I Exploring Non-locality in Many-Worlds Interpretation (MWI)

[Derek P]

In another thread, @Nugatory asked
"how exactly does it turn out that both spacelike-separated observers end up in the same world without some appeal to non-locality?"

Worlds are defined by their observed macroscopic values. They separate through decoherence. It is a physical process. Since decoherence is so fast, it effectively propagates as a sphere expanding at the speed of light. Thus the whole universe does not split, the splits are in the future light cones from the two observations. Only where the cones intersect have both worlds split. Alice can know that Bob's worlds have separated, her world is, however, not yet split by anything over at Bob's place.

 

[DrChinese]

Worlds are defined by their observed macroscopic values. They separate through decoherence. It is a physical process. Since decoherence is so fast, it effectively propagates as a sphere expanding at the speed of light. Thus the whole universe does not split, the splits are in the future light cones from the two observations. Only where the cones intersect have both worlds split. Alice can know that Bob's worlds have separated, her world is, however, not yet split by anything over at Bob's place.

Bob's world splits into 2 when he measures: one where you get (say) spin up and and another where you get spin down.

Alice's world splits into 2 when she measures: one where you get (say) spin up and and another where you get spin down.

Presumably these 2 expanding spheres (as you describe them) eventually meet. How do they know to sync up and produce the expected statistics? Especially since there must be many other expanding spheres meeting up all the time that are not entangled particles?

 

[Demystifier]

Locality and non-locality are properties of the 3-dimensional world. But according to MWI, there is no 3-dimensional world. According to MWI, the only object that exists is the wave function, which does not live in a 3-dimenional world. It lives in a 3N-dimensional world, where N is the number of "particles". Hence MWI is neither local nor non-local. It is alocal. [https://arxiv.org/abs/1703.08341 Sec. 3.1]

According to MWI, the 3-dimensional world is an illusion, very much like a 2-dimensional shadow is an illusion emerging from a real 3-dimensional object. Our 3-dimensional world sometimes look local (as in the classical limit), sometimes non-local (due to quantum entanglement), but both locality and non-locality are mere illusions, low-dimensional shadows of the multi-dimensional reality.

 

[Derek P]

Bob's world splits into 2 when he measures: one where you get (say) spin up and and another where you get spin down.

Alice's world splits into 2 when she measures: one where you get (say) spin up and and another where you get spin down.

Presumably these 2 expanding spheres (as you describe them) eventually meet. How do they know to sync up and produce the expected statistics? Especially since there must be many other expanding spheres meeting up all the time that are not entangled particles?

I don't understand you. The correlations already exist.

 

[Derek P]

Locality and non-locality are properties of the 3-dimensional world. But according to MWI, there is no 3-dimensional world. According to MWI, the only object that exists is the wave function, which does not live in a 3-dimenional world. It lives in a 3N-dimensional world, where N is the number of "particles". Hence MWI is neither local nor non-local. It is alocal. [https://arxiv.org/abs/1703.08341 Sec. 3.1]

According to MWI, the 3-dimensional world is an illusion, very much like a 2-dimensional shadow is an illusion emerging from a real 3-dimensional object. Our 3-dimensional world sometimes look local (as in the classical limit), sometimes non-local (due to quantum entanglement), but both locality and non-locality are mere illusions, low-dimensional shadows of the multi-dimensional reality.

Non-local wavefunctions are trivially expressible as a superposition of local, separable ones. So whilst "the" wavefunction may be non-local, it is always a superposition of 3-dimensional local states - ones that eventually go to make up the phenomenal worlds of MWI.

 

[Lord Jestocost]

So, as Nick Herbert has remarked: “The moral of Everett’s tale is plain: if you don’t want to split, stop looking at attribute-laden systems.”

 

[martinbn]

...According to MWI, the only object that exists is the wave function, ...

I must say i still don't understand what this means. What does it mean to exist when it refers to a wave function?

 

[DrChinese]

I don't understand you. The correlations already exist.

There is no requirement of that. Entangled particles need not share a common past. They don't even need to have existed at the same time. But they will eventually have a future overlapping light cone.

Which is precisely why I am asking how this works.

