Can We Distinguish Supposedly "Indistinguishable" Quantum Mixtures?
Here is a perplexing thought experiment (albeit sadly impractical, but more as a matter of reasonable duration than anything else.) It serves as “intuition pump” about possible knowledge in quantum mechanics, and the limitations of the density matrix. This TE suggests maybe we can distinguish between supposedly "equivalent" (experimentally non-distinguishable) quantum mixtures. Say, we want to distinguish between a continuing, random stream of mixed circular-polarized individual photons (equal mix of |R⟩ and |L⟩ states, Case I) versus a stream of mixed linear-polarized (Case II.) These mixtures are considered “indistinguishable” and have the same DM representation. There are several, in-line components so I hope a diagram is not needed (but I will out of courtesy when time permits.)
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.
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.
posted by Neil Bates at 09:55
10 Comments:
Got a drawing of this? It would help a bit. I'm reading and thinking about it.
Another thought I have is that "practicality" sometimes means something deeper than just practicality when it comes to quantum. Sometimes if something is impractical, it's not just really hard to do ... it's impossible to do because of some limit you are brushing up against. Which is why it's hard to do foundational quantum well without a strong dialogue between theorists and experimentalists.
I'm not really familiar with this type of scenario very closely but I'll think about it some more.
Thoughts from an amateur:
The HWPs show increment by nħ and -nħ respectively
Minor quibble: shouldn't that be 2nħ and -2nħ? (Since they're not just absorbing each photon's angular momentum, they're actually reversing it.
Important: even though HWPs increment angular momentum from CP photon passage, they do not "collapse" those photons into a mixture of RH and LH.
Are you sure? It seems to me that, if the HWPs serve as detectors of photon angular momentum, they have to collapse those photons.
To put it another way: The HWP's angular momentum values are entangled with the photon angular momentum values. To the degree that the HWPs' rotation is measurable by the experimenter, they're functioning as detectors, and do indeed collapse the photons into RH or LH. (This does not contradict the macroscopic observation that linearly polarized beams passing through HWPs stay linear, since the photons may be collapsed into RH and LH in roughly equal proportions.)
To the degree that the HWPs' rotation is not measurable by the experimenter, the HWPs stay in a superposition of left-spinning and right-spinning states, corresponding to the R/L superposition of the linearly polarized photons Any subsequent angular momentum measurement, that collapses the photons into a pure circular state, will do the same to the HWPs. Only then is the amount of AM incremented by the HWPs defined.
In your scenario the HWPs' rotation is only measurable after n-photon runs, but I think we can deal with that by treating it as a detector for the net angular momentum of n-photon ensembles, instead of individual photons. In that case, each ensemble is partially collapsed as it passes through the HWP, such that it acquires some net circular polarization. The polarization values of the individual photons in the ensemble aren't defined at that point, but they are entangled, so whatever they turn out to be, they'll add up to the net value measured from the HWPs.
Your Case I predictions seem reasonable to me. As for Case II...
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.
Expectation values are only averages, though. Just as, in Case I, you occasionally get a run of n photons that all happen to be RH, here you occasionally get a run of n photons that happens to have net measured angular momentum of nħ. Conservation of angular momentum isn't violated, since if you continued measuring N>>n photons, you'd get some runs with angular momentum in the opposite direction and it would all average out to zero.
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 the entanglement between HWP and photon AM resolves this. If we've measured the HWPs' AM before the photons hit the filter, then the photons aren't still linear. Our measurement has already collapsed them to circularly polarized states.
OTOH, if we haven't measured the HWPs' AM, then the HWPs themselves are in a R/L superposition until the photons hit the filter. Then the photons collapse into R, the first HWP collapses into 2R, the second HWP collapses into 2L, and it suddenly looks like the HWPs were incrementing the right amount of AM "all along."
This certainly looks like anticipation, but isn't it just another example of nonlocality in an entangled system? I don't think it's a true paradox, any more than the EPR thought experiment.
