PeterDonis said:
The "event ready" signal is generated by a combination of things: the 2 & 3 photons arriving at the BSM device within the same narrow time window, and the output of the BSM indicating the particular Bell state that the BSM is set up to detect. This happens at the BSM, not at the initial preparation.
@DrChinese is more familiar than I am with the specific papers describing the experiments, and I'm sure can give specific references to the descriptions in those papers that match the above.
Most folks probably don't know some of the details in these experiments, and most of the time they don't matter. Everyone gets the main idea. But as
@Nullstein shows us, the devil is sometimes in the details. So allow me to correct a few notions. What I am about to say below is the theoretical case of the perfect experiment, but swapping experiments are quite difficult to execute in practice so they won't work quite like I describe.
Imagine our same scenario, starting off with distant independent sources A and B generating entangled pairs (1 & 2) and (3 & 4) from Type I PDC (both H or both V). These pairs are produced spontaneously and asynchronously, no particular time of creation from either source. The initial quantum state is a Product State of 2 entangled pairs:
##\hat{\rho}=\hat{\rho}_{12} \otimes \hat{\rho}_{34}.## That of course is ONLY true when we have one pair from Source A and one pair from Source B that we choose to discuss as a candidate for a swap. We need to connect two pairs to have a set of 4 to make any sense of the pairs we start with.
What connects these pairs? Is it time of emission? No. Is it time of arrival of the 1 & 4 photons? No.
What connects them is the time the 2 & 3 photons cross the Beam Splitter (BS) during the Bell State Measurement (BSM). They must cross within a narrow time window, let's call it 10 nanoseconds to have a number. And they must be indistinguishable as to whether the source was A or B once they pass the BS. So same wavelength, etc. and no other identifying characteristics (cannot be polarized). This is how the pairs from different sources interact, the creation time of the pairs itself is not a factor.
When they come out of the BS they are routed to 2 Polarizing Beam Splitters (PBS) and then to an array of 4 detectors. Some Bell States can be distinguished by this method, but unfortunately not all can. There will always be a percentage of cases that cannot be used for a Bell test on 1 & 4 pairs. However, that does NOT mean they aren't entangled. They are, we just cannot determine how.
For our purposes, let's assume that the 1 & 4 measurement systems (PBS plus 2 detectors for each) are positioned such that when we have a successful BSM, the 1 & 4 detectors go off at the same time (within our designated time window). We add fiber cable to make that work out, and we place them at the same spot. Additionally, we do the same with the BSM detector array. We use fiber to adjust the travel time, and route them to the same location as the 1 & 4 detectors.
All 4 photons will arrive at location where all of the detectors are, and the photons will all arrive within the same time window. I am adding this little twist so you can see exactly what should be discussed when we talk about an ensemble or subset or subensemble. So what we expect, with a successful BSM, is that 4 detectors will click at almost precisely the same time. For the 1 & 4 photons, each will generate one click indicating their polarization relative to their respective PBS. The BSM detector array will register 2 clicks, one for the 2 photon and one for the 3 photon - but we won't know which is which.
So 4 "simultaneous" clicks means we have a successful BSM in this setup.
Here is where the labeling of ensembles gets confusing. Most everyone assumes we are ignoring numerous sets of clicks. That's not really accurate. We often get a click on the photon 1 detectors and a single click on the BSM detector array, but nothing in the time window for the photon 4 detectors. And we often get a click on the photon 4 detectors and a single click on the BSM detector array, but nothing in the time window for the photon 1 detectors. Since there is no 1 click to match with a 4 click, there is no way to make any sense of the earlier statement ##\hat{\rho}=\hat{\rho}_{12} \otimes \hat{\rho}_{34}.## For me, only cases in which the 1 & 4 photons are detected within the time window should be considered.
I would call that the full universe or data set. Note that this definition does not reference whether or not the BSM succeeded.
In theory:
for every single case where the 1 & 4 photons are detected within the time window: they are entangled. We may not know which of the 4 Bell states they are in - and therefore we can't perform a Bell test on them - but they ARE entangled. The new state is ##\hat{\rho}=\hat{\rho}_{14} \otimes \hat{\rho}_{23}.## If the experimental setup is perfect, then every single time we have clicks for 1 and 4, we will also have 2 clicks at the BSM array. So once again, every 1 & 4 pair is entangled. Of course, that wouldn't be true if we messed up the BSM setup so that the source was distinguishable (we could do that if we wanted). But assuming we are good scientists, there is no subensemble yet.
The subensemble comes when we ask what kind of entanglement did we get? Each of the 4 Bell states is equally likely, but usually only 2 can be identified. So half the results are thrown out, but of course they were still entangled. So when you talk about the subensemble, it is something quite different than many here are picturing. To me, the subensemble is the group for which we know the specific Bell state. The full data set are the cases in which we have matched clicks for 1 & 4. And in the ideal case, they are all entangled and there are also clicks at the BSM array.
