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Entanglement swapping doesn't violate locality

In his jihad against the principle of locality, Florin Moldoveanu has used entanglement swapping as a would-be argument. The claim he wants to fight against is that all correlations in the real world – in the successful approximation of non-gravitational quantum field theory – arise from the combination of quantum information's direct interaction (at one place) and the motion at most by the speed of light.

The misspelled word "implementation" on the picture isn't my fault. It's a fault of another anti-locality jihadist.

His situation is simple. (He doesn't have a picture and uses labels 1,2,3,4 for what is called A1,A2,B1,B2 on the picture above.) Two sources of entangled pairs of spin-1/2 particles (the gadgets at the bottom) create entangled spin-zero pairs. \[

\ket{\psi}_{A1+A2} = \frac{\ket{\uparrow_{A1}\downarrow_{A2}} - \ket{\downarrow_{A1}\uparrow_{A2}} }{\sqrt{2}}

\] Similarly for \(A\to B\). The internal members of the pairs A2,B1 propagate along the red lines towards the center where a joint measurement (the gadget at the center top) is being made.

The joint measurement is e.g. basically a measurement figuring out whether \(J_z\) of A2,B1 are the same or the opposite ones. Imagine that \((J_{A2,z}-J_{B1,z})^2\) is being measured. The result of this measurement is found. Because of the singlet-defining correlations between A1 and A2; and between B1 and B2, the known "relative qubit" produced by the joint measurement is translated to the knowledge of the "relative qubit" between A1 and B2.

In other words, A1 and B2 – propagating along the blue lines on the picture and later measured by Alice and Bob on the picture above – are entangled and the measurements will be correspondingly correlated. That's despite the fact that the blue lines A1 and B2 don't intersect each other. In fact, the big bang (a horizontal line starting the life of the world) could have taken place just beneath the production of the A1+A2 and B1+B2 entangled pairs.

When Alice and Bob measure A1 and B2, the past light cones of the individual particles don't intersect. However, the locality is still true and all the correlations are explained by not-superluminal propagation combined with interactions at one place. To explain the correlation between A1 and B2 as seen by Alice and Bob, we just need to use three interactions – the creation of the entangled pairs in the two bottom gadgets; and the correlation extracted from the joint measurement.

OK, a persistently annoying Romanian anti-locality warrior could say, but aren't the particles A1 and B2 a counterexample of subsystems that didn't interact in the intersection of their past light cones but that are correlated, anyway?

No. The point is that these particles are correlated only according to the observer who is aware of the result of the joint A2-B1 measurement in the middle of the picture. And this observer knows that A1+A2 form an entangled pair, and so do B1+B2. So my implication remains true. Exactly when the conclusion breaks down, the assumption breaks down, too.

First, let me talk about the particles A1,A2,B1,B2 from the viewpoint of an observer who doesn't know the result of the A2+B1 joint measurement. You may verify that according to this observer, there won't be any correlation between the measurements of A1 and B2 done by Alice and Bob. In particular, all four results for the 2 measured bits A1, B2 are equally likely. This is just a reflection of the fact that from a more objective perspective, we don't have a correlation between A1 and B2. We have a correlation between A1, B2, and the relative bit "whether A2 is equal to B1" measured by the joint measurement.

So for an observer not familiar with the joint measurement, there is no correlation. And it's allowed according to locality that there's no correlation. That's what locality predicts in the absence of the not superluminal contact.

On the other hand, the observer in the middle top is familiar with the result of the joint measurement. He knows basically "whether A2 and B1 are the same", if I use the classic bit language that is totally sufficient to explain the "propagation of correlations" in any particular experiment of this kind. But this observer in the middle unavoidably knows that A1 and A2 interacted in the past; and B1 and B2 interacted in the past. In fact, both A1+A2 and B1+B2 came from the entangled pair sources.

Because he knows about this past action of something (the two sources) on A1 and B2, he can't neglect it in his predictions of the correlations. He must realize that A1 and B2 aren't standalone particles. Instead, they are parts of the pairs A1+A2, B1+B2 created in the sources. And these A1+A2 and B1+B2 subsystems have legally interacted with each other. The joint measurement of A2 and B1 is this interaction because A2 belongs to the first subsystem and B1 belongs to the second. It took place exactly at the moment when the A1+A2 and B1+B2 pairs became able to influence/overlap each other.

