# What's wrong with this experiment showing that either FTL communication is possible or complementarity doesn't hold?

The assumptions are:

• Alice and Bob have perfectly synchronized clocks
• Alice and Bob have successfully exchanged a pair of entangled photons

The idea is simply to have Alice and Bob perform the Quantum Eraser Experiment (doesn't need to be the delayed choice).

Alice and Bob agree on a specific time when Bob's photon will be between the "path marker" (which is usually just after the slits) and the detector.

If Alice acts collpasing the wave-function on her photon, the interference pattern will disappear. If not it won't.

Alice and Bob can be spatially separated...

What am I misunderstanding?

The only meaningful difference from this spatially separated quantum eraser experiment to one done on tabletop is that you won't be able to use a coincidence detector, but that is not impeditive to identifying the interference pattern, just will make errors more probable. Which we should be able to deal with a appropriate protocol...

There is a experimental paper with a small amount of citations pointing out to the breaking of complementarity in a very similar setup: http://www.pnas.org/content/early/2012/05/23/1201271109

• If Alice and Bob are spatially separated, you cannot force the premise that they have perfectly synchronized clocks. Well, you can but that would be unphysical. – Jim Jun 20 '14 at 17:42
• Why shouldn't they be able to compensate the time difference? Is some kind of uncertainty involved? I can't remember anything (from my limited knowledge as I'm an engineer). – WalyKu Jun 20 '14 at 18:51
• @Jim One usually imagine that we're setting up a transmitter between Alice and Bob which functions as a continuous source of entangled particles on which the participant will act in an attempt to achieve super-luminal communications (once the system is established, which implies a one-time set-up delay in excess of $d/c$). If we also assume that the parties are at relative rest, then it is easy enough for the source to also transmit a clock signal to both parties and achieve synchronization. – dmckee Jun 20 '14 at 20:36
• Since I am out of my depth here I won't post this as an answer. I notice that all the descriptions of this experiment I read have the two separated detectors talking to a coincidence detector. If that thing is needed for the experiment to work, then the issue is resolved because you can't get the coincidence news any faster than a lightspeed signal. No, I don't understand why it would be needed, but I figure some bright bulb would have done the experiment without if it were possible... – dmckee Jun 20 '14 at 20:38
• @dmckee I have looked upon a lot of papers with experiments of delayed choice quantum eraser and in most of the setups the interference pattern only shows up in the coincidence detector. So you seem close enough. The only paper with disagreeing results was the one mentioned. The question that remains to me: if we were using one photon of the entangled pair for a local double slit experiment would it display the interference pattern or not? If it would, I cannot see why my experiment wouldn't work. – Flávio Botelho Jun 20 '14 at 21:01

This page has a good simple summary of the experiment, they present the following simple schematic of a delayed choice quantum eraser (the same as Figure 1 in the original paper by Kim et al.):

In this case, if the entangled "idler photon" is detected by Alice at D3 or D4 then she will know whether the "signal photon" went through slit A or slit B, but if the idler is detected at D1 or D2, then the which-path information for the signal photon is "erased". Alice would be free to replace the beam-splitters BSA and BSB with mirrors to make sure all the idler photons were directed at the middle beam splitter BS, ensuring they all ended up at D1 or D2 and had their which-path information erased. She would also be free to simply remove BSA and BSB so all the idlers would end up at D3 and D4, so none of them would have their which-path information erased. And for a large enough setup, she can in principle make either choice after Bob has already observed the pattern of the signal photons that have passed through the double slit and been detected at D0 (D0's position is supposed to be varied to see how many signal photons it detects at different positions, but feel free to replace D0 by the more traditional "screen" behind a double-slit if it makes things more clear, the point is that this is where you expect to see either an interference pattern or a non-interference pattern depending on whether you know which slit the signal photon went through).

The key as to why this doesn't allow for FTL or backwards-in-time signalling is that Bob never sees any interference pattern in the total pattern of signal photons going through the slits, he can only see an interference pattern if he looks at the subset of signal photons that he knows were entangled with photons that ended up at one of the two which-path-erasing detectors D1 or D2. If you graph what the original paper calls the "joint detection" of signal photons with idlers that all went to D1, you see an interference pattern, likewise if you graph the joint detection of signal photons with idlers that all went to D2, you see an interference pattern. But even if Alice makes sure all the idlers go to D1 or D2 by replacing BSA and BSB with mirrors, the total pattern of signal photons that Bob sees at D0 will be the sum of signal photons entangled with idlers that end up at D1 and signal photons entangled with idlers that end up at D2--and the sum of the two "joint detection" patterns is actually not an interference pattern, because the peaks of the D1 interference pattern line up with the valleys of the D2 interference pattern and vice versa. This is illustrated in another diagram from the page I linked to earlier:

You can also see it in the graph of the R01 and R02 interference patterns on the wikipedia page:

The original paper by Kim et al. notes this issue on p. 2, saying "It is clear we have observed the standard Young’s double-slit interference pattern. However, there is a $\pi$ phase shift between the two interference fringes."

So, regardless of what Alice does, Bob never sees an interference pattern if he just looks at the total pattern of signal photons arriving at D0, only after he has communicated with Alice and learned which idlers went to which detectors can he look at particular subsets of signal photons whose idlers all went to a particular detector, and see an interference pattern if that detector is D1 or D2.

