Entanglement, nonlocality, and the slickness of the MSQM folks

Update: See at the end of this post.


0. Context

This post began its life as a comment to Roger Schlafly’s blog post: “Smolin preaches nonlocality nonsense” [^]. However, at 7000+ characters, my comment was almost twice the limit (of 4k characters) there. So, I decided to post my reply here, as a separate entry by itself.

I assume that you have read Schlafly’s post in toto before going any further.


1. Schlafly’s comments:

Schlafly says:

“Once separated, the two particles are independent.”

The two particles remain two different entities, but their future dynamics also remains, in part, governed by a single, initial, entangling, wavefunction.

“Nothing you do to one can possibly have any effect on the other.”

The only possible things you can do to any one (or both) of the entangled particles necessarily involves their shared (single) wavefunction.

Let me explain. Let’s begin at the beginning.


1. System description and notation:

Call the two entangled particles EP1 and EP2.

If you want to imagine two different things physically being done to the two EPs, then you have to have at least two additional particles (APs) with which these EP’s eventually interact. APs may be large assemblages of particles like detectors; EPs are regarded as simple single particles, say two electrons.

Imagine a 1D situation. Initially, the EPs interact at the origin of the x-axis. Then they fly apart. EP1 goes to, say, +1000.0 km (or lightyears), and EP2 goes to -1000.0 km (or lightyears). Both points lie on the same x-axis, symmetrically away from the origin.

To physically do something with the EP1, suppose you have the additional particle (detector) AP1 already existing at 1000.0 +\epsilon km, and similarly, there is another AP2, exactly at -1000.0 - \epsilon km, where \epsilon is a small distance, say of the order of a millimeter or so.

Homework 1: Check out the distance from the electron emitter to the detector in the single-particle double-slit interference experiments. Alternatively, the size of the relevant chamber inside a TEM (transmission electron microscope).

The overall system thus actually has (and always had) four different particles, and in the ultimate analysis, they all have always had a single, common, universal wavefunction. (Assume, there is nothing else in the universe.)

But for simplicity of talking, we approximated the situation by eking out a two-particle entangled wavefunction for the EPs—just to get the discussion going.

All MSQM (mainstream QM) people blithely jump to and forth between abstractions in this way—between two abstractions of having basically different scopes. That’s not the trouble. The trouble is: They never tell you exactly when they are about to do that.

OK. Now, think of the 4-particles system-wavefunction as being built from four different 1-particle wavefunctions (via an appropriate linear superposition of all the appropriate product-states of the four 1-particle wavefunctions, with the proviso that the resulting single wavefunction must have enough generality, and that it obey the appropriate exchange-operator rules etc.).


2. The sense in which entangled particles approach independence—in their interactions with the other particles:

Each 1-particle wavefunction has an anchoring point in space.

[MSQM people never tell you that. [Google on “anchoring of” “potentials” or “wavefunctions” in the context of QM.]]

Each such a wavefunction very rapidly drops off in intensity from its anchoring point, so as to satisfy the Sommerfeld radiation condition. …May be there is a generalization of this principle for the many-particle situations; I don’t know. But I know that if the system-wavefunction has to be square-normalizable, then some condition specifying a rapid decay over space is what Sommerfeld the nature ordered.

[MSQM people never remind you of such a condition in any such contexts to you. [Google!]]

So, the 1-particle wavefunction for AP1 affects EP1 far, far more than it affects EP2. Similarly, the 1-particle wavefunction for AP2 affects EP2 far, far more that it affects EP1.

Homework 2: Find the de Broglie wavelength for an electron, and for a typical detector. Work it out on your own. Don’t cheat [^][^] !

In this sense, sure, what AP1 does to EP1 (and vice-versa) has overwhelmingly greater effect than what it does to EP2 (and vice-versa).

So, what Schlafly says (“Nothing you do to one can possibly have any effect on the other”) does have a certain merit to it, but only in a limiting and approximate (“classical-like”) sense.

In a certain limiting sense, the AP1 \Leftrightarrow EP1 and AP2 \Leftrightarrow EP2 interactions do approach full independence.

