# Bell Test statistics

I have some questions about the statistics of Bell Tests. For convenience I'm going to refer to the test described in this paper:

http://arxiv.org/abs/quant-ph/0205171

They show a coincidence rate of 300 counts with polarizers aligned. I'm wondering how many counts there would be if you took out both polarizers altogether, and how many of those would be coincidence counts. I'm interested in the ideal case of 100% detector efficiency and 100% downcoversion efficiency, but I also wonder how close the ideal case can be approached in practise.

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Your question looks like a sound one, but you will probably get better answers if you can ask it in a self-contained way. That way people don't have to try to extract the relevant bits of a paper before they understand what you're talking about. – Mark Eichenlaub Apr 24 '11 at 20:51
Another note: we like links to papers. Especially papers (like those on arXiv) that don't reside behind a paywall. But please point to the abstract page rather than a binary. – dmckee Apr 24 '11 at 20:54
Oops. I hope this is how you want the link:arxiv.org/abs/quant-ph/0205171 I'd like to ask the question in a self-contained way but I don't know exactly how to extract the relevant bits, so I thought I'd take this experiment as a prototype. For what it's worth, I think it's pretty nicely written up. – Marty Green Apr 24 '11 at 21:08
Just a note - I'm interested in a write-up similar to this but with a completely specified list of lab items necessary to recreate as a pedagogical tool for illustrating QM foundations. I mean with brands and part numbers/names for the required polarizers, etc. Please msg me if you know any good links. – BjornW Apr 24 '11 at 21:46
@Bjorn Wesen: This is similar Observing the quantum behavior of light in an undergraduate laboratory and it has a PARTS LIST AND COST OF EXPERIMENT – Helder Velez Aug 16 '11 at 16:35

I am not sure if the answer is still wanted...

Under your assumptions hypothesis 2 is correct. The state of photon pair in this case is $\left|\psi\right>=\frac{1}{\sqrt{2}}\left(\left|H_1H_2\right>+\left|V_1V_2\right>\right)$ The probability of a coincidence to be detected with polarizers oriented, say vertically, is $P_{VV}=\left|\left<V_1V_2|\psi\right>\right|^2=1/2$. This means, that half of the pairs give rise to coincidence counts, so the total pair production rate is 600 pairs/s. This may be understood like this: if the vertical polarizer is put in one channel, the presence of similarly oriented polarizer in the second channel will not reduce the number of coincidences since the photons are perfectly correalted in polarization, while of course it will cut off half of the single counts. The feature of this particular Bell state is that this is true for any orientation of polarizers, if both are rotated at the same angle.

The first hypothesis will be true, for example, for detection of pairs in completely unpolarized state described by the density matrix $\rho=\frac{1}{4}\left(\left|H_1\right>\left<H_1\right|+\left|V_1\right>\left<V_1\right|\right)\otimes\left(\left|H_2\right>\left<H_2\right|+\left|V_2\right>\left<V_2\right|\right)$, where only quarter of pairs will produce coincidences in the described situation. Locally, i.e. using data from a single detector only, these states are indistinguishable, since the reduced single photon density matrices are identical. However, the Bell state is polarized (in fourth-order in the field), which is revealed in intensity correlation or coincidence measurements. This is an example of what is sometimes called "hidden polarization".

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Thanks, Straups, for the delayed answer. I was going somewhere with this a year ago but I'm damned if I can remember where. – Marty Green Jul 15 '12 at 5:05

This was the first question I posted on StackExchange and nobody answered it so I will have to try. Looking at the ideal case, there are two plausible answers: 1200 or 600. Assume that the source produces only entangled photons and the detectors are 100% efficient. Without polarizers, every detection is a coincidence. Given that there are 300 coincidences per second with two aligned polarizers, how many are there with no polarizers?

Hypothesis 1: there are 1200 coincidences per second. Put a polarizer on one side and only half the photons get through. The number of coincidences goes down to 600. Put a polarizer on the other side: half the photons get through and the number of coincidences goes down to 300, as observed.

Hypothesis 2: there are 600 coincidences per second. Put polarizers on both sides. Since the polarizers are aligned, they knock out photons always in pairs. Since they each knock out half the photons, there are 300 coincidences per second at the detectors.

I cannot easily choose between these two hypotheses.

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