If you understood this, and found the answer, you will have gained a great insight in quantum theory in general, and in quantum optics in particular :-)
Not that I expected very much, but would that be all the gratitude I get for escorting you out of the "QM measurement" darkness?
Why does this only work if the incoming states on the beam splitter are 1-photon states ?
There are two basic misconception built into this question, one betrayed by "this" another one by "only". The two tie the knot of your tangle.
Your "this" blends together the results of actual observation (the actual counts and their correlation, call them
O-results[/color]) with the "results" of the abstract observable C (
C-results[/color]). To free you from the tangle, we'll need finer res conceptual and logical lenses.
The C-results are not same as O-results. There is nothing in the abstract QM postulates that tells you
what kind of setup implements C[/color] or, for a given setup, what kind of
post-processing of O-results yields C-results[/color]. The postulates just tell you C exists and it can be implemented. But to implement it, to perform operational mapping between formal C and experiment, you need a more detailed physical model of the setup, where at least part of the 'aparatus' interacting with the 'object' is treated as a physical interaction. In our case one needs QED applied to detectors, such as treatments in [4] or [9].
The first observation of such dynamical analysis is that the "trigger of DT" involves making a
decision how to define "trigger" and "no-trigger" O-results[/color] (which we can then use to define C-result). The number of photo-electrons ejected will have Poissonian distribution i.e. the (amplified) photo-current corresponding to r photo-electrons with probability p(r,n) = exp(-n) n^r/r!, where n=<r> is the average p-e count (and also a variance). This is
the most ideal case, the sharpest p(r,n) you can get[/color] (provided you have perfectly reproducable source pulses and precise enough detection windows so that the incident field intensities I(t) are absolutely identical between the tries). Note that EM pulse need not be constant in the window, only the integral of I(t) must be constant for the window to obtain the "ideal" p-e distribution sharpnes p(r,n). If there is any EM amplitude variation between the tries, the p(r,n) will be compounded (smeared or super-) Poissonian which has variance larger than n.
A common sleight of hand in pedagogical QO treatments (initiated by Purcell during the HBT effect controversy, the QO birthing crisis, in 1950s, later elaborated by Mandel and refined into a work of art by Glauber) is to point out one example which provides such nearly perfectly reproducable incident EM fields, the perfectly stable laser light (coherent light), and note that the (single mode) photon number observable [n]=[a+][a] of such source has also the Poissonian distribution of photons. From that, the pedagogues leap to the "conclusion" that O-results r are interchangeable with the [n]-results, the values of the observable [n] (the photon number) i.e. as if measurements of r is an implementation of the observable [n]. From this "conclusion" they then "deduce" that if we can produce Fock state as the incident EM field, thus have a sharp value for observable [n], we will have a sharp value for r. Nothing of the sort follows from the QED model of detection. The association between the [n]'s [n]-values and the measured O-values r is always statistical (the EM intensity 'I' fixes <r>=<r(I)> and its moments) and the sharpest association between the [n]-values and the O-values one can have is Poissonian.
The average r (parameter n), is a function of the incident light intensity I and of the settings on the "detectors" (bias voltage, temperature, amplifier gain, pulse anlyser, window sizes, etc.). Assuming we keep detector & window parameters fixed, <r>=n will be a function of the incident light intensity I
only, i.e. n=n(I).
The key observation about this function n(I) from the QED detection model [4] is that, for a given detector settings, the
n(I), thus the p(r,n(I)), is determined solely by the EM fields reaching the detector within the detection window[/color]. In particular, [4] being a relativistic model, given the incident fields, there are no effects on p(r,n) from any interactions occurring at the spacelike distances from the detector during the detection window.
Following the common convention, we can define O-result "no-trigger" to correspond to r=0 photo-electrons and "trigger" to correspond to r>0 photo-electrons (we're idealising here by assuming perfect amplification of the ejected photo-electrons into the measured currents). We'll define q = p(0) = p(0,n) = exp(-n) and p = p(1) = 1-q.
To obtain the operational interpretation of Glauber's [G1] observable (his single 'detection' rate observable, [G1(x,t)] = [E-][E+], where [E]=[E+]+[E-] is electric field operator [E] decomposition to positive & negative frequency parts [E+] (annihilator) and [E-] (creator)), we need another result of the dynamical analysis (cf. [4] 78-84). The desired behavior of [G1] is that it has <0|[G1]|0>=0, i.e. [G1]-value is 0 when no incident EM field interacts with the detector. Thus we want [G1] to count photo-absorptions of the incident field only. The dynamics for the detection, unfortunately yields only, and at best, the Poissonian r-counts. That means we will have O-triggers with no incident light (corresponding to vacuum rate n0=n(I0), I0 from hv/2 vacuum energy per mode) and absent O-triggers when the incident light is present (since p(0,n)>0).
