A Joke Nobody Gets

OK, this is based on a chicken joke I saw on FB (Jenny's Think Tank and Holistic Comedy Bar - drop in sometime if you can find it.) It's a concept joke, I hope it gets around. But instead of just telling you, I'm making it a riddle.
An entangled particle said to its partner across the lab, "how do I get to the other side?"
What did the other particle say back?

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.

My FQXi Essay About the Quantum Measurement Problem is Now Available!

The Foundational Questions Institute (FQXi, the "X" is there perhaps to represent the unknown, or maybe extra effort to try and find out what this is all about) promotes independent research into the ultimate scientific "why" questions. One of the reasons we need FQXi is given below from their summary, from http://fqxi.org:

EXPLORING THE FOUNDATIONS AND BOUNDARIES OF PHYSICS AND COSMOLOGY

FQXi catalyzes, supports, and disseminates research on questions at the foundations of physics and cosmology, particularly new frontiers and innovative ideas integral to a deep understanding of reality, but unlikely to be supported by conventional funding sources.

The money quote as they say:  "... but unlikely to be supported by conventional funding sources." Such sources are not likely to be much help to those like me who are"amateurs" in practical terms, whatever their abilities. (I do note there are other helpful groups, like the Society for Amateur Scientists.) One of the ways FQXi supports quality inquiry that is open regardless of professional affiliation, is through their periodic essay contests. The current FQXi Essay Contest wants answers to the question: "Is Reality Digital or Analog?" My submission to this contest was just accepted. The title: Our Non-Deterministic Reality Is Neither Digital Nor Analog: Experimental Tests Can Show That Decoherence Fails to Resolve the Measurement Problem. The paper itself is IMHO too long and maybe hard to format into this space, but here's the abstract and a link to the article (pdf download):

Is reality best described in digital or analog terms? In proper context, we are asking: what type of math is best for that purpose? However, I argue that our universe is genuinely non-deterministic, as conventional notions of quantum mechanics imply. Since mathematics is by nature deterministic, reality is not fully describable by any true mathematical model. The best answer to the original question is then, “neither – reality transcends mathematics.” It is argued that some popular attempts to avoid the quantum measurement problem, such as the decoherence interpretation, are flawed. The logical case for DI is flawed by the circular argument at its core. More importantly: some experiments are described, which could falsify the DI. If successful, they would show that we can recover superpositions supposedly lost to decoherence. Hence our finding definitive experimental outcomes instead of superposed results is not due to the effects of decoherence. Those definite, exclusionary results show a genuinely indeterminate character of the universe.

The complete paper is available here.

Previous visitors to this blog will recognize that this paper deals with one of my controversial pet peeves: the idea that decoherence helps solve the quantum measurement problem. My previous post about the subject, Decoherence Interpretation Falsified?, generated lots of intense discussion. Please, hop on over to FQXi and tell me what you think.

How the word "interference" causes confusion

It is good to check on whether some terms used in physics are "well posed" and do their jobs right. I believe one of the unfortunate, sloppy semantic habits in physics is use of "interference." The word came from observation of light and dark fringes in e.g. double-slit experiments. This can be explained by superposition (amplitudes add linearly) and squaring of summed amplitudes to get intensity. The classic experiments reveal a vivid depiction of this principle, but led to careless semantics that were wrongly tied down to the original context and to superficial demands of visible manifestation. For example, if the phases between waves changes around for various reasons, or they can't match up in a well-ordered or stable way (different frequency or polarization etc.) people loosely say "the waves don't interfere anymore." But of course the waves still superpose everywhere, the result just isn't a stable or evident pattern over time and space.

And as for polarization: yes, x and y polarized light do "interfere" in the genuine technical sense, properly enlarged to mean vector addition and not just "amplitude" as a scalar. Add x and y light in phase: we get a definite 45 degree wave. If added at pi out of phase, they make a 135 degree wave; or circular etc. in other phases. We can even get a pattern of varying polarization type on a screen from little x and y sources acting like slits, but it all has the same amplitude. Should we call them "state-fringes" or "type-fringes"? (And yet, we could pull back a bit and choose to pick out the x or y with filters. That seems to offer 'which way' information in a contradictory way, reminiscent of issues raised by the Afshar experiment.) But since that isn't the historically conditioned intensity fringes, this led to careless and misleading talk that "orthogonal polarized states don't interfere."

