Will quantum computers ever be possible?

[colorSpace]

There is no known fundamental requirement to a "limit on information" that a mathematical entity representing physical entities is to have. As I pointed out, a single point particle in a classical 3-dim space is already represented by a mathematical entity which requires an infinite amount of bits (namely a point in E^3, isomorphic to 3 real numbers).

I consider it an advantage of MWI to show the *hugeness* of hilbertspace as postulated by the quantum formalism (remember that my aim, with MWI, is to "get a better feeling of the workings of the quantum formalism", not to have a postulated "true worldpicture").

What I try to do is to show that it is conceivable to replace the global wavefunction (the vector in hilbert space describing the "state of the universe") by a set of other entities, the "kets of individual systems + label structure", so that we can consider all of these entities as localised in space, and to have their evolution be determined by only OTHER entities in space that are at the same spacetime locality. IF I can find such a way of building these entities and dynamics, then I am justified in claiming that the global wavefunction dynamics represents a local dynamics. I tried to illlustrate that with some analogy: the dynamics in Hamiltonian phase space "looks global", but it can be re-written in such a way that it only uses "localised entities" and "local interactions". I'm trying to do the same here for quantum dynamics.

There is a difference of many orders of magnitude between 3 real numbers and 20 Million complex numbers plus 10 Million unique id's! And one could debate that even the 3 real numbers for the 3D coordinates are "carried" along by the particle, as you have said happens for the A/B, x and y information.

I'm not really impressed by the "hugeness of hilbertspace", you have mentioned already that it has many dimensions. At this point it seems to be a) a purely mathematical construct for the convenience of physicists to do calculations, and b) you haven't shown yet (at least not in a way that I would be able confirm or reject) how the many dimensions of hilbertspace can be used to make the case that the information can be "carried" along in a way that maps to a 3D-local physical explanation. I'm not debating mathematical possibilities, from the beginning not, but the physical possibility of storing and handling this information. You haven't shown to me, yet, how this information can be stored and then used in a way that could be called 3D-local. "Dimensions" by themselves do not constitute usable information.

I'm especially curious about the 10 Million unique id's!

If you think I concede at this point, think again. We have merely come to the point where we can discuss the question I was asking, but there is no answer to it in sight yet, as far as I am concerned.

Hilbertspace is common to all interpretations of quantum mechanics, and other interpretations haven't been able to use it for a local explanation. You have shown me how MWI could attempt to provide a local explanation, *if* it could carry the information in a local fashion, *and* if it could then be used to do the "pairing-up" in a local fashion. But you haven't shown me any *specific* way yet how *either* of the latter would be possible in a meaningful physical way that makes sense, and that could be mapped to 3D-space in a way in which it can then be decided that it could be called "local".

You have only *claimed* that it *should* be possible because hilbertspace has so many dimensions.

I don't see why 3 dimensions shouldn't be enough. The problem is to come up with a way to store that information, even if it is just in the 'coordinates', that works as part of a meaningful physical process. BTW, is hibertspace euclidian, or curved? Is it generally acknowledged that hilbertspace has physical reality, or is that a specific theory?

Yet those were the objections I had from the beginning.

So you have merely reached the starting point for discussing the objections which I had from the beginning.

What else do I need to say to clarify that, if it isn't clear yet?

No, that is not true. When we did the split A-B, we "picked" a reference axis for the two entangled particles in the source. We saw that we could have picked any axis, but when assigning the labels A and B, we had to choose one of them. So we have arbitrarily associated, with the |u+A> state, a particular direction in space, shared with |v-A>.

Now, the numbers x and y are a result, PURELY LOCALLY AT BOB'S, of his measurement basis (his axis of analyser), and the axis fixed in the A/B-label. So, x and y are "generated" locally at bob's and he doesn't need anything about alice to do so.

In the same way, the numbers r and s are the result, purely locally at alice, of her measurement basis (her analyser axis) and the fixed axis in the A/B label (which comes to her with the v-ket).

So x,y,r, and s are determined locally.

Your response is evading a very simple point. I've said that the term (-xs + yr) is non-local as long as x and y refer to physical states at one location, and r and s refer to states at a different location. That changes when x, y, r and s are brought to the same location. It became clear only recently that the term (-xs + yr), in relation to A and B, is information that may resolve the problem. However you haven't shown yet how that is supposed to happen, and certainly not in a way that I could form an opinion about whether that might be a *physical* and *local* possibility, rather than just a theoretical one plainly *assuming* infinite storage capabilities, just because hilbertspace has so many dimensions.

Yes, of course it is "additional information". It is the "algebraic information" that is normally included in the form of the wavefunction, which must now be "distributed" amongst all its constituents. As such, it will determine part of the "local dynamics" (which would correspond to simple algebraic operations on the global wavefunction, such as multiplications, distributivity and complex sums), which has now to take care of this locally.

But the point is that we CAN construct such mathematical objects associated with the different localised states, and that at no point, we need to have dynamical rules which need "information from states at different locations" to have a dynamical change (of the kets, or of the additional information). In other words, you can build mathematical structures which are all the time indexable over space, with a dynamical rule which is also only function of the structures at the same locality, and which is isomorphic to the global wavefunction dynamics. If you can do that, then the global dynamics represents a local dynamics, and that was the aim of the exercise.

"Algebraic information"? Which "exercise"? You need to specify physical means in order to have a physical theory. I have from the beginning doubted that there is a *physical* way to carry that possibly huge amount of information along, and that it can then be used to do the "pairing-up"

Yes, so ? Hilbert space is HUGE. The "information" carried by the wavefunction (a point in hilbert space) is enormous. As such, you shouldn't be surprised that if you scatter this information over local structures, that they have to carry a lot of information.

How huge? You intend to store the information in terms of the coordinates of the wavefunction in space? What does "N" in 6N refer to? How many dimensions do you need to store 20 million complex numbers plus 10 million unique id's? And that number could easily be larger. How do you you store the equivalent of unique id's in a coordinate? How are those wavefunctions going to interact at all if they are so scattered in space? How will A/B, and the angles of measurement be translated into coordinates? does hilbertspace have dimensions that go from -1.0 to 1.0 like cos and sin?

Perhaps it would be asking a lot to explain that to someone like me, but I don't see any answer at all, not even one that I wouldn't understand.

Nobody required you to run the quantum universe on a pentium-3 machine with 128MB of RAM

The question of how to use that information to do the "pairing-up" is not just one of scale. If you make silly jokes, I have to assume that you don't actually have an answer.

And that means that apparently MWI isn't a viable physical theory. Just a game of playing around with hilbert dimensions in a purely mathematical fashion of thinking: 'As long as we have enough dimensions, we can do anything we want...'

As I said, hilbert space is really, really big. This is the problem on which quantum chemistry breaks its teeth btw., the huge "solution space". If this exercise can make you see the hugeness of hilbertspace, then it has already had a good effect!

But, as I repeated earlier, even a single point in euclidean space alrready carries "infinite information".

I'm not impressed by a purely mathematical possibility of infinite dimensions as a magic solution. On the contrary, if a space of infinite dimensions is required also as a *physical reality*, then I'm more tempted to think there isn't any viable theory at all.

So I hope I've made clear why I don't find all this convincing.

However I'm not really expecting any more insightful explanations, so perhaps this is the time to discuss the 3 particle GHZ entanglement case:

We haven't yet discussed any case for which Bell's theorem actually states that there can't be a local explanation. This is one, and I hope that my limited understanding of GHZ entanglement is sufficient to discuss the challenge it poses for a local explanation using "pairing-up" of local MWI-like splits.