Anton: thanks for catching the typo, the transferred AM in the HWPs should indeed be that be 2nħ and -2nħ because of the spin reversal. It is only nħ in the case of the CP filters, which is correctly noted. I need to note that quantity also in my IMHO even more important thought experiment at Proposal Summary. It stands corrected.
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@Anton: Are you sure? It seems to me that, if the HWPs serve as detectors of photon angular momentum, they have to collapse those photons.
Yes, it would seem that way, but we know that does not happen. First, you accept that experiments show how polarized light is affected by HWPs. Linear light stays linear, etc. A HWP reverses the coefficients of circular components, hence the sign of circularity of light. (Also, polarization axis is reflected around the HWP fast axis.)
You think maybe a single photon is treated differently, but experiments with sporadic individual photons give the same results as classical light in subsequent filters etc. A photon state a|R⟩ + bφ|L⟩ ---> b|R⟩ + aφ|L⟩, with φ the complex phase difference.
This is enabled by each HWP not being able to measure the spin of a single photon. They interact together as a form of weak measurement, and each affects the other minimally relative to measurable differences (I mean, measurable statistical deviations from deterministic photon state.) If the angular momentum uncertainty of a single HWP was small enough to allow measuring a few ħ, then the angle uncertainty would be large and thus ruin the definitive output (or so we suppose.)
We also know we can measure the transfer of angular momentum of an ensemble of photons because of the Robert Beth experiments in 1936. There is no contradiction between an effect on the HWPs and specific transformation of polarization because no individual photon has been characterized (yet however see re Proposal Summary for a possible way to get around that: use the same photon over and over again!) The effect of each |R⟩ and |L⟩ component should add separately, so millions of linear photons (a = b) should have no net effect.
There is not individual entanglement for the same reason. By the time enough photons have passed to affect the HWP, they have long since passed subsequent filters and there is no individual connection, either. Note that actually we are looking at collective consistency: The effect on HWP2 is consistent (and opposite) with the effect on HWP1, and that in turn is consistent with the effect on the circular filter. So with a run of no hits we get either 2nħ, -2nħ, nħ; or 0, 0, 0; depending on the mixture case. Remember also the ensemble angular momentum arguments.
So we can indeed find out that either millions of RH or LH or linear photons went through the HWPs, and that should be consistent with the effect on the circular filter following. (The HWPs serve as a "check" on the CF.) Now that I cleared up that problem, reread my argument for why absorption of many linear photons should not cause increment of angular momentum to the CF either. Even if it did, the HWPs would provide a consistency test. (Note also a symmetry argument using a sort of reverse case: wouldn't you expect that the half of circular photons absorbed by a linear filter would add spin to the LF? So they weren't "turned into linear photons" (without net spin), so why would linear photons "be turned into circular photons"?
Neil,
I need to note that quantity also in my IMHO even more important thought experiment at Proposal Summary. It stands corrected.
No problem. By the way, I have a couple of observations on that thought experiment. Dunno if they're worth anything, but would you rather I post them in this thread, or on that one?
Apologies for the length of this--I had to split it in half to fit posting limits.
First, you accept that experiments show how polarized light is affected by HWPs. Linear light stays linear, etc.
In the macroscopic case, yes.
You think maybe a single photon is treated differently, but experiments with sporadic individual photons give the same results as classical light in subsequent filters etc. A photon state a|R⟩ + bφ|L⟩ ---> b|R⟩ + aφ|L⟩, with φ the complex phase difference.
But are there any individual-photon experiments where the HWP rotation is actually measured between photons? I don't know of any. (Probably for good reason, since it's impossible to do this with any accuracy using current technology.)
In the single-photon experiments I can think of, the HWP isn't being used to actually detect polarization, but is simply there to alter polarization before it goes through subsequent filters and ends up at a detector farther down the line. You wouldn't expect wave function collapse at the HWP in that case.