 

[Derek P]

There is no requirement of that. Entangled particles need not share a common past. They don't even need to have existed at the same time. But they will eventually have a future overlapping light cone.
Which is precisely why I am asking how this works.

I don't see what the origin of the entanglement has to do with the correlations.

 

[Derek P]

I must say i still don't understand what this means. What does it mean to exist when it refers to a wave function?

It means that something exists that is fully described and not over-described by the wavefunction. For economy we equate the real entity with its description.

 

[martinbn]

It means that something exists that is fully described and not over-described by the wavefunction. For economy we equate the real entity with its description.

How is that specific to MWI?

 

[Derek P]

How is that specific to MWI?

I have no idea. Is it?

 

[DrChinese]

I don't see what the origin of the entanglement has to do with the correlations.

I'll try to say it another way. There are correlations in some worlds of Alice and Bob that match statistical expectations. How do the worlds know to match up properly when Alice and Bob's "expanding spheres" meet up? We have agreed that the source of the entanglement doesn't matter.

 

[zonde]

How do the worlds know to match up properly when Alice and Bob's "expanding spheres" meet up?

I would say that MWI wolds need a "snap" feature in addition to "split".
If I remember correctly there was something about it in this paper http://philsci-archive.pitt.edu/504/2/cracow.pdf

 

[Derek P]

I'll try to say it another way. There are correlations in some worlds of Alice and Bob that match statistical expectations. How do the worlds know to match up properly when Alice and Bob's "expanding spheres" meet up? We have agreed that the source of the entanglement doesn't matter.

They are already "matched up". Nothing new happens when they meet.

 

[Derek P]

I would say that MWI wolds need a "snap" feature in addition to "split".
If I remember correctly there was something about it in this paper http://philsci-archive.pitt.edu/504/2/cracow.pdf

TLDR. Please make your point clearly as I have no idea what a "snap" feature is

 

[DrChinese]

1. They are already "matched up". Nothing new happens when they meet.

Nugatory asked: "how exactly does it turn out that both spacelike-separated observers end up in the same world without some appeal to non-locality?"

You answered: "2. They separate through decoherence. It is a physical process. Since decoherence is so fast, it effectively propagates as a sphere expanding at the speed of light. Thus the whole universe does not split, the splits are in the future light cones from the two observations. "

Your explanations contradict. One says they start "matched up" (which is nonlocal); the other says they meet in a future light cone (where the speed of light is a limiting factor).

So, which?

 

[akvadrako]

The Deutsch-Hayden interpretation of WMI shows that it's local in spacetime. There have been some criticisms over the years, but after a thorough review of this topic, I'd say it's well established by now. Deutsch addresses most of the criticisms in his most recent paper on the topic.

The main idea behind WMI is that the wave function is what's real, so you must look at it's evolution to see how the splits match up (how they interfere).

 

[Derek P]

"matched up" (which is nonlocal)

Why on Earth do you say that? Please don't tell me Bell's Theorem!

 

[PeterDonis]

It would probably be helpful to this discussion to look at the math. Say we have two entangled qubits in the singlet state, and two measuring devices A and B (which stand for the obvious names everybody always uses in these scenarios). The initial state of the system is then

$$
\Psi_i = \frac{1}{\sqrt{2}} \left( |\uparrow>_1 |\downarrow>_2 - |\downarrow>_1 |\uparrow>_2 \right) |R>_A |R>_B
$$

where the meanings of the kets should be reasonably obvious ("R" stands for "ready to measure").

The two measurements are realized by two unitary operators $U_1$ and $U_2$ (for simplicity, we assume that the Hamiltonian is the identity other than these two operators, i.e., that time evolution outside the measurements leaves all states the same) which induce the following entanglement transitions

$$
U_1: \ \ |\uparrow>_1 |R>_A \rightarrow |\uparrow>_1 |U>_A \ \ ; \ \ |\downarrow>_1 |R>_A \rightarrow |\downarrow>_1 |D>_A
$$

$$
U_2: \ \ |\uparrow>_2 |R>_B \rightarrow |\uparrow>_2 |U>_B \ \ ; \ \ |\downarrow>_2 |R>_B \rightarrow |\downarrow>_2 |D>_B
$$

The final state of the system will therefore be

$$
\Psi_f = \frac{1}{\sqrt{2}} \left( |\uparrow>_1 |\downarrow>_2 |U>_A |D>_B - |\downarrow>_1 |\uparrow>_2 |D>_A |U>_B \right)
$$

(I have left out decoherence in all this by not including any kets representing the environment; that could be put back in but would not add anything useful to the analysis for this discussion. If you like, treat the final state as the state after everything has decohered.)