It's this interaction – the joint measurement of A2 and B1 – that (along with the correlations guaranteed by the pair sources) explains the future correlations between A1 and B2. Let me highlight the trade-off as a quote:

An observer either knows about the A2-B1 correlation, and then he has causal access to the sources (they are in the past light cone of this observer; as well as in the past light cone A1, B2, respectively), and then he can deduce a correlation between A1 and B2. He therefore knows that this correlation is one between the subsystems A1+A2 and B1+B2 that just touched, overlapped, interacted.

Or an observer doesn't know about the A2-B1 correlation. Such an observer predicts that there is no correlation between A1-B2. For such an observer, they're two random particles in two random pairs with random and independent first-bit assignment.

You can't eat a cake and have it, too. There is never any non-locality in Nature. Moldoveanu's wrong conclusion is obviously just an artifact of his incoherent usage of one observer (who knows the result of the joint measurement) to argue that there's a correlation; and another observer (who doesn't know the result of the joint measurement) to argue that the subsystems haven't been in a causal contact. If you consistently use the description according to a single observer, you will never encounter paradoxes or non-locality.
It works exactly the same for socks.

There's really nothing mysterious and nothing intrinsically quantum about the fact that known correlations between A1-A2, A2-B1, and B1-B2 translate to known correlations between A1 and B2, the endpoints of the A1-A2-B1-B2 chain. In fact, it works exactly the same with classical bits.

Imagine that the entangled pairs of particles are replaced by changing rooms in which someone named Bertlmann takes on socks and produces two feet with socks (of opposite color, red or green). Helpfully enough, the sources in the picture look like bare buttocks with legs. Fine. So A1+A2 are two feet of Dr Bertlmann with either red+green or green+red socks, and so are B1+B2, the feet of Mr Bertlmann, a twin brother of Dr Bertlmann (who didn't get a PhD because his PhD committee was meritocratic and wasn't struggling to increase the number of far leftists in the Academia).

Now, someone amputates the right leg of Dr Bertlmann and the left leg of Mr Bertlmann (I apologize to the latter) and compares the colors. Are they the same (red+red or green+green) or are they of opposite colors (red+green or green+red)? He learns the result and it allows him to say that the remaining two legs of the two Austrian men that he didn't amputate are also either the same or opposite, with the same result.

Would someone claim that there's some mysterious nonlocality between the socks A1 and B2 in this experiment? All the correlations result simply from known dressing habits of two men in changing rooms, locally, and a comparison of two socks in another room. No correlation ever arises without this contact of the bits at one place.

The explanation for the "propagation of information" works qualitatively in the same way for quantum bits as it does for the classical bits. No correlations may arise without a proper explanation based on the not superluminal propagation of information; and transmission or comparison of information done at one place. There are no counterexamples. It's silly to look for counterexamples. The right approach is to see that the principle is correct, find the proof, or understand the proof, and just stop talking about this trivial thing.

"Elementary" degrees of freedom are well-defined classically, not in quantum mechanics

In the classical case of socks, everyone recognizes the two changing rooms as the only true sources of correlations while the joint measurement is an act of passive learning about the information. In classical physics, the A1-B2 correlation is clearly a "derived quantity". Some correlation between A1 and B2 will exist because the formula for the correlation is a composite one, includes both A1 and B2, and the dependence on A2 and B1 is being canceled.

Using "XOR", the two-bit operator that is equal to 0 if the two bits are the same or 1 if they are not (it's the same thing as the addition modulo 2; and it's the isomorphic additive counterpart of the multiplication if 0,1 are renamed as +1,-1), we know that A1 XOR A2 = 1, B1 XOR B2 = 1 (sources), we find A2 XOR B1 by the joint measurement, and that's why we can calculate A1 XOR B2 as
A1 XOR B2 = (A1 XOR A2) XOR (A2 XOR B1) XOR (B1 XOR B2) = 1 XOR (A2 XOR B1) XOR 1 = A2 XOR B1.
This is a mathematical identity, not any evidence in favor of non-locality.

But for a given type of measurements in quantum mechanics, we may say exactly the same thing. Once it's determined what is being measured at all times, all the measured values are facts in exactly the same sense as in classical physics and the correlations predicted for them have the same interpretation as in classical physics (but different formulae if you want to predict them). The point is that to calculate (A1 XOR A2) XOR (A2 XOR B1) XOR (B1 XOR B2), we need to know the values from several places – A1,A2,B1,B2 etc. – and it's the past light cones of these spacetime events that is discussed in the principle of locality. Obviously, the past light cone A1+A2 and B1+B2 do overlap. And if we don't write A1 XOR B2 as the chain (A1 XOR A2) XOR (A2 XOR B1) XOR (B1 XOR B2) = 1 XOR (A2 XOR B1) XOR 1 or if we don't know the result of A2 XOR B1 in the middle, then it's right to say that there is no correlation between A1 and B2!