This is enabled by each HWP not being able to measure the spin of a single photon. They interact together as a form of weak measurement, and each affects the other minimally relative to measurable differences (I mean, measurable statistical deviations from deterministic photon state.)
(In the following, I interpret you to be saying above that we can't accurately measure the spin of a single photon via the HWP; measurements can be taken between photons, but their individual uncertainty is huge. Let me know if that's not what you mean.)
It's worth noting that even weak measurements collapse the measured system's wave function; they simply collapse it partially. (Okay, sometimes they un-collapse the wave function, but either way they're altering the relative amplitudes of the eigenstates.)
Suppose we measure the HWP's angular momentum increment from a single photon's passage to be in the right-hand direction, say--so that we would expect the outgoing photon to be left-handed, if there was no uncertainty in our measurement. Then, using your formalism above, the photon actually transforms as a|R⟩ + bφ|L⟩ ---> b' |R⟩ + a'φ'|L⟩, where a'>a and b'<b. For a strong measurement, a' and b' are as large and small, respectively, as possible: a' = 1 and b'=0. This corresponds to full collapse into a pure left-handed state. For a weak measurement, a' is close to but still larger than a, and b' is close to but still smaller than b.
That means that initially linearly polarized photons will leave the HWPs as slightly elliptical; their state is still a mixture of L and R, but it's no longer a perfectly even mixture.
Continued!
We also know we can measure the transfer of angular momentum of an ensemble of photons because of the Robert Beth experiments in 1936. There is no contradiction between an effect on the HWPs and specific transformation of polarization because no individual photon has been characterized
Quite true--note, though, that this is the very definition of entanglement. The individual photon polarizations are undefined, but their sum is constrained.
And I'm no longer sure that this is actually relevant, if I understand your setup correctly. Weak angular momentum measurements are permitted after every photon, and a strong polarization measurement is made at detector D on every photon, so there should be no entanglement between photons and no need to worry about multiphoton ensembles.
There is not individual entanglement for the same reason. By the time enough photons have passed to affect the HWP, they have long since passed subsequent filters and there is no individual connection, either.
Let's be careful here. Every photon that passes affects the HWP; it just doesn't noticeably affect the value of the weak measurement we make on the HWP.
And passage through subsequent HWPs and the CF will not destroy entanglement, because those filters' momenta are all being measured weakly. (Strong measurements destroy entanglement entirely, but weak measurements merely decrease it from its maximal state.) Instead, passage merely entangles the photon with all of those filters.
Entanglement only ends when the photon actually hits D--or fails to. D provides a strong measurement of polarization, so the photon collapses down to a pure circular state, and the entangled HWPs and CF collapse down to nearly pure angular momentum states of ± (2)ħ, with whatever sign is appropriate for the photon's measured polarization. (Only nearly pure, because the entanglement wasn't maximal. Thanks to our weak measurements, the angular momentum states of the filters are no longer quite guaranteed to match the measured polarization at D.)
So for large n, we should still get approximately 2nħ, -2nħ, nħ on the filters when D registers an n-photon run of no hits, even in the case where the individual photons were initially linear.
(Note also a symmetry argument using a sort of reverse case: wouldn't you expect that the half of circular photons absorbed by a linear filter would add spin to the LF? So they weren't "turned into linear photons" (without net spin), so why would linear photons "be turned into circular photons"?
Surely they were "turned into linear photons," without net spin, which is why the net spin they'd previously possessed had to be transferred to the LF? For that matter, the half of circular photons that passed the LF were also turned "linear"--if you measured their spins a second time, you'd find them randomized back to 1/2 right-handed and 1/2 left-handed.
(Note that if you shine a circularly polarized beam through a linear filter, all of its net angular momentum is transferred to the filter--not just half of it.)
Anton, others: I think it's better to post comments about that other experiment (repeated interactions of a single photon with a HWP) at that post. (BTW I plan to improve it shortly.) Here we do have a similar issue (spin transfer and polarization of photons) but it applies to many photons passing once each. They are however related, feel free to draw on the parallels.
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