As I've described briefly already, there are three particles entangled, at locations A, B, and C, again with variable measurement angles at each location. In GHZ, there is also the situation that for a specific combination of angles, the result at the third location is definite rather than probabilistic, similar to the spin in a two particle entanglement being always opposite, when the angles are the same.

So for a specific combination of angles at A, B and C, the results will have a specific relation to each other, whereas for other combinations the relation will be probabilistic.

Let's say the angle at C will be modified to cause either one or the other case.

A, B and C make their measurements, and the A and B send a message at speed of light to midpoint "AB". AB is closer to A and B, than to C. so when the messages from A and B arrive at AB, information from C won't be available yet.

Yet similar to the case described in your message #72, A and B meet with the corresponding options. The angles at A and B alone can't determine whether the outcome will be definite or probabilistic, so what we called "bobalice++" won't have any term like (-xs + yr) that cancels out. That means, if the pairing-up is done locally, "bobalice++" must be allowed to develop, even though when meeting C, that combination may turn out to be impossible, depending on the angle at C. Will bobalice++ then 'vanish' in some way, or not allowed to meet C? That would seem absurd.

That is the challenge.

 

[colorSpace]

[Continuation from the last message]

I have a feeling there might be a possibility that you could resolve the specific GHZ-entanglement challenge that I have outlined above, depending on how exactly GHZ-entanglement works, which I don't know well enough in detail.

This mostly since the case which I have outlined doesn't really reflect the argument of Bell's theorem. [or the GHZ extension of it].

I think if you were able to specify your claim of locally resolving the "pairing-up" in a way that could be consider both 'local' also in details like passing the information and using it to resolve the paring-up, and if it could also be considered physically 'viable', then you would need to put it to test with Bell's theorem and look exactly at the cases for which Bell's theorem says they can't be resolved with a local model, and see whether you are able to use that model to make the same predictions regarding probabilities in 2-particles (which in MWI there may be no directly corresponding concept for), and definite outcomes for entanglement of 3 or more particles.

However that is probably not an easy task, if possible at all (which can be doubted easily), and you don't seem to have done that yet.

[Edit:] My guess would be that as soon as your model is made physically viable, and also in a physically meaningful way "local", at that point it will fail to resolve Bell's theorem.

 

[vanesch]

I'm not really impressed by the "hugeness of hilbertspace", you have mentioned already that it has many dimensions. At this point it seems to be a) a purely mathematical construct for the convenience of physicists to do calculations, and b) you haven't shown yet (at least not in a way that I would be able confirm or reject) how the many dimensions of hilbertspace can be used to make the case that the information can be "carried" along in a way that maps to a 3D-local physical explanation.

Well, the hugeness of hilbertspace which you don't seem to realize comes from the superposition principle when different "subsystems" are considered.

Consider a classical system A that can be in 100 different states, and a system B that can be in 100 different states too. A's hilbert space has 100 dimensions, and so has B's hilbert space. So, if taken "together", we could assign a state to "A" and to "B" individually with 200 complex numbers, right ? Wrong. That's only the case when they are separated, non-interacting, independent systems. With these 200 numbers, we can only describe individually what is a state of A and what is a state of B. But the superposition principle allows for superpositions of this kinds of states too. In mathematical terms, the hilbertspace of A and B is A x B, which has 10 000 dimensions.

And here we see the need for labeling, if we insist on assigning states to systems A and B individually.

I'm not debating mathematical possibilities, from the beginning not, but the physical possibility of storing and handling this information. You haven't shown to me, yet, how this information can be stored and then used in a way that could be called 3D-local. "Dimensions" by themselves do not constitute usable information.

I think you think of a classical universe, where we have things like "point particles" in a 3-D space. A "local version" of quantum theory is not going to live into such a small apartment! To demonstrate locality, we have to show simply that there are OTHER objects than just "points" walking around in 3-D euclidean space. And those other objects can be just ANY mathematical entities, like vectors in hilbertspaces, index-spaces, everything you want. From the moment that one can construct a set of mathematical objects that is associated to a locality (local environment to a point in 3-dim space), no matter how complicated, in such a way that it only changes as a function of itself and other mathematical objects at that same locality, and is isomorphic to the original theory, then we can say that the original theory is local.

I'm especially curious about the 10 Million unique id's!

If you think I concede at this point, think again. We have merely come to the point where we can discuss the question I was asking, but there is no answer to it in sight yet, as far as I am concerned.

Hilbertspace is common to all interpretations of quantum mechanics, and other interpretations haven't been able to use it for a local explanation. You have shown me how MWI could attempt to provide a local explanation, *if* it could carry the information in a local fashion, *and* if it could then be used to do the "pairing-up" in a local fashion. But you haven't shown me any *specific* way yet how *either* of the latter would be possible in a meaningful physical way that makes sense, and that could be mapped to 3D-space in a way in which it can then be decided that it could be called "local".

But I DID show you how you can construct mathematical objects which are associated to a locality, and only use that local information (of itself, and of other constructions, associated to the same locality) to change itself (= dynamics).

I did show you how we can associate to each substate (which is a ket in a hilbertspace of a local system, like "particle u" or "bob's brain"), a set of indices containing complex amplitudes etc which makes up another mathematical object (kind of "structure" dataset as in computer science, but we could mold it into something else)... and this whole mathematical object (that is, the ket in the hilbert space, and this structure) wanders around in a 3-dim space for each "particle" or system. So we have, with each subsystem we consider (each particle, brain, ...) at least one such mathematical object walking around in space. It is when these objects meet, at a certain location (in 3-dim space), that they interact through
1) a unitary operator which acts upon the ket vectors to generate new ket vectors for these systems
2) some rules which combine eventually the amplitudes and labels, or which generate new labels and amplitudes in the datastructure.
You have only *claimed* that it *should* be possible because hilbertspace has so many dimensions.

The fact that this formulation is possible (where things that wander through space are hence "kets in their own hilbert space" and a datastructure), means that quantum theory is local, because "a local theory" is nothing else but the requirement that such a formulation is possible.

I don't see why 3 dimensions shouldn't be enough. The problem is to come up with a way to store that information, even if it is just in the 'coordinates', that works as part of a meaningful physical process. BTW, is hibertspace euclidian, or curved? Is it generally acknowledged that hilbertspace has physical reality, or is that a specific theory?

But we are talking about MANY hilbertspaces here of course: each particle its own (which it carries with it if you want to).

You seem to confuse the 3-dim euclidean space (which is the base space) and then the objects IN this space (which can have just any structure, and be in fact much "richer" than the 3-dim space itself). To demonstrate locality, you just have to show that you CAN have a 3-dim euclidean basespace. Whatever the objects that are carried along, doesn't matter. *locality* is a mathematical criterium !


Yet those were the objections I had from the beginning.

So you have merely reached the starting point for discussing the objections which I had from the beginning.

What else do I need to say to clarify that, if it isn't clear yet?

Your response is evading a very simple point. I've said that the term (-xs + yr) is non-local as long as x and y refer to physical states at one location, and r and s refer to states at a different location. That changes when x, y, r and s are brought to the same location. It became clear only recently that the term (-xs + yr), in relation to A and B, is information that may resolve the problem. However you haven't shown yet how that is supposed to happen, and certainly not in a way that I could form an opinion about whether that might be a *physical* and *local* possibility, rather than just a theoretical one plainly *assuming* infinite storage capabilities, just because hilbertspace has so many dimensions.

