a), yes. In the
other, no, because dr_a doesn't vary with p_a.

So you cannot claim that each click then corresponds to an event where
the source polarization is coincident with a because the source has fixed
polarization, and we changed a from that polarization.

Making the assumption that MDR_a is aligned with
the source polarization doesn't change the formulation
or affect cd_AB, so long as MDR_b = MDR_a.

We can ascertain for sure in either case whether
p_a = MDR_a and MDR_b = MDR_a. If so, and assuming
independence of A and B, then the product of the individual
probabilities for any given interval matches the qm
prediction for rate of coincidental detection --
that is, you get a maximum visibility Malus' Law
angular dependence for cd_AB.

... for a source with a FIXED polarization (of which we KNOW the polarization
and which is NOT randomly varying), it is clearly not true that on a click,
this means that the source polarization is parallel to the detected polarization.
So if it is not true for a fixed source, why should it suddenly become true for
a random varying source ?

If you assume that MDR_a (for fixed source) is parallel
to the source polarization, then if p_a is set at MDR_a,
then it would mean that on a click the source polarization is
parallel to the detected polarization. For a randomly
varying source any p_a = MDR_a, because dr_a doesn't vary
as you vary p_a.

For setups where dr_a and dr_b vary with the polarizer
setting, the assumption that MDR_a is aligned very close
to the polarization of the incident light seems warranted.

For setups where dr_a and dr_b don't vary with the polarizer
setting, the assumption that MDR_a is aligned very close to
the polarization of the incident light is a bit of a problem,
in that this assumption would seem to imply that the polarizer
won't transmit a wave (or wavetrain or wavepacket) of sufficient
amplitude to trigger a detection if it is offset from the
polarization of the incident light by more than say a few degrees.

So, I guess I'll shelve that assumption for the time
being -- unless you can think of (or think up) a setup
where a photon detector only registers when the analyzing
polarizer is aligned to within a few degrees of the
(known) polarization of the light incident on the
analyzing polarizer.

We did show, however, that in setups where,

dr_a = mdr_a(cos^2 |p_a - MDR_a|), and

dr_b = mdr_b(cos^2 |p_b - MDR_b|), then if

mdr_a = mdr_b = mdr, and MDR_b = MDR_a = p_a, then

(normalizing mdr to 1)

cd_AB = (dr_a)(dr_b) = (cos^2 |p_a - p_b|).

This ports to the Bell type setups
with randomly varying source polarization, and
duplicates the normalized qm prediction for rate
of coincidental detection as the product of
the individual detection probabilities -- which does
not contradict the locality assumption
(causal independence of A and B).

MDR_b = MDR_a = p_a is necessary. This indicates
a common or global polarization parameter. Since
the assumption that this global polarization
parameter is a property of the light incident
on the polarizers isn't contradicted by the
formulation, there's no reason to assume that
it isn't produced at the emitter.

So, do we get any points for locality here,
or what? :)

 

[vanesch]

We did show, however, that in setups where,

dr_a = mdr_a(cos^2 |p_a - MDR_a|), and

dr_b = mdr_b(cos^2 |p_b - MDR_b|), then if

mdr_a = mdr_b = mdr, and MDR_b = MDR_a = p_a, then

(normalizing mdr to 1)

cd_AB = (dr_a)(dr_b) = (cos^2 |p_a - p_b|).

Yes, I agree that in the special case where p_a = MDR_a (that the source is aligned with polarizer A) you can write your formula. I do not agree with your MDR_a = MDR_b formula however, for a randomly oriented source, because you've been cheating: you've redefined MDR_a (which was initially the common polarization of the light) into a parameter of the intensity ; intensity which is constant and hence a parameter which is degenerate. Indeed, for a randomly oriented source, WITHOUT your formulas, I can write: dr_a = 1/2 and dr_b = 1/2 and then I apply cd_AB = dr_a x dr_b = 1/4. What's wrong with THAT then ?

But I'll tell you why there is no reason to assume that the source is aligned with the polarizer if we have a random source. If a polarized source is aligned with the polarizer, then the intensity WITH or WITHOUT the polarizer is the same (so MDR_a is the intensity of the beam, with or without polarizer). If you have a randomly oriented source, then MDR_a is indeed independent of the direction, but only HALF of the intensity with a polarizer).
So it is not that you "cannot distinguish" a randomly polarized beam from one that "follows the orientation p_A = MDR_a": indeed what counts is the ratio of the intensity before and after the polarizer. In the case of a polarized source, this is given by Malus' law, in the case of a randomly oriented source, this is 1/2.

cheers,
Patrick.

 

[Sherlock]

Yes, I agree that in the special case where p_a = MDR_a (that the source is aligned
with polarizer A) you can write your formula.