There is a difference between quantum mechanics and classical physics. In classical physics, the joint measurement basically always measures the bits A2, B1 separately, and then does a classical operation (to find whether they're equal). The apparatus may later "forget" the individual values of A2 and B1 but they were there. In quantum mechanics, it's possible to make joint measurements that do not measure A2 and B1 separately. This is important. If we measure the qubits A2 and B1 separately, we induce collapses on both subsystems and the correlation between A1 and B2 only survives if we measure properties of A1 and B2 that commute with those we measured on A2 and B1.

However, quantum mechanics allows you to make the softer joint measurement of A2 and B1 that only induces a collapse of the relative information (the part of the wave function knowing whether the two systems have the same or opposite spins etc.). All these subtleties are absent in classical physics because observables always commute – and, equivalently, measurements never influence the state of affairs.

Efforts to sound interesting

I suspect that many experimenters etc. trying to hype these simple experiments as "evidence for non-locality" etc. actually know very well that they're fooling the readers. They know that they may become famous for emitting these "big statements" whether they are right or not. They are not right. It's similar to the "apparent debunkers of relativity" and other principles. I don't necessarily claim that e.g. Lene Hau is a great similar example but things like "changing the speed of light" sound attractive to many ears exactly because they seem to contradict some (sloppily understood or analyzed) principles that physics is based on and that physicists often boast about. There is never any actual contradiction which is clear if one looks carefully but many people's jobs may depend on the assumption that other people never look carefully.

The price we pay for this self-promotion is that people's misunderstanding of physics is being deepened.

Bonus: wormhole interpretation

The ER-EPR correspondence has a geometric interpretation of the entanglement swapping. The sources are actually producing wormholes with 2 openings (non-traversable wormholes, i.e. tiny Einstein-Rosen bridges). In each case, the two openings of the wormhole propagate and get separated. The joint measurement is a procedure that merges the two wormholes/tunnels, from A1 to A2 and from B1 to B2, into one "longer" wormhole A1-B2. It's still true that none of the openings ever propagates faster than light and all the correlations between some bits C,D depend on these bits' traceable, superluminal-motion-avoiding connection to some events in which they shared the location.

A concise explanation

The blog post above is longer than I originally thought and I have said the same thing in many ways so it may be confusing. If I had to offer just one solution that is short and clear, it would be the following:

There is no correlation between A1 and B2 independently of other measurements. The correlation only exists given a particular result of the joint measurement of A2 XOR B1. For this reason, the actual observed correlation that exists is not among two measured quantities A1 and B2 but among three measured quantities A1, A2 XOR B1, and B2. The correlation is such that
A1 XOR (A2 XOR B1) XOR B2 = 0
This actual correlation between three variables E=A1, F=A2 XOR B1, G=B2 doesn't violate locality because the measurements E and F (and similarly F and G) have common causes in the shared past light cones (the production of the pairs at sources). That's enough to explain the correlation (the non-factorization of the probability distribution for E,F,G). There doesn't have to be a shared event in the past between any pair of the variables in the list E,F,G.

In particular, there's no need for the past light cones of the measurements E,G to overlap (have common causes): the overlap of the past light cones of E,F and similarly F,G is enough to explain the correlation between E,F,G!

Indeed, the correlation between E,F,G above is easy to verify by commutativity and associatity of XOR:
A1 XOR (A2 XOR B1) XOR B2 =
= (A1 XOR A2) XOR (B1 XOR B2) =
= 1 XOR 1 = 0.
This works for colors of socks just like \(J_z\) of spin-1/2 particles.

(If we need to rephrase the correlation between three variables E,F,G as a correlation between two systems, because that's what I made my statements about, we need to take objects E+F, G or E, F+G, or F, E+G. In all cases, one of the two objects is composite and therefore extended, its past light cone is correspondingly wider, and overlaps with the past light cone in the pair, anyway. In this case and any case in Nature, any correlation is always explained by the subsystems' contact/interaction in the intersection of the light cones. That's what locality means and locality simply holds.)

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