Uh, yes, of course I assume the necessary "storage space" in the mathematical objects that wander around in 3-dim euclidean space. It is stored in the index space and the hilbert space that wanders around with each individual object of course. But your error is to call that "non-local". The (-xs + yr) is PRESENT locally when it is needed to do the annihilation of the alicebob++ term. So why do you insist on calling this non-local ?

"Algebraic information"? Which "exercise"? You need to specify physical means in order to have a physical theory. I have from the beginning doubted that there is a *physical* way to carry that possibly huge amount of information along, and that it can then be used to do the "pairing-up"

But if you want to, you turn every mathematical object into a "physical" one in theoretical physics ! That's your choice ! Where is the information of the location of a pointparticle "stored" in classical physics ? Is that "storage" physical or not ?

And that means that apparently MWI isn't a viable physical theory. Just a game of playing around with hilbert dimensions in a purely mathematical fashion of thinking: 'As long as we have enough dimensions, we can do anything we want...'

I'm not impressed by a purely mathematical possibility of infinite dimensions as a magic solution. On the contrary, if a space of infinite dimensions is required also as a *physical reality*, then I'm more tempted to think there isn't any viable theory at all.

Well, that's already the case in any classical field theory !
I don't make any difference between "physical" objects and mathematical objects. As the only objects we can ever think of or describe, are of mathematical kind, to me, we can at our likings, attach or not the label "physical object" to some of them. But all conceivable objects are of course primarily mathematical. I do not make any pre-hypothesis to which ones should be declared physical. That's up to a theory to decide and see if this corresponds to reality.

We haven't yet discussed any case for which Bell's theorem actually states that there can't be a local explanation. This is one, and I hope that my limited understanding of GHZ entanglement is sufficient to discuss the challenge it poses for a local explanation using "pairing-up" of local MWI-like splits.

As I've described briefly already, there are three particles entangled, at locations A, B, and C, again with variable measurement angles at each location. In GHZ, there is also the situation that for a specific combination of angles, the result at the third location is definite rather than probabilistic, similar to the spin in a two particle entanglement being always opposite, when the angles are the same.

Well, give me the starting state, and that's good enough.


So for a specific combination of angles at A, B and C, the results will have a specific relation to each other, whereas for other combinations the relation will be probabilistic.

Let's say the angle at C will be modified to cause either one or the other case.

A, B and C make their measurements, and the A and B send a message at speed of light to midpoint "AB". AB is closer to A and B, than to C. so when the messages from A and B arrive at AB, information from C won't be available yet.

Yet similar to the case described in your message #72, A and B meet with the corresponding options. The angles at A and B alone can't determine whether the outcome will be definite or probabilistic, so what we called "bobalice++" won't have any term like (-xs + yr) that cancels out. That means, if the pairing-up is done locally, "bobalice++" must be allowed to develop, even though when meeting C, that combination may turn out to be impossible, depending on the angle at C. Will bobalice++ then 'vanish' in some way, or not allowed to meet C? That would seem absurd.

That is the challenge.

If you can give me the begin state of the 3 particles, and how they interact with the 3 observers, we will work this out together...

 

[colorSpace]

Your message mixes several statements from me and you, and it looks as if some of my statements were yours. I think you need to edit it to fix that, before I write a longer reply. Also note that meanwhile I wrote a second message.

Regarding the term (-xs + yr): It was non-local (to me) at a point in our discussion where it wasn't clear that it was those values which you are going to bring along in the physical states of the objects. At that point it seemed that those values only described the probability of resulting physical states, rather than that they would be brought along as information. And the latter still isn't clear.

I'm actually still not sure if (and especially how) your model brings them along. To me it means that the states of the affected particles must reflect that they were influenced exactly by those values (x,y) or (r,s), in a way that they can be matched when they meet. I still don't see your model actually describing this. The fact that they appear as a factor in front of a mathematical term, doesn't mean that they cause a "match" when they physically meet. Other factors in that term may remove the information of which angle and which entangled particle this physical state corresponds to.

 

[colorSpace]

My first scan of your response indicates that it is very vague and general, where I am asking specific questions.

Your model doesn't seem to be worked out in detail yet, and I expect that is where its limitations will show up. It is those details that I had doubts about from the beginning, and I don't see a specific "solution" that answers those doubts. It still sounds like a claim that you think a solution should be possible, rather than that you actually have one. As is probably obvious, I don't understand Hilbertspace so I can't discuss with you on that level.

 

[colorSpace]

Quoting myself to highlight a point:

Regarding the term (-xs + yr): It was non-local (to me) at a point in our discussion where it wasn't clear that it was those values which you are going to bring along in the physical states of the objects. At that point it seemed that those values only described the probability of resulting physical states, rather than that they would be brought along as information. And the latter still isn't clear.

I'm actually still not sure if (and especially how) your model brings them along. To me it means that the states of the affected particles must reflect that they were influenced exactly by those values (x,y) or (r,s), in a way that they can be matched when they meet. I still don't see your model actually describing this. The fact that they appear as a factor in front of a mathematical term, doesn't mean that they cause a "match" when they physically meet. Other factors in that term may remove the information of which angle and which entangled particle this physical state corresponds to.

That is, in so far as they even affect the physical state. Your description seems to be a meta-description of which objects will be there, rather than a description of the physical states of those objects. To me, the whole description appears to be written from a non-local point of view. I can't see how these descriptions describe actual physical states of those objects, that is how is information of the angles and the entangled particle that they interacted with, how is that encoded in the physical state of that object.

That is, when the email arrives, how could one tell which angles and which entangle particle affected this email. Your description seems to describe only which version of the email will appear under which condition.

What I am looking for is a physical property of this "version" of the email that will decide whether it will interact with this "version" of Bob, And a physical process that use this physical property to make that happen.

Your description seems to only say whether under certain conditions they will match up or not, from a bird's eye point of view. It doesn't address in any specific way, at least not that I can recognize, how this information will be reflected in which physical property. Will it have some kind of vibration, will it appear in some special dimension, or... how I am I supposed to see this as a local *physical* state?

 

[vanesch]

My first scan of your response indicates that it is very vague and general, where I am asking specific questions.

Your model doesn't seem to be worked out in detail yet, and I expect that is where its limitations will show up. It is those details that I had doubts about from the beginning, and I don't see a specific "solution" that answers those doubts. It still sounds like a claim that you think a solution should be possible, rather than that you actually have one. As is probably obvious, I don't understand Hilbertspace so I can't discuss with you on that level.

It seems to me that to every technical explanation I try to give, you object that it is "vague and general" and doesn't answer your question, while I don't see how I can be more precise and clear (in the technical way) than I've been.

The point *I* am trying to make clear, which is what was at the origin of this discussion, is that the MWI view allows quantum theory to be seen as having the property of "locality", as this was put in doubt.

In fact, I even go further, and I try to convey the idea that the MWI view does nothing else but REVEAL the workings of the quantum formalism which everybody uses, but who might not be aware of it.

I'm trying to demonstrate that the machinery of the quantum formalism (the hilbert space formalism, with the kets, the unitary evolution operators and all that) CAN be formulated in a way which is "local", although this is admittedly a clumsy way of doing, and the global formalism which incorporates all these manipulations algebraically does it in a much more elegant way, and that it is only upon EXIT of the quantum formalism that explicitly non-local things seem to happen. This exit is what is the quantum-classical transition in Copenhagen for instance.
Indeed, from the moment that there are classical outcomes to mesurements, Bell's theorem is quite restrictive. Apart from superdeterminism, there's not much hope to have a local, classical explanation to the outcomes of quantum theory. That's what comes out of Bell's theorem.

