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DrChinese
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Perhaps I can help, at least discussing the cases of whether the 1&4 or 2&3 photons interact in any way. This is the 2012 paper by Zeilinger's team, in which the 2 & 3 interaction variable is the primary objective of the paper - and no interaction between 1 & 4.javisot20 said:
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Re: 1&4 photons interacting:
In no cases do the 1 & 4 photons ever interact. They are created at separate locations (different PDC crystals) about .5 meter apart and fed into fiber, and measured about 35ns later. There is no time or place for them to interact. From Figure 2 of the paper:
"Photons 1 and 4 are directly subject to the polarization measurements performed by Alice and Bob (green blocks".
In other experiments, the separation of photons 1 and 4 are more clearly delineated than in this particular experiment. But obviously: If the independent variable is what happens at the BSM, then any hypothetical interaction can't really matter to our conclusion.
Re: 2&3 photons interacting:
1. To execute a swap via the BSM, the 2 & 3 photons must arrive at the beam splitter within a narrow time window. The same time window is applied whether an Entangled State (ES) measurement is to result, or a Separable (non-entangled) State (SS) measurement is to occur. The decision to make it ES or SS is made randomly by automation on a case by case basis. The time window is measured by clicks at the BSM in 2 detectors.
2. There are 4 possible Bell states that result from a entanglement swap. For a variety of mostly technical reasons, only a single state is reported in the experiment. That is the |φ-> state. That state is indicated when the BSM registers either two H clicks or two V clicks at the BSM. Note that to get the two clicks |HH> or |VV>, that can result only from the 2 & 3 photons being both reflected or both transmitted at the beam splitter (BS). The only entangled stats being reported are from this one Bell state for ES scenario. No other Bell states are being combined with the entangled |Φ-> state numbers. Similarly, the SS scenario also looks at |HH> or |VV> results at the BSM. So the statistics are "apples to apples". The key here is that we are going to compare the ES and SS (entangled vs non-entangled) correlations. From the paper:
"After all the data had been taken, we calculated the polarization correlation function of photons 1 and 4. It is derived from their coincidence counts of photons 1 and 4 conditional on projecting photons 2 and 3 to |Φ−〉23 = (|𝐻𝐻〉23 − |𝑉𝑉〉23)/√2 when the Bell-state measurement was performed, and to |𝐻𝐻〉23 or |𝑉𝑉〉23 when the separable state measurement was performed."
3. Here is exactly how the physical variable changes that creates the ES (entangled) or SS (non-entangled) results: There are 2 Electro-Optical Modulator (EOM 1 and EOM 2, see figure 2) that together change the beam splitter between 2 possible configurations. To get ES outcomes, the beam splitter operates in a 50:50 mode - that is, 50% transmitted and 50% reflected. To get SS outcomes, the beam splitter operates in a 0:100 mode - that is, 0% transmitted and 100% reflected (i.e. a mirror).
There IS entanglement when the 2 & 3 photons can overlap in the beam splitter within the time window, but you cannot know whether both photons were transmitted - or both were reflected (50:50). I.e. the 2 & 3 photons are indistinguishable. There is NO entanglement when the 2 & 3 photons cannot overlap in the beam splitter within the time window, because both were reflected (0:100) before they could possibly cross. I.e. now the 2 & 3 photons are easily distinguishable according to which side's detectors click.
This is the only difference between the statistics reported in Figure 3 between the a) side [left 3 bars] and the b) side [right 3 bars]. The easiest to see is the middle of the 3 bars on each side. These are the associated click outcomes for the |RR>/|LL> correlations at the Alice and Bob stations. The middle bars have the Alice and Bob stations only measuring circular polarization R or L. When there is |Φ-> entanglement, Alice and Bob should get identical results on any same basis, so you would expect mostly RR> or LL> outcomes and few LR> or RL> outcomes. Without entanglement, Alice and Bob should not see any correlation. Note that correlation is calculated as C=(Matches - Mismatches)/(Matches + Mismatches) and can vary from 1 to -1.
When the Entangled State was selected randomly, you expect significant correlation (theoretically perfect would be C=1.0). When the Separable State was selected, you expect no significant correlation (theoretically perfect would be C=0.0).
The actual entangled (ES) correlation is about 0.603+/-0.071, while the actual separable (SS) correlation is about 0.010+/-0.072. These results are clearly saying: If a physical change is made at the BSM, then there is a corresponding observable change of the overall statistics - as predicted by QM.
So the answer is: When there is an entanglement swap, the 2&3 photons are allowed to physically interact (but they aren't if a separable state is to be generated). The only thing varied in the results a) vs b) is that setup at the BSM. So the difference in statistics, according to norms in experimental science, is the independent variable. Which is selected and chosen well after the Alice and Bob perform their measurements on photons 1 & 4.
You can make of this what you like.
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