The Glauber's ideal [G1]=[E-][E+] operationally corresponds to filtering out both types of r-results 'we are not interested in'. While detector designs (including pulse analyser & discriminator/PAD) perform this subtraction atomatically, they cannot compensate for the 'failed triggers'. To account for the failed triggers, detectors have a parameter Quantum Efficiency QE which is obtained (calibrated) as a ratio of vacuum filtered trigger rate and the average photon rate of the incident field. Thus, knowing the measured trigger rate R(I) and R0 (dark rate, the adjustable leftover from the built in vacuum subtractions), one can compute the average 'photon number' rate PN(I)=<[G1]>=<G> of the incident field as PN(I) = (R(I)-R0)/QE (cf. eq (2) [10]).
This relation among averages does not get around the Poissonian spread p(r,n) for the r-counts, thus of the dark triggers p(r>0,n0) and the missed triggers p(r=0,n>n0). Namely, even if the EM field has a perfectly sharp incident photon number within the detection window (as we have approximately in PDC on TR photon), the r-counts still have at best the Poissonian distribution, thus the variance of at least n. This implies a tradeoff between the TE (trigger efficiency, TE=R(I)/PN(I), which is different than QE=(R(I)-R0)/PN(I)) and the 'false' triggers for the r-counts, no matter what n we select or how we adjust n0 of our detector (n0=<r> for no incident field). Defining the average of r-count for incident field 'alone' as nf=<r>-n0, for given sharp [n] incident field we can maintain the fixed nf. We can still adjust detector sensitivity by tuning n0, thus adjust n=nf+n0, which adjusts the loss rate as LR=p(r=0,n)=exp(-(nf+n0)) and the false trigger rate FT=p(r>0,n0)=1-exp(-n0). If we reduce losses LR->0, then we need n0->inf, which causes FT->1, thus making nearly all triggers false. If we reduce false triggers via n0->0, then we are maximizing the loss rate to exp(-nf).
In particular, for single (on avg) mode absorption per window, nf=1, and reducing the false triggers to 0, will yield the loss rate (per window) at least LR=exp(-nf)=1/e=36.79%, which is well above the max loss rate for an absolutely loophole free Bell's test of (1-0.83)=17%. But, to avoid only the natural semiclassical models, the tests require a less demanding than 83% (limit for any conceivable local model) efficiency. They require at least 2/Pi=63.66% trigger efficiency i.e. the max loss allowed to eliminate natural classical models is LR=1-2/Pi=36.34%, which is almost there, yet it is 0.45% below the unavoidable (when false triggers are minimized) p-e Poissonian loss of 36.79%. Thus any optical Bell test will fail, falling short of eliminating the natural classical models by mere .45%, precisely because of the dynamically deduced statistical association between the r-counts (the O-triggers) and the photon numbers of the incident field (which [G1] counts via photon absorption counts).
As a mnemonic device, one can think of [G1] as corresponding to r-counts on 1-by-1 basis instead as a relation among averages (and moments) of the two distributions. For coherent or chatoic states this causes no problem, since averages and moments agree. But for the Fock state |1> (or similarly any Glauber "non-classical" states), [G1] has sharp [G1]-values 1, while the r-counts remain Poissonan with average 1 (which requires the lost counts to be at least 1/e=36.79%). As cited earlier [11], one can introduces a different kind of (
nonlinear[/color]) annihiliation operator E_ which does maintain consistency between the distributions of r-counts and these 'new-photo-counts', and consequently the result in [11] for the Fock state is also the Poissonian 'new-photo-count' distribution. The regular annihilator shows other strange properties, as well, if one takes it literally as 'photon absorption' operators [12], such as increasing the number of field quanta for super-Poissonian states (and even more so than the creation operator [a+] for some states!).
The operational meaning of Glauber's 2 point "correlation" G2(x1,x2)=<[G2]> where [G2]=[E1-][E2-][E2+][E1+] and its "non-locality" (convention) has been discussed at length already. Here I will only add that the same Poissonian r-counts limitations and the caveats apply when heuristically identifying, on 1-by-1 basis, the r-counts coincidences as the observable [G2]-values. The association is still statistical (in the sense of being able to map only the averages & the moments between the two). Additional important caveat here is that [G2] implementation requires non-local operations to subtract the rates of accidentals and unpaired singles (or losses from p(r=0,n>n0)), which now occur at different locations, thus any "non-locality" one deduces from it is just a matter definitions, not anything genuinly non-local (since one can graft the same non-local subtraction conventions to the semiclassical models and make them Glauber "non-local", too).