But the point is that *forcing an outcome* is *outside of the strict quantum formalism*, and that the quantum formalism ITSELF doesn't suffer this non-locality. MWI only *reveals* this. It doesn't impose anything, but because MWI doesn't *require* the forcing of an outcome (not more so than the unitary quantum formalism), it can escape Bell's theorem. It isn't MWI per se which does this, it is because MWI doesn't ADD something to the quantum formalism. As such, it allows one to analyse in more detail exactly how, at no point, "information at a distance" IS STRICTLY REQUIRED to obtain the apparent results of EPR experiments. So I don't have to build a *nice* model to show this, I only have to show that *A* model exists, which can carry around enough information LOCALLY, so that this local information is enough to obtain the EPR outcomes just as "non-local" quantum mechanics predicts.

However, I want to point out that this is not a local hidden variable theory as understood in Bell's theorem, simply because in Bell's assumption, there ARE objective and unique outcomes at Alice and Bob, while in MWI, we don't assume that. In MWI, BOTH outcomes "happen" at both sides, and the correlations only appear when we combine the "messengers" from both sides. It is the excape route from Bell's theorem.

So I've tried to show you that it is possible to have "local buckets of information" which travel with the particles, and which contain all that is necessary to calculate the correlations as observed in EPR experiments. These local buckets do in fact nothing else but carry around the necessary information to reconstruct the relevant parts of the global wavefunction, and the "treatments" (the "dynamics") they undergo is equivalent to the algebraic operations such as sums of vectors, distributivity and so on which is inherent to the operations in global hilbert space.

The fact that I can show you that such a "bucket" can exist and does what it has to do, is sufficient to show that the global quantum dynamics has the property of locality. Whether you LIKE this model or not, or whether you find it "plausible" or not, is really not an issue! It is the mathematical existence of the model which proves that the global quantum dynamics has a mathematical property of locality. It is a *mathematical proof* of a mathematical property. You are not supposed to argue about the elegance of the elements used in a proof, right ?

Of course, I assume a certain technicality on your side too, in order to be able to have a meaningful discussion.

So what I showed was that:

To an entangled state of a couple of subsystems, which is globally represented by an algebraic expression of the kind:

|u+>|v-> - |u-> |v+>

we CAN associate LOCALISED states, to the u-system and to the v-system individually, which contains:

1) the state information of course (|u+> and |u-> for the u-system), which is nothing else but elements in the hilbert space of states associated to the system U, and which we can imagine "being carried along with the U-system" (it is actually more subtle than this, but let us for the moment assume that the spatial degrees of freedom are as of yet classical - I can do it in more generality but you will be more lost)

2) but also extra information (the "bucket") which is encoded in the algebraic expression |u+>|v-> - |u-> |v+>.

It is of course number 2) which is crucial here, because in order to show locality, we cannot use the global wavefunction anymore. So we have to show that there CAN exist local structure of information which carry the same information and which act in the same way as does the global wavefunction.

You object to the amount of information here, but that's what I tried to say: the superposition principle, which requires that the hilbert space of a combined system is the TENSOR PRODUCT and not just the sum of the spaces of the subsystems, makes that the hilbertspace of a combined system is MUCH BIGGER than the set product of the subsystems. It is exactly this which is expressed by the possibility of entangled states: namely that it is not sufficient to have just "the state of A and the state of B", but that we can find A LOT OF COMBINATIONS of these "product states". This is encoded in the algebraic expression of the entangled state, and must hence be ENCODED LOCALLY if we want to find an equivalent local formulation. As there are a lot of possibilities (as this product hilbert space is very big), you shouldn't be surprised that this concerns a lot of information! But the point is that it is POSSIBLE to encode this locally.

We do this by saying that to the |u+>|v-> - |u->|v+> state (global state), we have that the u+ state "has to know" that it was paired up with a v- state of the v system, and that it had amplitude +1. We also have to add to the state v- that it was paired up with the u+ state, and that it had amplitude +1. Similar for the u- state, it has to know that the u- state was paired up with the v+ state of the v-system, and that it had amplitude -1.

This is the information we have now to include into the "bucket" that goes with the u-system. The u-system has hence as a "local description":
a u+ state with an indication A (label) shared with the v-system and amplitude +1
a u- state with an indication B (label) shared with the v-system and amplitude -1

The v-system has as a local description:
a v+ state with indication B shared with the u-system and amplitude +1
a v- state with indication A shared with the u-system and amplitude -1

I think I showed that this entangled state could only come about by an interaction of the u and the v-system when they were in the same locality.

From the moment that the u-system and the v-system separate, they carry this information along, and from that moment on, an action on the v-system will not be able to alter the information carried with the u-system. That is what is the requirement of locality.

Now, when Bob interacts with the u-system, he only has access to the information carried in the u-bucket (and his own) to determine the new state of himself and of the u-system.

If Bob has an analyser in a certain direction, then locally, at Bob's, where we also have the u-particle, using only the information in the u-bucket, we see that we can turn this into:

Bob interacting with the u+ state which becomes:
x |bobAC+> |u++AC> + y |bobAD->|u--AD>

and bob interacting with the u- state which becomes:
-y |bobBE+> |u++BE> + x |bobBF-> |u--BF>

This means that the "bob bucket" now becomes:
a bob+ state with labels A (from u, shared with v, amplitude +1) and C (shared with u), amplitude of C is x
a bob- state with labels A (from u...) and D (shared with u), amplitude of D is y
a bob+ state with labels B (from u shared with v, amplitude -1) and E (shared with u), amplitude of E is -y
a bob+ state with labels B and F...

The u-bucket becomes:
a u++ state with labels A (shared with v, amplitude +1) and C (shared with bob) amplitude is x
a u-- state ...
etc...

This is simply the local buckets which encode for the global state:

(x |bobAC+> |u++AC> + y |bobAD->|u--AD> ) |v-A>
- (-y |bobBE+> |u++BE> + x |bobBF-> |u--BF>) |v+B>

but the important part is that we could constitute the new bob and u buckets with just the information that was locally available at bob's when he did his measurement: that is, we only needed the u-bucket, and we only needed the local axis of bob wrt the states in the u-bucket.

It is exactly this which we can do at ANY interaction: we can, using the local information buckets of the systems present at a certain location, define the NEW buckets after interaction which determine the new states and the new amplitudes. We only need the contents of the buckets which are locally present to do this, and nevertheless, at any moment, we can transform them such that they remain at all moments "in sinc" with what one would have obtained using the global wavefunction.

When alice meets bob, the information in the alice bucket is then combined with the information in the bob bucket to find out what states can appear, with what amplitudes. It can happen that certain amplitudes become 0, such as xs - yr. But this is established by just combining the local information buckets carried along with each subsystem.

It should be clear that this way of handling things is *entirely equivalent* to the global wavefunction dynamics. As such, we have shown that the global wavefunction dynamics has the property of locality.

 

[vanesch]

What I am looking for is a physical property of this "version" of the email that will decide whether it will interact with this "version" of Bob, And a physical process that use this physical property to make that happen.

If I give you the rules of how to calculate the amplitudes, that's good enough, no ? There are no "internal gears and wheels", not more so than if you would ask the question:

"if I have a stone with electric charge q, and a electric field of intensity E, then I'm looking for the physical property that will make the stone interact with the field to have a force qE on it". There are no "gears and wheels", there's just the mathematical rule in this case that you have to multiply the charge with E-field. What property makes one have to multiply this ?

I cannot give a "mechanism" for the rules of calculation of the amplitudes. I can just give the rules.