This confusion about "interference" plays a major role in the fallacious claims that "decoherence" resolves the puzzle (to whatever extent) of collapse of the wave function by in some sense converting superpositions into mixtures. There are many faults with that interpretation. Briefly, it is a circular argument since it introduces at the outset what it is trying to explain: collapse is required as unacknowledged "selector" to introduce quantum statistics into the density matrix along with the classical statistics. Then such a hybrid DM is used to seem to explain the collapse it already incorporated.

DI proponents often say that "decohered states don't interfere anymore." That harks back to old, careless classical talk of vividly evident fringes, which are equivalent to the orderly statistics of ensembles with consistent phase relations. First, REM that classical interference wasn't about "statistics" at all, but amplitudes and their squares per se. The conceptual models of QM say the superpositions always exist together until "collapse" somehow both concentrates and isolates a given state, allowing for statistical outcomes. Second, it is misleading to talk of "interference" when comparing waves to each other that occur in different instances in an ensemble of trials. Superposition is about waves existing and adding together here and now, not a collective. Third, once we understand the universality of superposition, we realize that claim of non-interference is actually false. Instead, decohered states do "interfere" in the technically correct sense of following the superposition principle. If the phase difference is phi1 one time and phi2 another time, so what. The states still would superpose accordingly each time. That is regardless of whether the overall patterns made either by their squared sums in any one case, or by the statistics from collapse applied to ensembles of such superpositions was coherent and orderly; versus whether the patterns (merely) simulated the behavior of genuine mixtures. 

Proposal summary: can we find circularity of a single photon along a range of values?

Suppose a photon is a superposition of e.g. 0.8|RH> and 0.6 |LH>. Standard QM says we can only test for the 64%/36% chance of getting full "RH" or "LH" as projected eigenstates. But let's use mirrors to send a single photon round and round many times through a pair of half-wave plates ("pair", to revert the photon to original state for entry.) HWPs flip the rotation of polarized light. They accumulate angular momentum thereby (Beth experiment 1936 etc.) That AM is proportional to the circularity of the light times the number of photons. Hence it shows intermediate values for elliptical light etc.

From particle indistinguishability: the same photon going through each HWP many times (being reverted to original state each pass through two HWPs) should have the effect of the same number of like photons, going through once each. Hence, I argue, we should get a range of results for circularity (intermediate for elliptical, no net AM change for linear, etc.) for one photon as for many. But that contradicts standard theory, so what happens?

Decoherence Interpretation Falsified?

[Note: this post has been revised here and there, off and on. I gave up specifying each change in []s; just NB that it's been changed. The essential point is the same, but with a major new caveat about just where the conversion from superposition to "mixture" is imagined to take place. See also note below about the controversy provoked at another blog. Perhaps most important: I am defending orthodox QM from the Decoherence Interpretation, not challenging it with some alternative theory etc. Traditional QM says that wave functions continue to evolve together in superposition until some mysterious measurement process selects and localizes an eigenstate - as per the projection postulate. Hence, we have no idea why Schroedinger's Cat doesn't stay both alive and dead.  DI enthusiasts like W. Zurek say that decoherence gives an "natural" explanation (in some sense which is often unclear) of why we don't find such macro superpositions. I propose that my experimental setup would distinguish the two ways of imagining the evolution of wave functions. Hence, they would no longer be mere "interpretations" - but readily subject to experimental testing. I continue to refer to DI as "interpretation" for convenience and by tradition. I also just posted similar to sci.physics and sci.optics.]

"Decoherence" is both a real phenomenon, and part of an interpretation purporting to tell us why don't observe macroscopic quantum superpositions like Schroedinger's Cat. ("Explanation" is too strong, often avoided even by supporters.) Some would say, the DI avoids the paradoxical quantum measurement problem of "collapse of the wave function."  This supposedly comes about because, roughly, decoherence and entanglement of the phase relations between superpositions (like "dead" + "alive") effectively converts them into mixtures (like classical particles, roughly speaking.) Some say the unactualized alternative slips into another universe. (All easy to find online.) I don't agree, making rebuttal at other posts here and elsewhere. I am heartened that Roger Penrose made similar complaints at e.g. Shadows of the Mind, and critics like N P Landsman have picked at various loose ends and problems. Yet few DI advocates are swayed by critics. As Landsman writes: "Like capitalism, decoherence seems here to stay."