To use the language that the MSQM people typically use, the reason put forth is that AP1 and AP2 never directly interacted with each other.

Actually, they all always had interacted with all the others—but in this case, only dimly so. So, as we would say to describe the same point: Due to the Sommerfeld radiation condition, AP1 \Leftrightarrow AP2 interaction always was, remains, and assuming that they don’t leave their fixed positions at \pm 1000.0 km so as to go nearer to each other, it will also always remain, very negligibly small.


3. The entangled particles’ dynamics continues to be influenced from the initial entanglement:

However, note that as EP1 and EP2 travel from the origin to their respective points (to their respective positions at \pm 1000.0 km), this entire evolution in their states (consisting of their “travel”s/displacements) occurs at all times under an always continuing influence of the same, initial, 2-particle entangled part of, the 4-particle system wavefunction—its deterministic time-evolution (as given by the Schrodinger equation).

Since the state evolution for both EP1 and EP2 was guided at each instant by the same 2-particle entangled part of the same wavefunction, the amount of distance does not matter—at all.

Even if their common entangled wavefunction initially has almost a zero strength at the distant points \pm 1000.0 km away, once EP1 and EP2 particles begin moving away from the origin, their states evolve deterministically (obeying the time-dependent Schrodinger equation). As they approach the two \pm 1000.0 km points respectively, the common wavefunction’s strength at these two points accordingly increases (and the strength of that portion of the same wavefunction which lies in the space near the origin progressively decreases). That’s because the common entangling part of the system wavefunction, is composed from two 1-particle wavefunctions, one each for EP1 and EP2, and each of these two 1-particle wavefunction has the respective current positions of EP1 and EP2 as their reference (or anchoring) point. Why? Because the potential energy has a singularity in their current point positions, that’s why.

So, all in all, yes, the nature of what EP1 can at all do in its interaction with AP1 is still, in part, being governed by the deterministically evolved state of the initial, single, 2-particle entangling wavefunction. [That’s how even the MSQM folks put it. Actually, it’s a 2-particle part of the 4-particle system wavefunction.]

So, the net result at the +1000.0 km point is that, when seen in an approximate manner, EP1 seems to be interacting with AP1 (or, AP1 with EP1) in a manner that seems to be completely independent of how  EP1 interacts with AP2 and EP2—i.e., there is almost no interaction at all.

Similarly, the net result at the -1000.0 km point is that, when seen in an approximate manner, EP2 seems to be interacting with AP2 (or, AP2 with EP2) in a manner that seems to be completely independent of how  EP2 interacts with AP1 and EP1—i.e., there is almost no interaction at all.


4. The paradox we have to resolve:

We thus have two apparently contradictory ways of summarizing the same situation.

  • Since the two EPs have gone so farther apart, and since AP1 and AP2 never “interacted” strongly with each other (or with EP1 and EP2), therefore, EP1’s behaviour should be taken to be “independent” of EP2’s behaviour, when they are at the \pm 1000.0 km points. Their behaviour should have nothing in common.
  • Yet, since EP1 and EP2 were initially entangled, and since both their respective state-evolutions were governed by the common, single wavefunction entangling them, therefore, their behaviour must also have something in common.

Got it?

How do we resolve this paradox?


5. What kind of things actually happen:

Suppose the interaction of AP1 with EP1 is such that we can say that it is EP1’s spin-property which gets measured by AP1.

Here, imagine an assemblage of a large number of particles, acting as a spin-detector, in place of AP1. (We will continue to call it a single “particle”, for the sake of simplicity.)

Suppose that the measurement outcome happens to be such that EP1’s spin is measured at AP1 to be “up” with respect to a certain z-axis (applicable to the entire universe).

Now, remember, measurement is a probabilistic process. Therefore, the correct statement to make here is:

If (and when) AP1 measures EP1’s spin, the outcome is one (and only one) of the two possibilities: either “up”, or “down.”

In other words, it is always possible that EP1 interacts with AP1, and yet, the action of EP1’s spin influencing some large-scale configuration changes within AP1 (an event which we call “measurement”) never actually comes to occur. This is possible to. However, if a measurement does occur, then the outcome is one and only one of those two possibilities.