Your description seems to only say whether under certain conditions they will match up or not, from a bird's eye point of view. It doesn't address in any specific way, at least not that I can recognize, how this information will be reflected in which physical property. Will it have some kind of vibration, will it appear in some special dimension, or... how I am I supposed to see this as a local *physical* state?

That's what I mean: there is no underlying "gears and wheels" of a physical theory. There's just the rules of the mathematical manipulations. Again, why does one have to multiply a number (charge) carried with the stone with another mathematical structure (a vector at a point in space), to find something like a "force" acting upon the stone ?

Is the "charge" stored in a vibration of the stone ? Is the E-field stored in a silicon memory ? What is the physical mechanism that makes us multiply the charge with the E-field to find the force ?

 

[neu]

I disagree, if a interpretation make certain claims and predictions and these predictions and claims are disproven by experiment, they are no longer valid. THAT'S SCIENCE, a process of advancing.
Clinging to a theory cause it appeals you is religion, would you say ID's "interpretation" of how life came to be cannot be "disproven" either, when it claims Earth is 6000-10000 years old and we got fossil records who disprove this?

I'm sorry for quoting an old post but I think this is important.

Firstly, it is impossible to define criteria for distinguishing science from non-science. You're statement of what you believe to be science ammounts to a falsificationist argument that a theory must be falsifiable and science lies in tests which aim to refute it. By this reasoning many aspects of creasionism, astrology may be qualified as "scientific" and on the contrary some scientific methods as non-scientific e.g. creationism's claim regarding the Earth's age is refutable, and hence "scientific".


Secondly, you say clinging to a theory because it appeals to you is religion. There are numerous examples of scientists "clinging on" to their theories after apparent refutation (at which point, by your reasoning, they do so religiously), only later to have this refutation proved invlid and their theory reconfirmed.
A famous example is the violation of Newtonian mechanics by the orbit of Mercury observed in the 19th Century later proved a phenomenon of special relativity by einstein.

Sorry to diverge from the topic but i had to say it!

 

[colorSpace]

If I give you the rules of how to calculate the amplitudes, that's good enough, no ? There are no "internal gears and wheels", not more so than if you would ask the question:

"if I have a stone with electric charge q, and a electric field of intensity E, then I'm looking for the physical property that will make the stone interact with the field to have a force qE on it". There are no "gears and wheels", there's just the mathematical rule in this case that you have to multiply the charge with E-field. What property makes one have to multiply this ?

I cannot give a "mechanism" for the rules of calculation of the amplitudes. I can just give the rules.

We are running into a situation where we have to consider the state of our discussion, and how to continue it, if we do.

From my point of view, the impression is that you try to sell the idea that the concept of local splits in MWI gives you a way to explain entanglement in a local fashion for free.

I don't buy especially the "for free" part, and so I don't buy the whole package.

In addressing the measurement-problem in the Copenhagen interpretation, and its randomness, AFAIK, MWI has criticized CI for not having gears and wheels in the 'collapse' concept. So MWI should take such questions seriously.

I will write more later, I'm busy today.

 

[vanesch]

From my point of view, the impression is that you try to sell the idea that the concept of local splits in MWI gives you a way to explain entanglement in a local fashion for free.

I don't buy especially the "for free" part, and so I don't buy the whole package.

The property of locality is a mathematical property of a theory. It essentially means that you can find a map f from E^3 into a set of mathematical structures (ANY structures) in such a way that the temporal evolution of, say: f(p) (p is a point in E^3) and f(p) is a mathematical structure is only a function of all the f(q) with q in a neighbourhood of p in E^3, and also such that physical observations and so on done at point p in E^3 are only a function of the mathematical structures f(q).

If you can show that such a map f exists, and that it gives rise to a dynamics which is equivalent to the dynamics of your theory, then your theory has the property of locality.

It is this mapping which I established, by showing that we could associate mathematical structures (kets + a label system) to points in space (or neighbourhoods of points in space) in such a way that this is equivalent to the dynamics of the quantum formalism of non-relativistic QM in the Schroedinger representation, as long as the hamiltonian (or its integration, which is the unitary time evolution operator) is build up using only local interactions.

So once this has been shown (as in "mathematical proof") you can go back to standard quantum formalism with its global hilbert space, you KNOW that it has the property of locality - in the same way as you can continue doing Hamiltonian dynamics in 6N dim phase space, knowing that it has a local representation in E^3, even if that seems more clumsy to work with.

In addressing the measurement-problem in the Copenhagen interpretation, and its randomness, AFAIK, MWI has criticized CI for not having gears and wheels in the 'collapse' concept. So MWI should take such questions seriously.

The problem in CI is that no rule exists to prescribe when a physical interaction is to be treated one way ("measurement") or the other ("dynamics"). I don't have such a problem in what I presented: the rules are very simple, and universal.

But what is more, when considering a collapse, such as in CI, THEN you cannot find such a "local" map f(p) anymore. It is impossible to find a local representation, no matter how clumsy, in which a genuine collapse occurs. As such, the projection, in CI, is strictly non-local.

Now, why do we cling so much to locality ? What could one care ? In a purely Newtonian setting, not much. Newtonian mechanics is non-local. Forces act between "things at a distance". There exists a nice, consistent extention of Newtonian mechanics which includes quantum effects: it is Bohmian mechanics. It is explicitly non-local.

The problem we have with non-local theories is when we want to go relativistic. Because in relativity, all mathematical objects representing physical things have to live on the spacetime manifold - it is the basic idea of relativity. As such, no "non-local" objects are allowed. If you introduce them in relativity, you run in all kinds of paradoxes, such as being able to kill your grandpa and so on. This is why "locality" is such an important thing. You can hope to extend a local theory into relativity. You know that a non-local theory will bring you troubles.

Now, you can say, if we find experimentally that nature is non-local, then so be it. Right. But we've seen that the ONLY aspect of non-locality in quantum theory comes about by the interpretation we give to it. From the moment that there is a projection, there is non-locality (no hope to implement a relativistic version). But if we don't do projections, and we keep with the unitary dynamics, then we have seen that locality is preserved. So it is premature to say that quantum mechanics is non-local, as the non-locality is imposed only by an aspect which is discutable: projection.
Given that in MWI, you DON'T use projection, you can still keep the locality property, and as such, the extension to a relativistic theory.

 

[colorSpace]

The problem we have with non-local theories is when we want to go relativistic. Because in relativity, all mathematical objects representing physical things have to live on the spacetime manifold - it is the basic idea of relativity. As such, no "non-local" objects are allowed. If you introduce them in relativity, you run in all kinds of paradoxes, such as being able to kill your grandpa and so on. This is why "locality" is such an important thing. You can hope to extend a local theory into relativity. You know that a non-local theory will bring you troubles.

You are touching an interesting topic here. The exact nature of relativistic causality in regards to non-locality, which becomes especially relevant in regards to the question of whether quantum tunneling could be FTL.

My understanding is that entanglement, even as seen non-locally in any 'Single World' Interpretation, is not a problem for relativity since it is symmetrical in regards to particle A and B. That is, even if some observer sees the measurement of A as occurring first, and another will see the measurement at B as occurring first, the symmetric nature of entanglement will allow both to see a consistent picture.

In regards to our discussion in general, I need to "slow down" for a few days, to get a larger picture and see how to proceed.

[Edit:] BTW, what do you mean with "projection"?

 

[vanesch]

Concerning the GHZ state, as I don't know exactly what you're after, I worked out, in global notation, what happens to 3 observers when they measure such a state. We'll introduce labels and so on later, once I know what you're after, because it becomes quite tedious.

