Can We Distinguish Supposedly "Indistinguishable" Quantum Mixtures?
First, the photons pass through two half-wave plates in series. Each HWP is free to rotate and serves as a detector of angular momentum (as per the Beth experiment etc.) These plates can't detect the angular momentum from passage of a single photon, but at some large number n (maybe in the millions, depends on sensitivity and note that ħ is very small) we will know that a given number of circular-polarized photons have passed. Important: even though HWPs increment angular momentum from CP photon passage, they do not "collapse" those photons into a mixture of RH and LH. A HWP preserves the amount of circularity of light but reverses its sign. Hence RH becomes LH and vice versa, and e.g. a linear state exits as linear (but orientation angle may change.) The photons then encounter a filter CF that passes only left-circular light, followed by a general photon counter D as detector. Filter CF is about as sensitive as the HWPs and we can measure its angular momentum changes too.
In Case I, sometimes (albeit rarely) we encounter a run of n right-photons in a row, where n is large enough to measurably affect angular momentum. All of this run are absorbed by CF, so a run of “no counts” is correlated with a subsequent find of increment by nħ in the angular momentum of CF. The HWPs show increment by 2nħ and -2nħ respectively (remember, the first one showed direct effect from the RH photons and reversed them, so HWP2 has a negative AM change.) In a case of n LH photons, we should get switched results from the HWPs, no change in CF (all passed it), and n counts at D. Of course, we need to know when to wait for enough angular momentum to accumulate for a measurement. This is hard, since we need to set aside a block of n photon shots and hope we're lucky to have such a run. It won't happen often!
However, a run of no counts in Case II can only come from some linear photons just happening to be absorbed in CF “as if” they were |R⟩ photons. One's intuition from QM is to expect originally linear photons absorbed at CF to be effectively detected "as" LH with the same effect there, but there are problems.The lost set of n linear photons had no net angular momentum expectation value, so it seems we violate conservation of angular momentum if CF increments as before. Also, the HWPs are not supposed to "anticipate" what will randomly happen later to photons that are still linear (superposed R and L states.) During this run, they should not increment AM like they would for genuine RH photons. (I think including HWPs is crucial to driving home the perplexing nature of the situation.)
Yet if CF is not affected as before, we can distinguish Case I from Case II (eventually!), which is supposed to be impossible. Furthermore, distinguishing the Cases means that the DM is an inadequate representation. This is a paradox, and it’s hard to know what would happen if tried. In any case, it shows the limitations of not considering special subsets of random collections – a shortcoming of much thinking about quantum (and other venues of?) statistics.