I thought of an experiment we could do to demonstrate the weakness of the DI.  It shows we can recover information that would have been lost if indeed "decoherence converts a superposition into a mixture" as some have IMHO too-boldly proclaimed. That's better than just arguments per se against DI. Briefly, it shows that true but decohered superpositions would produce one set of results in this system, but a true mixture (or anything, by definition, "indistinguishable" from a mixture) would (if present where detectors are usually deployed) produce a different set of results. Hence they can't both describe the situation. Since I gather there is agreement ("conventional QM") that the first outcome is the correct result, that causes difficulties for the DI. (Note: the importance of this argument may go beyond DI, if such information "should" simply have been "lost" period, apart from any interpretative framework.) The predicted outcomes are already derivable through known quantum optics, so I assembled a case from existing knowledge. (It still should be empirically verified.) I hold therefore that the "quantum measurement paradox" remains unresolved, perhaps the deepest mystery about the nature of our world.

My proposal is fairly easy to describe. (The math is less simple but not hard to work through.) I use some ASCII conventions for now from tech issues, so "*" for multiplication where needed etc. Synopsis: even if we completely scramble their relative phases over a history of instances, and then recombine split waves; we can recover their original amplitudes when the secondary outputs are recombined again in a subsequent beamsplitter. This would not be possible from a genuine mixture, as opposed to an apparent one as the case below is revealed to be. Hence decoherence does not always convert a ensemble of superpositions of random phase into a mixture.

Consider a Mach-Zehnder interferometer (as shown above) with a first beamsplitter BS1 that does not [was "need not"] divide intensity equally. For sample values we'll use intensity along bottom leg L1 = 64% and top leg L2 = 36%. Hence, relative amplitudes are: a = 0.8 and b = 0.6. We use a 90 degree (i) phase change at each half-silver and treat full reflections and transmissions as not changing phase (per custom of Roger Penrose, OK as long as consistent.) We can represent what happens to a single photon entering BS1 as: transmitted state a|1> goes along the bottom, and reflected state ib|2> along the top.

However, suppose some interaction/s introduce a new phase shift φ to the wave in L2, as complex angle. [Shift φ was "u" before font change, and added cleanup of this section.]  That changes the phase in L2 to iφb|2>. Then we recombine the beams at BS2, which is a 50/50 splitter/recombiner. It combines relative phases as did BS1, with output from the lower face of BS2 called channel CA2; and from the other face: channel CB2. (This keeps numbering consistent and allows easy reference to original output from BS1.) I will just use "s" for sqrt(0.5) ~ 0.7. A half-silver mirror reduces intensity to 1/2 and thus multiplies amplitude by s. Hence [equations adjusted to new phase standard, pardon some earlier untidiness],



(1) CA2 superposition = s[(ia|1> + iφb|2>],
      CB2 superposition = s[(a|1>  -   φb|2>].

We find intensity (and photon statistics if we collected at this point) by inserting into

(2) I = A^2 + B^2 + 2AB cos theta,

showing involved net (superposed) amplitudes A and B that are comprised from combinations of |1> and |2>. If a = b and u = 0 and thus net phase between legs is i, then the intensity out of CA2 = 1/4 + 1/4 + 1/2 = 1, and out of CB2 = 1/4 + 1/4 - 1/2 = 0. That is equivalent to bright and dark fringes. If a = 0.8 and b = 0.6, we get CA2 = 0.32 + 0.18 + 0.48 = 0.98 and CB2 = 0.32 + 0.18 - 0.48 = 0.02. Hence, lower contrast fringes.  If there is a further phase difference introduced between |1> and |2>, then the relative intensities change accordingly as can be calculated (but still must add to one of course.) If we introduce photons one by one into such a device, the statistics of detection are the same. To make the argument and experiment about "the wave function of a single photon," that is what we'd do. (Despite some states of unclear photon number, an effective "one photon at a time" *can* be introduced into such a device. It means basically, one net "click" from arrays of ideal photon detectors covering all avenues of escape.)