Before constructing operational rules for your C observable on the |Psi.1>, we'll look at the actual observation results. The O-results of the superposed state will be (T,R) pairs: (0,0), (0,1), (1,0), (1,1). If the average r on DT for the single state |T> is <r>=n, then the <r> for the superposed state |Psi> = (|T>+|R>)/s will be n/2 for individual DT and DR "trigger" probabilities (per window or per try).
Assuming DT and DR are at spacelike distance during the detection windows (defined via DG events) and that the PDC pump is stable enough so that within the sampling windows we get repeatably the 'same' EM fields, (i.e. at least the same Integral{I(t)*dt}) on DT and on DR in each window (so we can get the sharpest possible distribution of the p-e r-counts predicted by the QED model), and that the light intensity is low enough so that we can ignore dead time on the detectors, the probabilities of
the four kinds of O-triggers are simple products p00=q^2, p01=p10=pq and p11=p^2[/color]. In short, whatever interactions are going on at the spacelike location DR, it has no effect on the evolution of the fields and their interactions at the DT location, thus no effect on the probabilities of the r-counts on DT. This is, of course, the same result that the finite detectors and finite R & T fields model predicts, as described [post=538215]earlier[/post].
Now, finally, your observable C. Nothing in the QM axioms specifes or limits how the C must be computed from the r-counts, and certainly does not require that computed C-values are the same as O-values (the observed r-counts) on 1-by-1 basis. QM only says such C exists and it can be mapped to the experimental data. The fact that C "predicts" indeed the avg r-counts <r> for setups with "mirror" or with "transparent" have no implication for the setup with PBS. Similarly, the fact that via (AJP.9) one could write down your C in a concise form, implies nothing regarding the operational mapping of C to our setup, and implies nothing about the O-values for the setup (since AJP.9 plane wave operators don't describe DT and DR setup with the |Psi.1> incident fields). On the other hand, the Glauber's model [4], augmented with the finite detectors & EM volumes does predict proper O-values p00...p11, and does provide a simple operational mapping for your C.
Note first that the r-counts for the actual finite DT and DR are not exclusive, whether |Psi> is a single photon or multi-photon or partial photon state (no sharp [n]). Thus your requirement of exclusivity "only for single photon state" is an additional ad hoc requirement, an extra control variable for the C-observable mapping algorithm, instructing it to handle the case of EM fields with <[n]>=N=1 differently than case N != 1 (N need not be an integer). Nothing in the r-counts, though, is different in any drastic way (other than the difference in n=<r> used in p(r,n)).
Note that your C is not sensitive to PBS split ratio i.e. since |T> and |R> are basis in the 0 eigenspace of C, any Psi(a,b) = a|T> + b|R>, will yield C=0, which simplifies the C mapping algorithm since it doesn't have to care about the values of a and b, but at most it needs to take a note that a beam splitter is there so it can enforce C=0. As luck would have it, though, from the r-count probabilities p00...p11, when either a->0 or b->0, the proportion of (1,1) cases automatically converges to 0 (or background accidentals, globally discarded for C, same as for [G2]), which were the cases of the "mirror" or "transparent" setups, thus no special C-algorithm adjustments are needed for all 3 of your setups.
For N=1, one could thus simply compute C-values by treating the (1,1) O-values as (0,0) O-values, discarding them (it discards double O-triggers the same way that the triple and quadruple O-triggers are discarded in Bell tests, i.e. by definition, cf.[post=531880]Ou, Mandel[/post] [5]). Note that the fact that |a|^2+|b|^2=1 for C, has no operational mapping implications on the variability of the number of 'C-values obtained' (and the total of C-values which gave no-result, such as (0,0)), since the any 'results obtained' are normalized to the "results obtained" total (for all eigenvalues), hence we get C=0 for 100% of the obtained results for any a,b, just as the observable C "predicts". For N!=1, the algorithm will report result of (1,1) as C=1. The (0,0) cases, as in [G2] observable, are always reported as no-result (disposal of the unpaired DG trigger singles built into the Glauber's QO subtraction conventions).
The C-algorithm is non-local, by its convention of course, but that doesn't contradict any QM postulates which only say that observable C exists and it can be computed (but not how). Even without your ad hoc exclusivity requirement for N=1, even the finite-detector/EM augmented [G2] algorithm is already non-local as well due to the requirement for the non-local subtractions.
[10] Edo Waks et al. "High Efficiency Photon Number Detection for Quantum Information Processing"
quant-ph/0308054
[11] M. C. de Oliveira, S. S. Mizrahi, V. V. Dodonov "A consistent quantum model for continuous photodetection processes"
quant-ph/0307089
[12] S.S. Mizrahi, V.V. Dodonov "Creating quanta with 'annihilation' operator"
quant-ph/0207035