Now, in MWI terminology, the two terms in $\Psi_f$ represent two "worlds". But there is nothing to "match up" in these two worlds: they already include, by construction, the correlations between the A and B measurements, because each term already says that the two results are opposite. So there is nothing else that needs to happen for those correlations to be there; they're there because of how the unitary operators that realize the measurement interactions affect the state. And each of those operators is local: it only acts on the part of the state that is at the measurement location (A or B).

Given the above, I think the answer to @Nugatory's original question is that the unitary evolution of the overall state enforces the correlation between the A and B measurements; each "copy" of A and B must find out (once an ordinary light-speed or slower communication has happened between the two) that the other measurement got an opposite result to theirs, because that's built into the term in $\Psi_f$ that describes their "world". And the process that produces this is perfectly local. What it isn't, at least in the terminology of Bell's Theorem, is "realistic"; at least, that's the usual description of how the MWI evades the "non-locality" horn of the dilemma posted by theories that violate the Bell inequalities.

 

[Derek P]

@PeterDonis Thanks.

 

[bhobba]

I must say i still don't understand what this means. What does it mean to exist when it refers to a wave function?

Now that is something a philosopher would argue about for a lifetime.

All I will say is its like probability - does probability exist? For a roulette wheel as found out by some enterprising physics students it is an objective property of the roulette wheel:
https://en.wikipedia.org/wiki/The_Eudaemonic_Pie

But like interpretations of QM who really knows. You can just about take any view you like. In many worlds it's real in some sense - but I have to say having studied that interpretation from Wallace I could not explain to you in what sense. Strange.

Thanks
Bill

 

[Demystifier]

I must say i still don't understand what this means. What does it mean to exist when it refers to a wave function?

Do you know what does it mean that a particle "exists" in classical mechanics, or that electromagnetic field "exists" in classical electromagnetism, or that spacetime curvature "exists" in general relativity? It means that it is there in the ontological sense, even if we neither measure nor calculate it. According to MWI, the claim that wave function "exists" in QM means exactly the same.

 

[martinbn]

Do you know what does it mean that a particle "exists" in classical mechanics, or that electromagnetic field "exists" in classical electromagnetism, or that spacetime curvature "exists" in general relativity? It means that it is there in the ontological sense, even if we neither measure nor calculate it. According to MWI, the claim that wave function "exists" in QM means exactly the same.

That is what I understand by exist, but it requires that you have a space-time first. Look, you say that it means that it is there in the ontological sense. And earlier you said that according to MWI there is no three dimensional world. Also all of the above, particles and fields, can have energy and momentum and can interact with each other and so on. None of that makes sense about the wave function.

I suppose my main problem is that the statement "object A exists" means that it exists in space-time and interacts with other objects. What does all that mean for the wave function?

 

[Demystifier]

That is what I understand by exist, but it requires that you have a space-time first. Look, you say that it means that it is there in the ontological sense. And earlier you said that according to MWI there is no three dimensional world. Also all of the above, particles and fields, can have energy and momentum and can interact with each other and so on. None of that makes sense about the wave function.

I suppose my main problem is that the statement "object A exists" means that it exists in space-time and interacts with other objects. What does all that mean for the wave function?

You are right, the wave function does not exist in space-time. And according to MWI, at the fundamental level there is no space-time, there is only wave function. If it doesn't make sense to you, then blame MWI, not me.

Anyway, when I try to make sense of MWI, I use the analogy with Plato's cave where the things one observes are mere shadows of the true things.

 

[martinbn]

You are right, the wave function does not exist in space-time. And according to MWI, at the fundamental level there is no space-time, there is only wave function. If it doesn't make sense to you, then blame MWI, not me.