We don't need to make any assumptions about the source polarization.

We're just counting photons (detections) for specific intervals
and polarizer settings.

The formulation is general. It should apply to any setup where
you're counting photons and you have crossed linear polarizers analyzing
light from a single emitter.

The goal was to see if the qm prediction (wrt Bell-type setups)
can be written as the product of the individual probabilities at the
spacelike separated detectors. Apparently it can. So, this
would seem to support the idea that they are causally independent
wrt each other.

Of course this still doesn't tell us anything specific about the
physical nature of quantum entanglement.

But, the idea that the entanglement is created at the emitter
can't be ruled out on the basis that the only way you can get
the qm prediction in the form of the product of the
individual probabilities is if spacelike separated
events are causally affecting each other superluminally.

That is, you can make the assumption that the observed entanglement
is (at the level of the light) due to a global parameter of the incident
light, and that this parameter is produced at emission, and that
assumption won't alter the results of the formulation.

I do not agree with your MDR_a = MDR_b formula however, for a randomly oriented source, because you've been cheating: you've redefined MDR_a (which was initially the common polarization of the light) into a parameter of the intensity ; intensity which is constant and hence a parameter which is degenerate.

MDR_a hasn't been redefined. MDR_a is the polarizer setting, p_a,
associated with the maximum detection rate, mdr_a,
for a given interval. (Maybe the notation is confusing.
The uppercase MDR_a means a polarizer setting, and the
lower case mdr_a means the photon flux or detection rate
at that setting.)

It means the same thing in either the random or fixed setup.

We found that when dr_a varies as you vary p_a, then the
full (qm) visibility coincidence curve requires
that p_a = MDR_a = MDR_b.

With a random source, dr_a is the same for any p_a.

So, with a random source, any p_a = MDR_a = MDR_b.

Indeed, for a randomly oriented source, WITHOUT your formulas, I can write: dr_a = 1/2 and dr_b = 1/2 and then I apply cd_AB = dr_a x dr_b = 1/4. What's wrong with THAT then ?

It isn't general.

This is:

dr_a = mdr_a(cos^2 |p_a - MDR_a|)

dr_b = mdr_b(cos^2 |p_b - MDR_b|)

If we find that dr_a = mdr_a for any p_a, then
any p_a = MDR_a (and we might infer that the incident
light is randomly polarized at the source -- at least
for the A side).

Now, if dr_a and dr_b are the same for
any p_a and p_b (and especially
when p_a = p_b), then MDR_b = MDR_a, mdr_b = mdr_a,
(then we can infer that not only is the source
random, but also that our setup is ok), then (if dr_a
and dr_b are causally independent of each other)
the rate of coincidental detection,
cd_AB, should vary from mdr --> 0 as,

cd_AB = mdr(cos^2 |p_a - MDR_a|)(cos^2 |p_b - MDR_b|),

which for a random source reduces to,

cd_AB = mdr(cos^2 |p_b - p_a|),

as |p_b - p_a| varies from 0 --> pi/2.

For a setup where dr_a and dr_b vary as you vary
p_a and p_b, respectively, we should find that

cd_AB = mdr(cos^2 |p_b - p_a|)

only when p_a = MDR_a = MDR_b.

But I'll tell you why there is no reason to assume that the source is aligned with the polarizer if we have a random source. If a polarized source is aligned with the polarizer, then the intensity WITH or WITHOUT the polarizer is the same (so MDR_a is the intensity of the beam, with or without polarizer).

We don't need to know the detection rates without the polarizer(s).
The formulation doesn't require it.

If you have a randomly oriented source, then MDR_a is indeed independent of the direction, but only HALF of the intensity with a polarizer).

It doesn't matter what the photon count is sans polarizer(s).

MDR_a is the p_a where dr_a = mdr_a. mdr_a is the maximum dr_a.

We want a generalized formulation for calculating cd_AB as the
product of dr_a and dr_b.

In order to do that we need a generalized formulation for dr_a and dr_b.

We're not referencing the emitter or the light in the
formulation. We're just counting detector registrations wrt
different polarizer settings and (keeping the duration of the
runs at the different polarizer settings constant).

We might infer that a source is random or fixed from the results, and
for convenience we refer to the setups as random or fixed source, but
we don't need to.

So it is not that you "cannot distinguish" a randomly polarized beam from one that "follows the orientation p_A = MDR_a": indeed what counts is the ratio of the intensity before and after the polarizer. In the case of a polarized source, this is given by Malus' law, in the case of a randomly oriented source, this is 1/2.

In the formulation I presented, we can see that the photon count
without polarizers is not necessary.

What counts is the relationship between the photon count associated
with p_a and the photon count associated with p_b.