Now suppose, to take the description further, that AP1 does indeed end up measuring EP1 spin. (That is to say, suppose that such a thing comes to occur as a physical fact, an irreversible change in the universe.)

Assume further—for the sake of pedagogic simplicity—that the EP1’s spin is measured to be “up” (and not “down”).

Suppose further that the interaction of AP2 with EP2 is such that we can say that it is EP2’s spin which is the property that gets measured by AP2—if there at all occurs a measurement when EP2 is near or at AP2. Again, remember, measurement is a probabilistic process. The correct statement now to make is:

If (and when) AP2 ends up measuring EP1’s spin, then, since the EP1 and EP2 are entangled, the outcome at the -1000.0 km point has to be: “down” (because we assumed that EP1’s spin was measured as “up” at the +1000.0 km point).

Note, the spin of EP2 is certain to be measured “down” in our case—provided it at all gets measured during the interaction of EP2 with AP2.

But note also that since AP2’s state is not entangled with AP1’s (they were too far away to begin with), just because AP1 does end up measuring EP1’s spin (as “up”) does not mean that AP2 will also necessarily measure EP2’s spin at all—despite the interaction they necessarily go through. (All four particles are, in reality, interacting. Here, AP2 and EP2, being closer, are interacting strongly.)


6. The game that the MSQM people play (with you):

Now, the whole game that MSQM (mainstream QM) physicists play with you is this.

They don’t explain to you, but it is true, that:

The fact

“AP1 interacted with EP1 to measure its spin state”

does not necessitate the conclusion

“AP2 must also measure the spin-state of EP2 in the same experimental trial“.

The latter is not at all necessary. It does not have to physically take place.

If so, then what can we say here? It is this:

But if (and when) AP2 does measure the spin-state (and no other measurable) of EP2, then the measured spin will necessarily be “down”.

The preceding statement is true.

This is because angular momentum conservation implies that if any one of the spins is measured as “up”, then the other has to get measured as “down”. This necessity is built right in the way the single entangling wavefunction is composed from the two 1-particle wavefunctions. It is the property of the initial entangling wavefunction that it has zero net spin-angular momentum. It gets reflected also in the measured read-outs with equal probability if two measurements at all take place at symmetrically far away points, so that the local patterns of the common wavefunction themselves must be symmetrically opposite. (Only a symmetrically opposite pair of 1-particle wavefunctions can together conserve angular momentum for the 2-particle entangling wavefunction.)

The slickness of MSQM people consists of refusing to make you realize that the common (entangling) wavefunction must, of necessity, arise from such symmetry conditions as just mentioned, and that it must also evolve perfectly preserving this symmetry throughout the Schrodinger evolution. Further, their slickness consists of making you believe that if AP1 does indeed physically measure EP1’s spin as “up”, then AP2 is also mandated to physically end up measuring EP2’s spin, in each and every trial.


7. How the MSQM people maintain their slickness, while presenting experimental data:

When they do experiments, they actually send entangled particles apart, and measure their respective spins at two equal distance apart and similarly tilted detector-positions.

What their raw data shows is that when the AP1 measures EP1 to be in the “up” state, AP2 may not always show any measurement outcome at all. Also, for all other three possibilities. (AP1 says “down”, nothing at AP2. AP1 says nothing, AP2 says “up”. AP1 says nothing, AP2 says “down”.)

What the MSQM folks do is, effectively, to simply drop all such observations. They retain only those among the raw data-points which have one of the two results:

  • EP1 actually measured (by AP1) to have the spin “up”, and EP2 actually measured (by AP2) to have the spin “down” in a single trial, or
  • EP1 actually measured (by AP1) to have the spin “down”, and EP2 actually measured (by AP2) to have the spin “up” in some other, single, trial.

So, their conclusion never do highlight the previously mentioned four possibilities.

No, they are not doing any data-fudging as such.

The data they present is the actual one, and it does support the theory.

But the as-presented data is not all the data there is—it’s not all there is to these experiments. And, so, it is not the complete story.