OK, but then it cannot be the same as the existence of the electromagnetic field. Also it is an abstract notion. How can ontological existence of everything else, particles and so on, come from that!

p.s. I am not blaming anyone, just trying to understand what the statement is.

 

[Demystifier]

How can ontological existence of everything else, particles and so on, come from that!

See the analogy with Plato's cave I mentioned in the revised post above. According to MWI, the other things (particles, fields, spacetime, ..., tables, chairs, ...) do not exist in the fundamental ontological sense, but only in some emergent sense.

 

[martinbn]

OK, is the wave function a function? If yes, what is the domain of that function? Without any reference to space, time, particles, fields, chairs.

 

[bhobba]

OK, but then it cannot be the same as the existence of the electromagnetic field.

I thought Wheeler and his well known PhD student called the existence of EM fields into question - but of course at a cost. Physics gives and takes away.

The reason we like to think EM fields are real is Noether. Since it has a Lagrangian it has momentum and energy that most think as real and conserved - of course as that PhD student explained in his famous lectures even the simple concept of a flat table breaks down when looked at carefully. This is the very issue I have with some people that like to try and pin down what science is - it has a way of turning around and biting you in the rear.

The more I keep reading it I think Dirac is closer to the truth than many, maybe even that PhD student in his more philosophical moments:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.485.9188&rep=rep1&type=pdf

Thanks
Bill

 

[Demystifier]

OK, is the wave function a function? If yes, what is the domain of that function?

Yes, it is a function $\mathbb{R}^{3n+1} \rightarrow \mathbb{C}$, where $n\in \mathbb{N}$.

 

[bhobba]

OK, is the wave function a function? If yes, what is the domain of that function? Without any reference to space, time, particles, fields, chairs.

Oh come on - you are an advanced enough mathematician to know about Rigged Hilbert Spaces and how that branch of math would answer such a question. As I said these things are not that easily pinned down. Even the fields of QFT are not as simple as usual fields like EM fields - they are quantum operators. The mind boggles about the 'reality' of those.

Thanks
Bill

 

[martinbn]

Yes, it is a function $\mathbb{R}^{3n+1} \rightarrow \mathbb{C}$, where $n\in \mathbb{N}$.

Now you are cheating. Why 3 and 1 in the 3n+1 and not say 7 and 2? And what are they anyway without saying anything about space and time?

 

[martinbn]

Oh come on - you are an advanced enough mathematician to know about Rigged Hilbert Spaces and how that branch of math would answer such a question. As I said these things are not that easily pinned down. Even the fields of QFT are not as simple as usual fields like EM fields - they are quantum operators. The mind boggles about the 'reality' of those.

Thanks
Bill

No, I am not asking about subtleties in the use of functional analysis in quantum mechanics. It's just that if space-time is to come after in some emergent way, then what is the wave function a function of? Or what Hilbert space (or rigged Hilbert space) it is an element of?

 

[Demystifier]

Now you are cheating. Why 3 and 1 in the 3n+1 and not say 7 and 2? And what are they anyway without saying anything about space and time?

Good question! Any physical theory has some free parameters that must be chosen such that the theory fits the observations. The numbers 3 and 1 fit the fact that we observe 3-dimensional space and 1-dimensional time.

 

[Demystifier]

No, I am not asking about subtleties in the use of functional analysis in quantum mechanics. It's just that if space-time is to come after in some emergent way, then what is the wave function a function of? Or what Hilbert space (or rigged Hilbert space) it is an element of?

Mathematical physicists tend to overlook the essence of physics and pay attention to mathematical subtleties. I like you for being the exact opposite of that, for being a physical mathematician.
(Something like Roger Penrose.)

 

[bhobba]

Now you are cheating. Why 3 and 1 in the 3n+1 and not say 7 and 2? And what are they anyway without saying anything about space and time?

Yes I was glib in my answer, and Demytifyer is right my math background sometimes smacks me in the face . I don't want to say I was a smart ass - but I was on rereading it.

Cosmology isn't really my thing but my reading of some of the latest stuff is it started from this thing called the false vacuum that has its own parameters - that's the parameters used in labeling however you want to expand the state. Just a thought. But I think its more QFT than QM which makes it harder to decipher. I guess right now its a bit of a mystery. Maybe someone more conversant in cosmology can chime in.