This way of looking at it has revealed that wrt rate of
coincidental detection, cd_AB, the full visibility Malus' Law angular dependence can be reproduced in a special case of a fixed
source -- which special case corresponds to any and all cases
of a random source.

 

[Sherlock]

Saying things like that the polarization is "irrelevant" for P(A,B) is clearly
wrong as you now see yourself, from the moment that the source is polarized.

It cannot be "irrelevant" for one case, and "of course dependent" in the other,
when the analyzing technique is identical (and only the source changes, which was
exactly what we were trying to analyze).

So clearly P(A,B) cannot be only a function of theta, but must depend also on the
angle between the polarization direction of the source and a and b, in this case.
But according to your claims, it is always the same value, no matter what the
incident polarization, as long as theta is a constant.

We can suppose that the primary rotational plane of
the incident light is aligned with the polarizer setting
that produces the maximum detection rate when the
detection rate varies with the polarizer setting.
If the detection rate is the same when the polarizers
are aligned, then we keep one polarizer fixed at the
max rate setting and offset the other to get coincidental
detection rates at various values for Theta (|p_a - p_b|).
If we offset the fixed polarizer from the max rate
setting so that the rate at the fixed polarizer is
a certain percentage of the max rate (n%mdr) less than
the max rate, then Theta will produce a similar angular
dependence skewed to fit into a range of values for cd_AB from
mdr - n%mdr --> n%mdr.

So, once the setting for the fixed polarizer has been
established, then the only thing determining the rate of
coincidental detection is Theta.

In the setup where detection rate doesn't vary
with the polarizer setting, specific values for
Theta produce the same cd_AB no matter what the
setting of the fixed polarizer. So, is the source
polarization relevant wrt determining cd_AB in this
type of setup?

 

[vanesch]

We can suppose that the primary rotational plane of
the incident light is aligned with the polarizer setting
that produces the maximum detection rate when the
detection rate varies with the polarizer setting.

That is not correct. We can only say so when, upon rotation, the detection rate varies from a maximum to a minimum which is 0. In that case, the incident light is completely polarized (with fixed polarization).

If the detection rate is the same when the polarizers
are aligned, then we keep one polarizer fixed at the
max rate setting and offset the other to get coincidental
detection rates at various values for Theta (|p_a - p_b|).

Again, this is not correct. It is only in the case of entangled photon pairs. In the case of classical, identical light, it is only correct when the light is fully polarized. For instance, it is not correct when you shine unpolarized light onto a beam splitter and look at the two outcoming beams (which DO have identical polarization at each instant of course because being a split beam). When you perform this experiment, you find rates which are INDEPENDENT of theta (as well, independent rates individually, as coincident).

If we offset the fixed polarizer from the max rate
setting so that the rate at the fixed polarizer is
a certain percentage of the max rate (n%mdr) less than
the max rate, then Theta will produce a similar angular
dependence skewed to fit into a range of values for cd_AB from
mdr - n%mdr --> n%mdr.

Again, this is not, in general, correct. Let us take incident light of fixed polarization. When the angle p_b is such that it is perpendicular to this direction, NO counts are seen at B, so no coincidences are observed.

So, once the setting for the fixed polarizer has been
established, then the only thing determining the rate of
coincidental detection is Theta.

Yes, that is true of course, FOR A GIVEN POLARIZATION PROPERTY OF THE SOURCE. If the source is polarized along a certain direction L, and you have fixed p_a, then of course you will find a coincidence rate which is only dependent on theta (because you have FIXED L and p_a).
However, for another value of L, and another value of p_a, you will find ANOTHER FUNCTION of theta.
Also, for a randomly polarized beam (no L anymore) and a given p_a, you will again find another function of theta. This time, however, by symmetry, for a different value of p_a (but the same "kind" of random polarization), you will find the same function of theta. However, that function of theta doesn't have to be the one you found for a polarized source. And it doesn't have to be the same if our random polarization is: 1) random and independent on both sides A and B ; 2) a mixture of identical polarizations at A and B 3) an entangled polarization at A and B.
You have 3 different formula for the 3 different cases.

In the setup where detection rate doesn't vary
with the polarizer setting, specific values for
Theta produce the same cd_AB no matter what the
setting of the fixed polarizer. So, is the source
polarization relevant wrt determining cd_AB in this
type of setup?

It is, in the derivation of cd_AB.

Again, explain me the difference between:

1) the production of entangled states (which do give us a cos^2(p_a - p_b) dependence, but that's a pure quantum prediction

2) a (polarized or unpolarized) classical beam, impinging on a beam splitter with the transmitted beam sent to Alice and the reflected beam sent to Bob.

I don't see, in your approach, how we can arrive at DIFFERENT predictions for the cases 1) and 2).

cheers,
Patrick.