Now, what if we introduced complete decoherence into the picture, in a manner like Chad Orzel uses (and similar in effect to found elsewhere) to model decoherence in e.g the post at http://scienceblogs.com/principles/2008/11/manyworlds_and_decoherence.php ? (We had quite a debate there and elsewhere. I admit being testy sometimes but think I'm in the right in the end. [I add, that Chad seems not to be an advocate of the strongest claims about DI. Commenters suggested also http://www.ipod.org.uk/reality/reality_decoherence.asp by Andrew Thomas.]  Now we have to integrate over a range of randomly varying φ, and divide by that full range to get the mean value. This is easy for a uniform distribution of phase differences, since

(3) integral (Eq. 2) d theta = (A^2 + B^2)theta + 2AB sin theta + C.

We pick a range that completely scrambles the phases ("complete decoherence") such as between +/- pi, substitute into the integral, and divide by the range 2pi. Then (since sine of each limit = 0) we find the result out of either channel is simply A^2 + B^2 = (a^2 + b^2)/2. This destroyed the statistical interference pattern of photon hits, even in the case a <> b. The output acts like a "mixture" of photons each exiting BS2 from either CA2 or CB2 with 50/50 chance, but not "both at the same time." [fixed poor wording]

A strong DI follower would say (following an ensemble interpretation): what would have been a coherent superposition is now a mere "mixture" despite being comprised from interacting waves. They would follow an essentially positivist yet post-modern tack that "we couldn't tell the difference, so the output should be regarded as being the same as a 'real' mixture." Hence, somehow we don't have to worry about why a hit occurred at the CA counter instead of the CB counter, when under old-fashioned (!) QM there are still wave amplitudes (usually) at both counters - and a mysterious collapse was still needed to sweep the whole big mess into one little atom that absorbed it all. Their argument sounds circular (what causes any "statistics" in the first place instead of distributed amplitudes, to allow comparing one set of stats to another etc.), and I'm fortified by seeing similar misgivings from e.g. Roger Penrose. But DI is popular because it lets the perplexed brush off their worries about paradoxical features of reality. I can't blame them for wanting to try, but Nature is what it is ...

So, is the DI view really apt - even in its own terms? I think not - but [added] that depends on a crucial distinction.  Instead of intercepting photons and collecting statistics right out of BS2, let's instead recombine outputs CA2 and CB2 into BS3. Since amplitudes are again reduced by s and reflection multiplies angle by i, the new output that combines the beams is like this:

CA3 = s[iCA2 + CB2] = s[s[(a|1> - φb|2>] + s[(-a|1> - iφb|2>]] = -φb|2>
CB3 = s[CA2 + iCB2] = s[s[(ia)|1> + iφb|2>] + s[ia|1> - iφb|2>]] = ia|1>

Note that since φ is just a variable complex angle, the amplitude ratios are the same each trial from BS3, and therefore the final average over a range of φ will reflect this as well. So, from BS3 we recover the respective original amplitudes a and b (and hence, same original statistics) that came out of BS1! Of less importance IMHO is that we can later recover the phase relation, that entered BS2. That information was hidden in the relationship between the wave outputs from BS2. It would not show in a raw statistics of hits if we used detectors right around BS2 instead of letting the wave continue on through BS3.

This result could not happen from a mixture that came out of BS2, since photons that came from either CA2 or CB2 but not "both at the same time" would just scatter as individuals from BS3. Their statistics from BS3 would be 50/50 output instead of a^2, b^2. [However, if we imagine that "a mixture" enters BS2 instead, then there is no discrepancy.] This does demonstrate the continued wave nature of the output from BS2, despite total mixing of phases which in some perspectives destroys the superposed character of the photon wave function.  The recovery of BS1 exit amplitudes from BS3 shows that. Does one of the deep mysteries of reality remains a challenge?

IMHO, Bye bye decodance! [snark snipped for peaceful purposes!]

--
Regards, and Happy New Year (and New Decade, so they say !)

Neil Bates

[Notes: My post provoked an unfortunate reaction at another blog. It was acerbically critiqued at, with key comment here: Neil Bates Owes Me $160.  In that comment Prof. Chad Orzel of Uncertain Principles sci-blog admitted that my math was not wrong as he had claimed. He still doesn't think it proves any good point about quantum measurement, and he may be right (not that it's always clear what the implications of a quantum experiment are anyway!) But readers should decide that for themselves.