Thanks
Bill

 

[DrChinese]

Why on Earth do you say that? Please don't tell me Bell's Theorem!

You said they were "matched up" at spacetime separated spots. How?

 

[PeterDonis]

You said they were "matched up" at spacetime separated spots. How?

Based on my post #20, I think the answer is that there is no "matching" because there doesn't have to be.

It might also be worth pointing out that neither A nor B can know how their results match up until ordinary light-speed-or-slower information has traveled between them; nothing they can measure or observe before that will tell them what result was obtained at the other measurement, or even whether the other measurement was made or not. So although there is information "stored" in the wave function about which result was obtained, that information is not accessible until ordinary classical information arrives to allow it to be "decoded". Deutsch, in a paper on information flow in entangled quantum systems, calls this "locally inaccessible information":

https://arxiv.org/pdf/quant-ph/9906007.pdf

 

[DrChinese]

Based on my post #20, I think the answer is that there is no "matching" because there doesn't have to be.

It might also be worth pointing out that neither A nor B can know how their results match up until ordinary light-speed-or-slower information has traveled between them; nothing they can measure or observe before that will tell them what result was obtained at the other measurement, or even whether the other measurement was made or not. So although there is information "stored" in the wave function about which result was obtained, that information is not accessible until ordinary classical information arrives to allow it to be "decoded". Deutsch, in a paper on information flow in entangled quantum systems, calls this "locally inaccessible information":

https://arxiv.org/pdf/quant-ph/9906007.pdf

Thanks for these points.

I keep trying to figure out how MWI can be local only, and it still makes no sense to me. If Alice makes a measurement, she gets V in one world and H in another. In the new V world, every other spot knows it is to act as if Alice got V. So that makes Bob's later measurement match the appropriate statistics. In the new H world, every other spot knows it is to act as if Alice got H. So that also makes Bob's later measurement match the appropriate statistics.

I just don't see how that is local, if Alice and Bob are far apart. They must be accessing the same information so that everything sorts into the proper world.

 

[PeterDonis]

I just don't see how that is local, if Alice and Bob are far apart.

In the description I wrote down in post #20, the non-locality is in the quantum state; each of the terms obviously entangles two subsystems that are spacelike separated.

However, the state I wrote down in #20 is in the Schrodinger picture. Deutsch's claim in the paper I linked to is basically that, if you use the Heisenberg picture instead, the non-locality disappears; all of the correlations go into the measurement operators that describe each qubit separately, and which get "carried along" by the qubits so that they can be viewed as local.

 

[DrChinese]

Deutsch's claim in the paper I linked to is basically that, if you use the Heisenberg picture instead, the non-locality disappears; all of the correlations go into the measurement operators that describe each qubit separately, and which get "carried along" by the qubits so that they can be viewed as local.

I saw that in the paper you cited, but it seems like some fancy footwork to call that "local". Seems global to me.

 

[Derek P]

Thanks for these points.

I keep trying to figure out how MWI can be local only, and it still makes no sense to me. If Alice makes a measurement, she gets V in one world and H in another. In the new V world, every other spot knows it is to act as if Alice got V. So that makes Bob's later measurement match the appropriate statistics. In the new H world, every other spot knows it is to act as if Alice got H. So that also makes Bob's later measurement match the appropriate statistics.

I just don't see how that is local, if Alice and Bob are far apart. They must be accessing the same information so that everything sorts into the proper world.

The correlations in each world are of the "pair of gloves" kind. There is no need to share information.

It's very simple. The original state of the entangled pair is
a|h>₁|v>₂ + b|v>₁|v>₂ + c|v>₁|v>₂ + d|h>₁|h>₂.
All four product terms are needed because Alice and Bob may use different orientations and there are four possible measurement results.

Consider just one of the product states, for example the first. Unitary evolution is deterministic. So by interacting with this state Alice will definitely detect H and Bob will definitely detect V. It's a predetermined matching pair of gloves! The result is an HV world. The other terms give the VH, VV and HH worlds.

 

[DrChinese]

The correlations in each world are of the "pair of gloves" kind. There is no need to share information.

