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I'd like to discuss 3 different examples of photon polarization correlations where we have 2 different, independent sources S1 and S2, each sufficiently distant to each other that the sources themselves are not directly responsible for the correlations.
I. 2 lasers S1 and S2 both have V polarizing filters over their output streams, which are not entangled in any way. We will label the stream of photons from S1 as being P1, and the stream from S2 as being P4 (the reason for labeling like this should become clear).
a. We measure the V polarization of P1 and P4, and they are each vertically polarized of course: |VV>. The correlation of P1 & P4 is 1.00 in principle.
b. Question: What is the correlation of the polarizations of P1 & P4 at other angles?
c. Answer: on same mutually biased bases to V, they will have various correlations between 0 and 1.
II. 2 lasers each drive a PDC crystal S1, S2 producing the entangled Bell state |VV>+|HH>. We will label the stream of photon pairs from S1 as being P1&P2, and will label the stream of photon pairs from S2 as being P3&P4. P1&P2 are entangled, P3&P4 are entangled, but there is nothing in particular that connects P1&P2 to P3 or P4, and vice versa. No swapping occurs.
a. We measure the polarization of P2 and P3, and it happens they are each vertically polarized: |VV>. The correlation of P1 & P4 is 1.00 in principle.
b. Question: What is the correlation of the polarizations of P1 & P4 at other angles?
c. Answer: on same mutually biased bases to V, they will have various correlations between 0 and 1.
III. Same setup as example II, but there is swapping of entanglement via a Bell State Measurement (BSM) on P2 and P3 - this done as a delayed choice by the experimenter. After the swap, P1 & P4 are entangled in a Bell state as revealed by the BSM.
a. We measure the polarization of P2 and P3, and it happens they are each vertically polarized: |VV>. The correlation of P1 & P4 is 1.00 in principle.
Note that although we know both are V, we cannot distinguish between them as to which is P2 and which is P3. This is required for a successful swap.
b. Question: What is the correlation of the polarizations of P1 & P4 at other angles?
c. Answer: on ALL SAME BASES, the correlation of P1 & P4 is 1.00 in principle. Perfect correlations!
IV. But what if it becomes possible to distinguish the P2 and P3 photons, perhaps by some kind of intentional tagging by the experimenter? In this case, there is no swap.
b. Question: What is the correlation of the polarizations of P1 & P4 at other angles?
c. Answer: on same mutually biased bases to V, they will have various correlations between 0 and 1.
The statistics look exactly the same as Example II. This
It should be clear that there can be correlations of varying types according to which type (per the examples) of streams we are comparing. But only 1 of the 4 examples above produces "perfect" correlations. Of course, real-world correlations are not "perfect". And even though in the ideal case, QM does make an exact (not statistical) predictions for Example III: you need to accumulate a sufficiently large dataset to be sure of what you've got. This is as @PeterDonis has correctly pointed out on a number of occasions.
Assuming you follow the above examples:
i) We know entanglement swaps (via a BSM on P2 & P3) can be executed after P1 and P4 cease to exist. (This is the delayed choice scenario.)
ii) We know entanglement swaps can be performed such that P1 & P4 have never interacted, and have never interacted with any common 3rd system in the past. (This is the strict locality test.)
iii) We know that the entanglement swap fails if there is distinguishability of the P2 & P3 photons. (As demonstrated in various experimental realizations such as THIS).
iv) We know Einsteinian causality should not allow a later choice (to execute a swap or not) to create correlations after the fact.
Something's gotta give! What gives? That's my question.
I. 2 lasers S1 and S2 both have V polarizing filters over their output streams, which are not entangled in any way. We will label the stream of photons from S1 as being P1, and the stream from S2 as being P4 (the reason for labeling like this should become clear).
a. We measure the V polarization of P1 and P4, and they are each vertically polarized of course: |VV>. The correlation of P1 & P4 is 1.00 in principle.
b. Question: What is the correlation of the polarizations of P1 & P4 at other angles?
c. Answer: on same mutually biased bases to V, they will have various correlations between 0 and 1.
II. 2 lasers each drive a PDC crystal S1, S2 producing the entangled Bell state |VV>+|HH>. We will label the stream of photon pairs from S1 as being P1&P2, and will label the stream of photon pairs from S2 as being P3&P4. P1&P2 are entangled, P3&P4 are entangled, but there is nothing in particular that connects P1&P2 to P3 or P4, and vice versa. No swapping occurs.
a. We measure the polarization of P2 and P3, and it happens they are each vertically polarized: |VV>. The correlation of P1 & P4 is 1.00 in principle.
b. Question: What is the correlation of the polarizations of P1 & P4 at other angles?
c. Answer: on same mutually biased bases to V, they will have various correlations between 0 and 1.
III. Same setup as example II, but there is swapping of entanglement via a Bell State Measurement (BSM) on P2 and P3 - this done as a delayed choice by the experimenter. After the swap, P1 & P4 are entangled in a Bell state as revealed by the BSM.
a. We measure the polarization of P2 and P3, and it happens they are each vertically polarized: |VV>. The correlation of P1 & P4 is 1.00 in principle.
Note that although we know both are V, we cannot distinguish between them as to which is P2 and which is P3. This is required for a successful swap.
b. Question: What is the correlation of the polarizations of P1 & P4 at other angles?
c. Answer: on ALL SAME BASES, the correlation of P1 & P4 is 1.00 in principle. Perfect correlations!
IV. But what if it becomes possible to distinguish the P2 and P3 photons, perhaps by some kind of intentional tagging by the experimenter? In this case, there is no swap.
b. Question: What is the correlation of the polarizations of P1 & P4 at other angles?
c. Answer: on same mutually biased bases to V, they will have various correlations between 0 and 1.
The statistics look exactly the same as Example II. This
It should be clear that there can be correlations of varying types according to which type (per the examples) of streams we are comparing. But only 1 of the 4 examples above produces "perfect" correlations. Of course, real-world correlations are not "perfect". And even though in the ideal case, QM does make an exact (not statistical) predictions for Example III: you need to accumulate a sufficiently large dataset to be sure of what you've got. This is as @PeterDonis has correctly pointed out on a number of occasions.
Assuming you follow the above examples:
i) We know entanglement swaps (via a BSM on P2 & P3) can be executed after P1 and P4 cease to exist. (This is the delayed choice scenario.)
ii) We know entanglement swaps can be performed such that P1 & P4 have never interacted, and have never interacted with any common 3rd system in the past. (This is the strict locality test.)
iii) We know that the entanglement swap fails if there is distinguishability of the P2 & P3 photons. (As demonstrated in various experimental realizations such as THIS).
iv) We know Einsteinian causality should not allow a later choice (to execute a swap or not) to create correlations after the fact.
Something's gotta give! What gives? That's my question.