• Imagine Bob didn't get entagled photons from Alice because of a mishap, would he see the intereference pattern? – Flávio Botelho Oct 16 '14 at 16:24
• @Flávio Botelho - The entangled photon pairs come from some source in between Bob's position at D0 and Alice's position near the set of detectors D1/D2/D3/D4. Do you mean that this source sends photons to Bob which aren't members of entangled pairs? Or do you mean that the source still creates entangled pairs, but the "idler" members of the pairs either don't reach Alice, or she doesn't send Bob information on which detector each idler was seen at? – Hypnosifl Oct 16 '14 at 16:48
• 1st Option. The source sends photons to Bob which aren't member of an entangled pair. Would Bob see an interference pattern? – Flávio Botelho Oct 16 '14 at 23:11
• A follow up to the last question if the answer was yes (which I hope was, or else I am completely lost). Then now, we now make the source only send entangled photons. Alice won't do anything to them, it is as if Alice didn't exist. Now, would Bob see the interference pattern? – Flávio Botelho Oct 16 '14 at 23:12
• @Flávio Botelho - Yes, with non-entangled photons Bob would see interference in the total pattern of photons at D0. But if the source creates pairs of photons that are entangled in such a way that, at the time they are created, it would in principle be possible to measure the idlers in a way that would allow a person to determine which slit the signal photons went through, then there will be no interference in the total pattern of signal photons, regardless of what actually happens to the idlers (i.e. regardless of whether there is an 'Alice' to measure them, or what she does to them). – Hypnosifl Oct 16 '14 at 23:20

If I understand correctly, Alice and Bob generate an entangled pair of photons and each take one. Alice does something with hers (you specified a quantum eraser experiment at a particular time, but I won't make that assumption) while Bob does a standard double-slit experiment with his.

A minor point here is that with one photon you'll never get interference fringes, just a dot, but they can work around that by repeating the experiment many times, with Alice trying to send the same information each time, until a pattern has built up on Bob's screen.

The outcome of this experiment depends on whether any measurement of Alice's photon could reveal which slit Bob's photon went through. (A more precise description of the experimental setup would settle this question.) If it can, then Bob will not see an interference pattern no matter what Alice does. This is because for the interference pattern to disappear it's only necessary that which-path information be recorded somewhere and in principle accessible to measurement; it doesn't matter whether the measurement actually happens.

If it can't, then Bob will see an interference pattern no matter what Alice does, because any measurement Alice could perform will not collapse the part of the wave function related to the two slits. Collapse isn't all or nothing; only the part of the wave function associated with the measured quantity collapses.

These two cases are actually two ends of a continuum; if measurements on Alice's photon can reveal partial information about the path, then Bob will see a pattern intermediate between the fully interfering and non-interfering patterns. But in no case does the pattern depend on what Alice actually does with her photon, only on what it could tell her in principle.

In the paper "Time-resolved double-slit experiment with entangled photons" (mentioned in your answer), although the text for Fig. 4 says "Interference pattern fringes move as the phase is changed remotely by the QWP", it appears that they are talking about fringes that only appear after postselecting with D1 or D2, as in the usual quantum-eraser setup. Note that the fringes in Fig. 4(B/D) are labeled "heralded by D1/D2", and Fig. 4(A) shows a total detection envelope with no fringes.

I only read the abstract of the paper mentioned in your question, but it only alleges a violation of the principle of complementarity. Unlike the uncertainty principle, the complementarity principle isn't a foundational principle of quantum mechanics and never had a standard mathematical formulation. If a particular attempt to formalize it turns out to be wrong, that's not a problem for quantum mechanics.

• Wow, that is exactly my understanding minus your last paragraph. The missing part for me, that explains everything is your last paragraph. Thanks! But a lot of people around here seem not to have the same "opinion"? They always argue that the coincidence detection is fundamental, when for me it appears as a technical solution not a theoretical necessity. – Flávio Botelho Oct 16 '14 at 23:16
• @Flávio Botelho - here is a nice summary of the paper you mentioned in your original question as violating complementarity, the author of the summary seems to feel it's ambiguous whether this really qualifies as violating the idea behind complementarity, though as benrg points out, complementary is usually just stated as an intuitive guide rather than given a strict mathematical definition. – Hypnosifl Oct 17 '14 at 4:11
• To add to the above, p. 65 of the book Decoherence and the Quantum-To-Classical Transition, viewable on google books here, does suggest that at there have been more technical definitions which try to formalize the intuition behind complementarity--it says that physicists Wootters and Zurek did a quantitative analysis of "the fundamental trade-off between which-path information and interference and thus of the intimate connection between complementarity and entanglement." – Hypnosifl Oct 17 '14 at 4:30
• And this paper gives some more detail: "Wootters and Zurek also gave an information-theoretic characterization of the trade-off between the quality of the path determination and the concurrent degradation of the interference contrast ... they demonstrated that Bohr’s strict complementarity is compatible with a notion of graded or quantitative complementarity". – Hypnosifl Oct 17 '14 at 4:33
• So, I would guess that if you applied the Wootters/Zurek technical statement about the tradeoff between which-path information and interference patterns to any of these experiments, you would find that they do always satisfy this definition of "complementarity". – Hypnosifl Oct 17 '14 at 6:08