It's very simple. The original state of the entangled pair is
a|h>₁|v>₂ + b|v>₁|v>₂ + c|v>₁|v>₂ + d|h>₁|h>₂.
All four product terms are needed because Alice and Bob may use different orientations and there are four possible measurement results.

Consider just one of the product states, for example the first. ...

There are no coefficients a/b/c/d that can reproduce the quantum expectation values because entangled pairs are not Product states. An entangled pair is hh + vv (or hv+vh). Now I realize that you are attempting to account for the basis being different for Alice and Bob, but their results are not independent of each other. When expressed in that manner, there are correctly still only 2 terms - and not 4 as you show.

It should be obvious that the "pair of gloves" analogy cannot work any better here than in other interpretations.

 

[PeterDonis]

There are no coefficients a/b/c/d that can reproduce the quantum expectation values because entangled pairs are not Product states.

I'm not sure what you're trying to say here. Entangled pairs are linear combinations of product states with appropriate coefficients--basically you choose two of a/b/c/d to be 1 (or you can throw in phase factors if needed) and the other two to be zero.

 

[PeterDonis]

The original state of the entangled pair is
a|h>₁|v>₂ + b|v>₁|v>₂ + c|v>₁|v>₂ + d|h>₁|h>₂.

I think you have a typo in here; shouldn't it be a|h>₁|v>₂ + b|v>₁|h>₂ + c|v>₁|v>₂ + d|h>₁|h>₂?

 

[Derek P]

There are no coefficients a/b/c/d that can reproduce the quantum expectation values

Sure there are. I can even tell you what they are within a phase factor
a = ½ cos(α-β)
b = ½ sin(α-β)
c = ½ cos(α-β)
d = ½ sin(α-β)

because entangled pairs are not Product states.

Umm, a|h>1|v>2 + b|v>1|v>2 + c|v>1|h>2 + d|h>1|h>2 isn't a product. It's the sum of products.

An entangled pair is hh + vv (or hv+vh).

Yes when measured in the same basis. And what @PeterDonis said. It's just that two coefficients are zero.

Now I realize that you are attempting to account for the basis being different for Alice and Bob, but their results are not independent of each other.

That's right. They are trigonometric functions of the two angles of measurement. The difference between them to be precise.

When expressed in that manner, there are correctly still only 2 terms - and not 4 as you show.

Only when α=β. To get from that case to the general case, one or other "h" must be resolved in the other observer's basis.

It should be obvious that the "pair of gloves" analogy cannot work any better here than in other interpretations.

Well no. The pair of gloves analogy works just fine in any interpretation which manages to coax a mixed state out of the superposition, whether it's a proper one as in Copenhagen or an improper one as in Many Worlds..

 

[DrChinese]

I'm not sure what you're trying to say here. Entangled pairs are linear combinations of product states with appropriate coefficients--basically you choose two of a/b/c/d to be 1 (or you can throw in phase factors if needed) and the other two to be zero.

Yes, 2 are zero. And with phase factors, you could also expand those to get 4 terms. Since the phase term is dependent on the specific difference between the angle settings (basis), how does the world know to split in a fashion that reproduces the statistics?

In other words: the coefficients can reproduce the statistics (if you vary them suitably) but you must know both Alice and Bob's settings first. Then the formula is no longer local.

 

[DrChinese]

Sure there are. I can even tell you what they are within a phase factor
a = ½ cos(α-β)
b = ½ sin(α-β)
c = ½ cos(α-β)
d = ½ sin(α-β)

Now the terms are dependent on something which is non-local, which is what we sought to avoid. Sorry I don't think I was clear on this point.

Looking at it another way: Bob's setting might not have even been determined at the time, so no way the values could be determined.

 

[PeterDonis]

the coefficients can reproduce the statistics (if you vary them suitably) but you must know both Alice and Bob's settings first. Then the formula is no longer local.

Ah, got it.

 

[Derek P]

I think you have a typo in here; shouldn't it be a|h>₁|v>₂ + b|v>₁|h>₂ + c|v>₁|v>₂ + d|h>₁|h>₂?

Yes, thanks. Though to be consistent with my later post, it's the third state that should be changed.