The (or one of the) GHZ states is:

|+++> + |--->, or |1+> |2+> |3+> + |1-> |2-> |3->.

Now, consider that particle 1 goes to Alice (for short, a) which puts her analyzer to such an angle that we have cos theta = x and sin theta = y.

Particle 2 goes to bob, with cos = r and sin = s

Particle 3 goes to celine, with cos = u and sin = v.Globally, the state then evolves, after the 3 did their measurements, into:

( x|a+> + y|a->) (r |b+> + s |b-> ) (u |c+> + v|c-> ) +
(-y|a+> + x|a->) (-s|b+> + r|b->) (-v |c+> + u |c-> )

which becomes, after working out:

(xru - ysv) |a+ b+ c+> +
(xrv + ysu) |a+ b+ c-> +
(xsu + yrv) |a+ b- c+> +
(xsv - yru) |a+ b- c- > +
(yru + xsv) |a- b+ c+> +
(yrv - xsu) |a- b+ c-> +
(ysu - xrv) |a- b- c+> +
(ysv + xru) |a- b- c->

We've dropped a 1/sqrt(2) factor from the start, so these complex amplitudes, squared, give us the final probabilities to find the triples of outcomes, given the settings of the axes. I hope I didn't make any mistakes.

In order to do this in the "local way", I would like you to indicate me which "path" you think will bring my explanation in trouble, as I'm not going to do it in thinkable ways (too much typing!).

 

[colorSpace]

Note that meanwhile I wrote message #102, preceding your last message.

 

[colorSpace]

You know, I feel like repeating this question I asked in message #83:

what does it mean for Bob to have u as a factor?

So meanwhile I understood very well that the amplitudes are sin or cos of a measurement angle, and so, numbers between -1 and +1. In the Copenhagen Interpretation, they refer to the probabilities of measuring a specific value.

But what does it mean in MWI for Bob+ to have a factor of, for example, -0.3 ?
What does it physically mean? Bob multiplied with -0.3 ?

The problem seems to me that with local splits in MWI, each state must then carry along this amplitude, plus a reference to the particle that was entangled, in some physical form, so that when these states meet, it can be decided physically, for example, whether there will be 4 versions of Bob (++, +-, -+ and --), or 2 versions of Bob (+- and -+).

But how can Bob, and the email he receives, carry along a mathematical number that doesn't have a Unit of Measurement or anything? A mathematical number is an abstraction. (As Hilbert space is called "abstract Hilbert space" and can be used apparently for very different purposes of calculation, a general mathematical tool.)

You have repeatedly asked me to think of it as mathematical values. And it seems to be just part of a mathematical formula. But how can that suddenly be carried along as a physical state? I have a good idea of how information can be carried in computers, etc. It always takes volume and time, and no matter how many dimensions there are, under normal circumstances it would require a certain volume to carry information.

So does Bob somehow have a "basket" for mathematical numbers?

Those are the questions that trouble me here, may they be due to my lack of understanding for quantum physical concepts, or not.

Plus, in just 'slightly' more complex cases, it would be quickly millions of numbers and references. All that information, somehow encoded in physical states, must apparently be present, in some physical form, at the very edge of any particle that is subject to influence from the measurement of the entangled particle. And at the very edge, they have to start creating another two Bobs, or not.

I can so easily see Bob being sliced by the email in two parts, each of which easily survives as a fully functional and conscious human being. Unless the email decides to leave it at two million Bobs, instead of four million.

 

[Enigma007]

I couldn't have said it better myself. Glad to see someone else thinking along those lines.

Enigma Valdez

CAH: I think you're misinterpreting V's POV and running the risk of sounding a little pompous. (Sorry, but writing 'science' in capital letters doesn't convince me of your argument.)

Everyone has different interpretations of QM, but we're all using the same equations. (I think that was V's point.)

 

[vanesch]

So meanwhile I understood very well that the amplitudes are sin or cos of a measurement angle, and so, numbers between -1 and +1. In the Copenhagen Interpretation, they refer to the probabilities of measuring a specific value.

But what does it mean in MWI for Bob+ to have a factor of, for example, -0.3 ?
What does it physically mean? Bob multiplied with -0.3 ?

It means that there is a probability of (-0.3)^2 = 0.09 for a bob-awareness to experience the state Bob+ in the frame of MWI. The specific wordings can change according to the specific flavor of MWI, but at the end of the day, that's what it means: if you are "a" bob, what's the probability that you experience the body state described by "bob+".

The problem seems to me that with local splits in MWI, each state must then carry along this amplitude, plus a reference to the particle that was entangled, in some physical form, so that when these states meet, it can be decided physically, for example, whether there will be 4 versions of Bob (++, +-, -+ and --), or 2 versions of Bob (+- and -+).

Yes. But it seems that you want to limit this "physical form of memory" to some kind of classical state, like a particle configuration, or a field form or something like that, which is of course absurd because we are here OUTSIDE of a classical state description. A classical state description is just ONE ELEMENT of the entire "state description" (it is a ket), so obviously you won't find any "place" INSIDE that classical state description to "record" the information needed.

All these extra (non-classical) notions of "amplitudes" and "which particle is paired up with which" are the consequence of the superposition principle, which is exactly what is non-classical in QM. The superposition principle tells you that you can combine different classical states into new states. There are miriads of ways to do this, and that's what global hilbert space is all about: the set of all possibilities of combinations. So a quantum state is a specific combination of classical states. It is the ground axiom of quantum theory. So this means, that the information of what exact combination of classical states must be "somewhere", but for sure, it cannot be INSIDE a classical state. It is what is in global hilbertspace encoded in the components (complex numbers) of the state vector (the amplitudes in a "classically-looking" basis), and it are these complex numbers which are scattered throughout the different "information buckets" if you insist on "local" structures which are attached to the different neighbourhoods in E^3.

Now, when we have only few components, as we usually do in the examples, then this set of components is simply encoded in an algebraic expression (a way of algebraically writing down the global wavefunction), and we can then use a few labels and complex numbers attached to states to set up "local buckets" attached to positions in space, which travel around.

So you have to understand very well that this "explosion of information" is present from the moment that you do quantum mechanics, due to the superposition principle which allows you to combine, in miriads of ways, different classical states.
The superposition principle, applied rigorously, allows a classical state with amplitude 0.2+0.1i which is made up of my body being at the grocery store and your body lying lazily in your bed (which is a classical state) together with an amplitude of -0.3-0.4i of my body jogging in the park and your body playing the piano, together with an amplitude of -0.01+i 0.001 of my body traveling in a trans-atlantic airplane and your body driving a sports car etc...

This is the superposition principle, rigorously applied to the classical states of the universe. It is the cornerstone of quantum theory. Now, it is very well possible that quantum theory doesn't apply in this way to systems containing airplanes, bodies, and all that, but by lack of any other theory, in MWI, we take it that quantum theory applies, as a working hypothesis. So that the superposition principle applies, and hence that these superpositions exist as possible states. Now, where is this information "stored" ? In hilbert space of course, not in any of the classical states themselves.

But how can Bob, and the email he receives, carry along a mathematical number that doesn't have a Unit of Measurement or anything? A mathematical number is an abstraction. (As Hilbert space is called "abstract Hilbert space" and can be used apparently for very different purposes of calculation, a general mathematical tool.)

"bob receiving an e-mail" is by itself a classical state. Everything with "units" are descriptions of classical states. The superposition principle COMBINES classical states with complex numbers.