Even though him checking some things could have avoided misunderstandings, I don't blame him for most of that. First, my presentation was not adequately clear for various reasons. Also, it was not fair for me to fish for a response from him (in the hope of garnering attention and "publicity") as even some part of referencing his former posts. I didn't mean to anger anyone or make a feud. Sadly it did, and I apologize for that. My main reason for noting his MZI example was it corresponding to how I could show there is a distinction between superpositions and mixtures at the critical juncture. Also, it or similar is in his new book which is selling well and deservedly so.

As to where the "mixture" is to be imagined found. If we can tell the difference between a real mixture and the actual superpositions, even after decoherence - then the claim "there's no way to tell the difference" is wrong. Chad Orzel used the similar phrase "since the end result is indistinguishable from a situation in which you have particles that took one of two definite paths" in the above link.

However, he was imagining a mixture occurring even before we recombined into BS2. Now we have to ask: in which treatments of this problem, would the output from BS2 be considered a mixture? Well, in order to have any significance to the measurement problem in the case of detectors just past BS2 - the mixture would have to be present outside BS2. But then, the result (50/50) would not agree with this experiment. OTHO - (and this is a supreme irony) - if you take the mixture as present before entering BS2, then entering photons would turn back into a superposition anyway! In that case we'd get my result, but it wouldn't help show selection at detectors. So DI either fails to be relevant to detection issues, or it is factually wrong.

BTW I wish no continuing "feud" as such, just fresh looks at this problem. tx!]

A quantum measurement paradox: the reallocation by measurement problem

I spend lots of time thinking about puzzling foundational issues in quantum mechanics (search for "quantum measurement paradox" and I often show in the top five), and do I have a deal for anyone else like me. I have been thinking of this problem for awhile, and the post Many Worlds, Many Headaches on Uncertain Principles stimulated me to post a version of it. If you want something really strange and "after the fact" about measurement, please consider my following proposal: In a Mach-Zehnder interferometer, insert a gray filter G into leg L2. Given a traditional stream of light, that alters the amplitudes delivered to the beam-splitter/recombiner R. With a 25% transmitting filter, the L2 amplitude is 0.5 of that in L1 (which itself is sqrt (0.5) of the original pre-split input.) Hence, with symmetrical R, we get an output mix rather than all A channel output. We can adjust R to a compensatory split so that output is again all A channel. Using individual photons, the statistics should be the same. FEL optical physicist Michelle Shinn of J-Lab agreed with me that's so, despite the weirdness of the photon's wave function in L2 being attenuated by the chance that it could have been absorbed, even if it wasn't (well, superposition of absorption and not-absorption in the dye molecules in the filter, etc, right?) Also, as G gets darker, this has to be the limiting factor approaching the results of an opaque stop in L2. But what happens if we can find out whether a photon has been absorbed in the filter?

Consider an opaque stop: the stop clearly "reallocates" the WF all into L1, in a manner akin to the Renninger negative result problem, even though no actual "measurement" is taken। But there, a photon will just never get through. However, G may or may not absorb a photon, something we can in principle check on (There are semi-transparent optical detectors, no? Just consider film for example.) Now, while G is still "deciding" (in a state of superposition) whether it will absorb or not, it makes sense to consider the L2 wave to be attenuated relative to L1. Maybe that's the normal time scale to allow interference in R before that happens. But, after a certain time, if we check G carefully to look for evidence of absorption, it should be settled: absorption or not. If it did, there's no paradox. But if we find "no absorption," why in the world should the L2 wave continue attenuated? The measurement result was "no" for G, so there is no longer "a chance" that the photon might end up there. The filter might as well have been clear glass, right? If so, then the interference at R would be different (it would follow normal equal-balance rules instead.)

The really weird thing is, that reallocation should take place as soon as the absorption/detection issue is settled. If so, we could manipulate the pattern of hits (with sequential photon shots) at the output by looking for evidence of absorption in the filter, which would start rearranging the WF as per Renninger etc. In principle, there's nothing to stop this from being a true FTL signal, since manipulating G (or perhaps the distance to R) causes noticeable effects (not distant signal correlations) at R. Sure, that's problematical, but you can't just blow off the supposed effect on the WF of the negative measurement in G, can you? Have fun.
(I also just put this up on sci.optics, sci.physics, etc.)