And, the part dropped-out of the final datasets sure tells you more about demystifying entanglement than the part that is eventually kept in does. It is this same—mystifying—data that gets presented in conferences, summarized in textbooks and pop-sci articles (including those on the Quanta Magazine site), and of course, in the pop-sci books (by all authors writing on this subject [Google (verb)!]).

Just hold the above discussion in mind, and see how it straightens out everything.


8. Summary of what we saw thus far:

A measured value is decided only in an act of measurement—if any measurement at all occurs during the ongoing interaction of a particle and a detector.

The respective probabilities for each of the two possible outcomes (in the spin “up” or “down” type of two-state situations) have already been decided by the deterministic time-evolution (the Schrodinger-evolution) of the initial, 2-particle entangling, part of the 4-particle system wavefunction.

If the AP1 detector is oriented to measure EP1’s spin as “up” with a P % probability, then EP2’s spin is necessarily “shaped” by the same wavefunction as to be inclined to be measured by AP2 as “down”, with the same P % probability—provided that:

  1. AP2 was in all respects identical to AP1 (including their orientations—say, placed in an exact mirror-symmetrical arrangement), and
  2. AP2 does at all end up measuring EP1. It might not, always.

Existence of an entanglement between EP1 and EP2 does not necessitate that if AP1 measures the spin-property of EP1 (w.r.t. a certain axis), then AP2 for the corresponding EP2 (coming from the same trial) must also measure the spin-property of EP2 (w.r.t to the same axis).

But if AP2 undergoes a measurement process too, then the outcome is determined, due to the commonality of the single entangling wavefunction (including the spinor function) which is shared by EP1 and EP2. And it works out as: if the first is “up”, the second must be “down”, or, vice versa.

 


Note: I am not sure if I noted in the NY resolutions post or not. But I’ve decided that I may not add a songs section every time—but sure enough I will, if one is somewhere at the back of the mind.

This topic is not difficult, but it is intricate. Easy to make typos. Also, very easy to make long-winding statements, not find the right phrases, ways of expression, metaphors, etc. So, I think I should come back and revise it after a few days. I should also give titles to the sections and all … But, anyway, in the meanwhile, do feel free to read.


History:

— 2020.01.03 12:15 IST: Initial posting.
— 2020.01.03 13:44 IST: Correction of typos, misleading statements. Addition of section titles, and a further section on the comparison with classical diffusion systems.
— 2020.01.03 15:33 IST: Added the section: “One last comment…”.
— 2020.01.03 17:03 IST: Further additions/corrections. Now am going to leave this post in this shape for at least a couple of days or more. But looks like it’s mostly done.
— 2020.01.04 14:18 IST: Nope. In simplifying everything as much as possible, it seems to me that I ended up getting off the track, and thus wrote something which is, I now think, was wrong. The error was confined to section 9.

The wrong part was important. I will have to look into the maths involving the spin property once again (and in fact learn more about it and many-particle systems in general), and further, I will have to integrate it with my new approach. Only then would I be able to come back on this point. It may take me quite some time to finalize such an integration, may be weeks, may be months.

My plan all so far was to leave the spin property of QM systems alone, and present the new approach only for spin-less systems. (That’s what I did in the Outline document too.) Yet, yesterday, somehow, I got tempted at covering the spin and the new approach together, right on the fly, and ended up writing a bit inadvertently adopting an ensembles-based interpretation. I thus sounded a bit too much like the Bohmian approach than what my approach actually should be like. (I know it from some other points of view that there are going to be important differences in my approach and the Bohmian one.)

All this, I realized, completely on my own, without any one prompting me or providing any feedback (not an indirect one, say as through the “follow-up” sort of channels), only this morning. So, I am deleting what earlier was the section 9.

The section 10 was not wrong as such. But its contents were prompted only by the topic covered in section 9. That’s why, though section 10 was essentially correct, I am also deleting it. I will cover both their topics in future.

In case any one is at all interested in having the original (erroneous) version of this post (with sections 9. and 10.), then I could share it. Feel free to approach me via an email or a comment.

As to any other errors/ambiguities/ill-expositions, I will let them be. I am done with this post. Time to move on.