You have repeatedly asked me to think of it as mathematical values. And it seems to be just part of a mathematical formula. But how can that suddenly be carried along as a physical state? I have a good idea of how information can be carried in computers, etc. It always takes volume and time, and no matter how many dimensions there are, under normal circumstances it would require a certain volume to carry information.

Yes, but these are classical concepts. By definition, a complex superposition of classical states is not encoded in those states themselves, but "outside". So of course you won't find "memory space" in your computer or e-mail server that contains this information.

So does Bob somehow have a "basket" for mathematical numbers?

Those are the questions that trouble me here, may they be due to my lack of understanding for quantum physical concepts, or not.

Yes, bob does have a basket for mathematical numbers in his quantum-mechanical description, by postulate of the superposition principle.

Plus, in just 'slightly' more complex cases, it would be quickly millions of numbers and references. All that information, somehow encoded in physical states, must apparently be present, in some physical form, at the very edge of any particle that is subject to influence from the measurement of the entangled particle. And at the very edge, they have to start creating another two Bobs, or not.

Yes, it is HUGE. But it is not MWI's fault. It is the superposition principle's fault. Just sit down and think a minute HOW MANY different combinations can exist if you allow for complex superpositions of all thinkable classical states of the universe (or the system you are considering) in all possible ways, with complex coefficients! It is mind-boggling. Maybe it is simply not true. Maybe there is a limit to the superposition principle. We haven't discovered it yet. But IN QUANTUM THEORY, we make as a theoretical working hypothesis, that it is strictly true. And then you get a glimpse of the hugeness of the number of quantum states (which is nothing else but the hugeness of hilbert space, which is simply a mathematical way of bookkeeping all those states).

MWI simply REVEILS this hugeness. But it is already there because of the postulates of quantum theory.

This is why I insist on MWI: it is a good revelator of many aspects of the quantum formalism. I don't "believe firmly" in it as a genuine "reality" (although it is possible that things are finally that way...). But it is a great way to get a feeling for how the quantum formal machine works.

In usual settings, we limit the superposition principle to very modest situations, which don't reveil what it actually says.

Consider a classical point particle. Normally, a classical state corresponds to a single point in space. Well, the superposition principle allows us to COMBINE all these classical states into a big superposition, each one with its own complex number. But we usually look upon this as "assigning a complex number to each POINT IN SPACE", and then we call this a "wave". But what really happens, is that we combine all classical situations, with the particle being at different points in each different classical state. The problem is that when thinking of "waves" we go classical again.
But now, think of 2 particles. Each individual classical state of 2 particles is given by TWO positions in space. Well, the quantum-mechanical superposition gives an amplitude to EACH of these classical states: state 1: particle 1 at position P, particle 2 at position Q. state 2: particle 1 at position R, particle 2 at position S ...
each of these has its own complex number.

And now we don't have 2 fields anymore, we have a function of 2 positions psi(P1, P2). This is already MUCH MORE than 2 functions psi1(P1) and psi2(P2). There are many more psi than there are combinations of psi1 and psi2.

You immediately see the explosion of possibilities when you go to bigger classical systems...

 

[colorSpace]

It means that there is a probability of (-0.3)^2 = 0.09 for a bob-awareness to experience the state Bob+ in the frame of MWI. The specific wordings can change according to the specific flavor of MWI, but at the end of the day, that's what it means: if you are "a" bob, what's the probability that you experience the body state described by "bob+".

At this point I have a very basic MWI question, and it is probably better to ask it first, before I go through the rest of your answer.

Of course even in MWI the number 0.09 would appear in statistical situations.

However:

In a single world interpretation, there would be only one Bob (perhaps unfortunately ), and 0.09 would describe the probability of him measuring the + state rather than the - state. That's easy to understand. (Unless perhaps one asks which physical state is anchoring the 0.09 probability, but that is, AFAIK, the whole system non-locally, at least in Bohmian mechanics, via the quantum potential).

However in MWI, AFAIK, there would be at least two Bobs, Bob+ and Bob-, each fully conscious. In the case that there are really (just) two, where does the 0.09 go, once the measurement has been done? And if there are two, both fully conscious, why wouldn't then the probability of being one or the other be 50% (0.5) ?

However if the 0.09 is reflected in the quantities of Bob, then there would have to be for example 9 Bob+, and 91 Bob-. Now it would be clear that the probability of being Bob+ is 0.09. However, now Bob+ doesn't have to carry the 0.09 with him anymore (except to resolve entanglement).

Are there two Bob's, or many? I hope that is a valid question, otherwise I would wonder whether MWI is a physical theory at all, and not just a calculation method.

 

[vanesch]

At this point I have a very basic MWI question, and it is probably better to ask it first, before I go through the rest of your answer.

Of course even in MWI the number 0.09 would appear in statistical situations.

However:

In a single world interpretation, there would be only one Bob (perhaps unfortunately ), and 0.09 would describe the probability of him measuring the + state rather than the - state. That's easy to understand. (Unless perhaps one asks which physical state is anchoring the 0.09 probability, but that is, AFAIK, the whole system non-locally, at least in Bohmian mechanics, via the quantum potential).

However in MWI, AFAIK, there would be at least two Bobs, Bob+ and Bob-, each fully conscious. In the case that there are really (just) two, where does the 0.09 go, once the measurement has been done? And if there are two, both fully conscious, why wouldn't then the probability of being one or the other be 50% (0.5) ?

However if the 0.09 is reflected in the quantities of Bob, then there would have to be for example 9 Bob+, and 91 Bob-. Now it would be clear that the probability of being Bob+ is 0.09. However, now Bob+ doesn't have to carry the 0.09 with him anymore (except to resolve entanglement).

Are there two Bob's, or many? I hope that is a valid question, otherwise I would wonder whether MWI is a physical theory at all, and not just a calculation method.

This is where different flavors of MWI enter. If you consider a large (infinite) amount of "bob-consciousnesses", which are distributed over the bob-body states in amounts proportional to the quantity given by the amplitude squared, then you have the flavor which is called the many minds interpretation.

If you consider that there is only one "true" bob state, and that the rest are "zombies" which, however are behaviourally indistinguishable, then this is the branching probability for a state to receive the "true bob state", that is, the bob that will be subjectively conscious.

You can find still other alternatives ; I have my own. But all this doesn't matter. It comes down that for a particular bob-consciousness to "experience" a particular bob-state, the probability is given by this square of amplitude.

In fact, for an external "bob" it doesn't matter ; it only matters for yourself. I like to give the following hypothetical classical example.

Imagine that the world is classical, and that it is possible to make perfect copies of a body. Now, you wander into the institute where they have such a copy machine, you lie down on the bench of the scanner, the scanner passes over your body - while you are perfectly conscious, awake and you don't feel anything (like an NMR scanner or anything). You get up, talk a bit with the doctor who is responsible, walk out of the institute again and go home.

Yet, in the 3 experimental installations next doors, they've received the scanning information, and produce 3 copies of your body, with brain and memory state and everything. Let us assume that these bodies are just as "conscious" as you assume other people you meet daily, are.

These 3 copies have a weird "experience": they remember coming in the hospital, lying on the bench, and suddenly find themselves in a strange machine with several scientists around them. Of course, they didn't "come in the hospital": their bodily material was in fact stored in bottles of the copying machines ; it is only because they have a copy of your memory in their brain, that they are TRICKED into thinking that they were coming into the hospital in the morning.

You, the original conscious you, are not aware of their existence, and at no moment, you experienced something strange when you were under the scanner.

But from an external point of view, there are now 4 copies of your body: 3 who have been fabricated in a machine, and one who was lying on the copying machine's scanner table.

