- #151
Sherlock
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- 0
Thanks for the corrections on details, some
comments below:
We wouldn't use a beamsplitter as the source, since we
wouldn't get coincidental detections this way. Would we?
cd_AB only goes to 0 when p_a = MDR_a = MDR_b (with dr_a = dr_b
for corresponding p_a and p_b, and ranging from some maximum
count, mdr_a = mdr_b, to a minimum of 0.
If you offset L from p_a by some angle < pi/2 (and it has
to be < pi/2 because detection at A initiates the coincidence
intervals), then moving p_b through a 90 degree rotation away
from alignment with p_a, then p_b is never perpendicular to L.
So, if L is offset from p_a, then cd_AB never reaches the
max cd_AB (which it would if p_a was aligned
with L), and cd_AB never reaches 0 because when
p_b and p_a are perpendicular there will still be >0 probability
of detection at A and >0 probability of detection at B.
So we can see how the upper and lower limits of the range of
cd_AB are determined by how much L is offset from p_a.
Ok, and there is a case where fixed polarization duplicates
the curve for random (entangled) polarization -- namely, when
p_a = MDR_a = MDR_b where dr_a and dr_b vary from
mdr_a = mdr_b to 0.
But it happens that it is the same function that we get
for a special case of a fixed source, using the
same formulation for fixed and random (entangled) setups.
(Question: since that special case of a fixed source
seems to violate a Bell inequality, and since violation
of Bell inequalities is sort of a rough entanglement
witness, then can we say that in the special case
of fixed source the results are entangled? According
to the usual ways of evaluating entanglement the
answer would seem to be no -- but, according to my,
admittedly hurried, calculation you do get a violation
-- following the CHSH method. I don't know how to think
about this -- maybe I just did it wrong.)
The relationship between L and p_a affects
the range of cd_AB for a fixed source.
But once we have set p_a, then the values
we get within the range determined by |L - p_a|
are completely determined by changes in
|p_b - p_a|.
The *variability* of cd_AB *within a given range*
is independent of the source polarization.
In the random entangled setup the source
polarization wrt a given coincidence interval
is unknown. But the range of cd_AB wrt a set
of runs is constant.
It wouldn't be applied to case 2). Beamsplitters (as the
single emitter between A and B) don't produce coincidental
photon detections (p_a and p_b wouldn't be analyzing the same
thing during a coincidence interval). The approach applies to setups
where crossed linear polarizers are analyzing light from a single
emitter -- setups where it might be said that p_a and p_b are analyzing
related or the same rotational properties of the incident light during
a given coincidence interval.
Anyway, I agree with you that modeling randomly
polarized entangled light in terms of a common source polarization,
L, seems impossible using extant geometric models of the emitted
light. (I've got some ideas on how this might be done without
the usual contradictions, but must take some time to explore them.)
But, have we not shown that (1) the cd_AB for the entangled
setup can be put into the form of the product of the
individual probabilities at A and B, corresponding to a
special case of the fixed setup, (2) that the global
variable affecting *range* of coincidental detection is
|Lambda - p_a|, and (3) that the global variable affecting
*rate* of coincidental detection within a given range is
|p_a - p_b|? (Note that p_a is the convention for the
polariizer setting at the end that initiates any, ie. all,
coincidence interval(s).)
There's nothing new here, just laying some groundwork.
And, thanks again for taking the time to talk through some
of this stuff. After reading up a bit on entanglement I'm
quite sure that I understand it much less than I thought
I did when I first jumped into this thread. :)
comments below:
vanesch said:
We wouldn't use a beamsplitter as the source, since we
wouldn't get coincidental detections this way. Would we?
vanesch said:
cd_AB only goes to 0 when p_a = MDR_a = MDR_b (with dr_a = dr_b
for corresponding p_a and p_b, and ranging from some maximum
count, mdr_a = mdr_b, to a minimum of 0.
If you offset L from p_a by some angle < pi/2 (and it has
to be < pi/2 because detection at A initiates the coincidence
intervals), then moving p_b through a 90 degree rotation away
from alignment with p_a, then p_b is never perpendicular to L.
So, if L is offset from p_a, then cd_AB never reaches the
max cd_AB (which it would if p_a was aligned
with L), and cd_AB never reaches 0 because when
p_b and p_a are perpendicular there will still be >0 probability
of detection at A and >0 probability of detection at B.
So we can see how the upper and lower limits of the range of
cd_AB are determined by how much L is offset from p_a.
vanesch said:
Ok, and there is a case where fixed polarization duplicates
the curve for random (entangled) polarization -- namely, when
p_a = MDR_a = MDR_b where dr_a and dr_b vary from
mdr_a = mdr_b to 0.
vanesch said:
But it happens that it is the same function that we get
for a special case of a fixed source, using the
same formulation for fixed and random (entangled) setups.
(Question: since that special case of a fixed source
seems to violate a Bell inequality, and since violation
of Bell inequalities is sort of a rough entanglement
witness, then can we say that in the special case
of fixed source the results are entangled? According
to the usual ways of evaluating entanglement the
answer would seem to be no -- but, according to my,
admittedly hurried, calculation you do get a violation
-- following the CHSH method. I don't know how to think
about this -- maybe I just did it wrong.)
vanesch said:
The relationship between L and p_a affects
the range of cd_AB for a fixed source.
But once we have set p_a, then the values
we get within the range determined by |L - p_a|
are completely determined by changes in
|p_b - p_a|.
The *variability* of cd_AB *within a given range*
is independent of the source polarization.
In the random entangled setup the source
polarization wrt a given coincidence interval
is unknown. But the range of cd_AB wrt a set
of runs is constant.
vanesch said:
It wouldn't be applied to case 2). Beamsplitters (as the
single emitter between A and B) don't produce coincidental
photon detections (p_a and p_b wouldn't be analyzing the same
thing during a coincidence interval). The approach applies to setups
where crossed linear polarizers are analyzing light from a single
emitter -- setups where it might be said that p_a and p_b are analyzing
related or the same rotational properties of the incident light during
a given coincidence interval.
Anyway, I agree with you that modeling randomly
polarized entangled light in terms of a common source polarization,
L, seems impossible using extant geometric models of the emitted
light. (I've got some ideas on how this might be done without
the usual contradictions, but must take some time to explore them.)
But, have we not shown that (1) the cd_AB for the entangled
setup can be put into the form of the product of the
individual probabilities at A and B, corresponding to a
special case of the fixed setup, (2) that the global
variable affecting *range* of coincidental detection is
|Lambda - p_a|, and (3) that the global variable affecting
*rate* of coincidental detection within a given range is
|p_a - p_b|? (Note that p_a is the convention for the
polariizer setting at the end that initiates any, ie. all,
coincidence interval(s).)
There's nothing new here, just laying some groundwork.
And, thanks again for taking the time to talk through some
of this stuff. After reading up a bit on entanglement I'm
quite sure that I understand it much less than I thought
I did when I first jumped into this thread. :)
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