Now, because of the asymetry of the setup, an external observer could give "extra credibility" to the fact that the "original you" was still the "original you" leaving. But that's because we have a scanning room and 3 fabrication labs.

It wouldn't come into your mind to say that you have 25% probability to be one of the 4 bodies, although there are now 4 identical bodies with identical memories around (except for the last event, which is: how did I get transported from the sofa into the machine ? for 3 of them). You would give 100% "chance" for you to "branch" into the body that was lying on the scanner table and went home, and 0% "chance" for you to become one of the 3 copies, right ?

I personally (it is my "version" of MWI) see the branching in the same way: an "original" presents itself before the split, and "branches" into one of the different alternatives, which is its "conscious continuity" ; the other branches are "copies". The "probability" of branching is then given by this square of amplitudes ; that is: the probability of the "original you" to "be" this or that branch. There is no a-priori reason why all branches should be equi-probable.

The many-minds version does in fact do something similar, only, many many minds which are equi-probable to be "yours" split in proportion to the square of the amplitude. As such, you being equiprobably distributed over them, your chances to branch this or that way are also proportional to the square of the amplitude.

EDIT: I would like to add something. Although it is fun (according to your hang for weird science fiction) to speculate about different versions of MWI, with consciousnesses and all that flying around, my personal stance on this is that one shouldn't delve too deeply into this. As I've repeated many times, I like MWI because it gives one a great feeling about the principles and workings of the quantum formalism. Instead of not knowing what one is handling when one does a quantum calculation, one can imagine, for a moment, by placing oneself in an MWI viewpoint, that one is really dealing with physically meaningful things.
But let us not forget that MWI, for all it is worth, is applying the axioms of quantum theory strictly, all the way, and hence seriously outside of the proven scope of quantum theory. So maybe (probably ?) MWI is placing too much faith in the correctness of quantum theory. Then, maybe quantum theory really is correct. Who is to say ?

So instead of speculating about a *change* in quantum theory (such as the CI does) without any serious formal backup, in MWI we *speculate* about the correctness of quantum theory beyond its proven scope. It is as if we speculated, only knowing Newtonian mechanics, of applying Newtonian mechanics everywhere.

I prefer to speculate about the extrapolation of an existing theory, than to invent properties of a yet-to-find theory which should behave this and so - but that doesn't mean I forget that I'm speculating nevertheless.

Again, don't see MWI as "a theory of the universe", or "a way of life" or "how nature really is", but rather as a viewpoint that helps you understand the machinery of our current (maybe limited) quantum theory.

 

[colorSpace]

But all this doesn't matter.

(Excerpt from the last message illustrating MWi flavors.)

Well, it sounds like there really is tendency to see MWI as a calculation method, rather than as a solid physical theory.

I think it matters a little in regard to the local split in MWI, and where the factor "0.09" remains after the split.

I am now even more confident that I have a valid question there, rather than just a lack of understanding.

Let me try to bring it to a point: Usually the wave function appears to be "anchored' in actual physical states. For example in the double slit experiment, the wavefunction is a result of the experimental configuration, specifically whether both slits are open, or just one of them. Although the wavefunction may have more information than we can measure, we can still assume (I think) that it is anchored in the physical states of the current situation.

It is a complex (in multiple senses) mathematical function with many terms, depending which mathematical model is used to formulate it. None of the mathematical terms is expected to have reality, they just establish the mathematical relation between possible measurements and the current physical state.

And when a measurement is made in an entanglement experiment, the wavefunction of course depends also on the measurement angle. Still each term is a mathematical consequence of current physical states, including the current physical state of the angle of the measurement device. This is where I see the wavefunction is "anchored" (for lack of an established term).

In a single world interpretation, or MWI with global splits (splitting the whole universe at once), the result is explained by a mathematical relationship of current physical states.

However with local, not yet paired-up splits, this result is postponed to the "meeting point".

And now it is difficult to see the mathematical relationship as being anchored in a current state. Where has the measurement angle gone? The device didn't come along, and also may have changed its angle meanwhile.

That is, the "email", as it travels through the internet, needs additional physical states that anchor the wavefunction so that it can carry the additional information (possibly huge), as the wavefunction itself is just an abstract formulation of the potentials of making a measurement. Hilbert space is a space in which this mathematical function is calculated, not a space in which physical states exist, AFAIK.

I'm not sure whether you haven't thought about all this, because you just see it as a calculation method, or whether you are still trying to leave it out of the discussion for simplicity, in spite of my many questions, or whether you just don't have an answer.

However I now really see this as a crucial question, if MWI really wants to claim that there is a local *physical* theory, rather than merely a mathematical or information-theoretical statement of illustrating the categorical difference between "correlation" and "signal sending".

In terms of a *physical* theory, I still see this only as the claim that a theory should be possible, and it would remain to be seen if an actual, physically valid, "implementation" would require compromises, as I would expect based on the need to carry potentially large amounts of information, which would negate the possibility to store this information in a way that the theory could still be called local, as all the information needs to be available at the very edge of a light cone, so to speak.

I said in a previous discussion, the universe doesn't send around infinite amounts of information [about all its internal past states] at the very edge of any light cone. Now you have reduced this requirement slightly, but not really that much.

 

[colorSpace]

I think you may get the answer from the textbook

Introduction to Quantum Mechanics, 2nd Edition
http://www.cocomartini.com/rainyland/product_info.php?products_id=1237

See any help.

hehe ^_^

Congratulations to your first post!

"Hehe".

 

[vanesch]

Let me try to bring it to a point: Usually the wave function appears to be "anchored' in actual physical states. For example in the double slit experiment, the wavefunction is a result of the experimental configuration, specifically whether both slits are open, or just one of them. Although the wavefunction may have more information than we can measure, we can still assume (I think) that it is anchored in the physical states of the current situation.

This is already a strange statement. Imagine I send an electron through a 2-slit setup, and then let it evolve afterwards for about a year. In the mean time, I blow up my setup and so on. Nevertheless, very very far from here, someone might do an experiment on the electron, and its wavefunction will determine the outcome there. The outcome there is not "anchored" in the state of my slit setup, which is blown up in the mean time: the electron wavefunction had to "carry that information" with it.

Now, you don't have much difficulties imagining this, because the electron is a simple point particle, and the "superposition of states" is in this case equivalent to a kind of "classical wave in space". But, as I pointed out, this only works for single point particles, and is in fact very unfortunate. People tend not to understand the REAL meaning of the superposition principle, because they confuse "superposition of classical location states" and "classical waves".

So your electron, which is, classically speaking a POINT particle, has to carry with it the memory of the state of your slit system when it passed through it, a year ago.

It is a complex (in multiple senses) mathematical function with many terms, depending which mathematical model is used to formulate it. None of the mathematical terms is expected to have reality, they just establish the mathematical relation between possible measurements and the current physical state.

But this is a statement that I don't understand. To me the ONLY things that can have "physical meaning" are mathematical objects, in the frame of a physical theory. So, saying that it is "just a mathematical construction" to me, is the first prerequisite for something to be able to become charged with "physical meaning". I cannot personally conceive something that has "physical meaning" if it is not, in the first place, a mathematical object.

And when a measurement is made in an entanglement experiment, the wavefunction of course depends also on the measurement angle.

? Before the experiment, the wavefunction DOESN'T depend on the measurement angle of course. The measurement angle enters the game only:
1) in CI: by PROJECTING the wavefunction on one of its components, when it is written in the basis corresponding to the angle at Bob, for instance.
Of course, once this is done in a strictly global way, Alice only has this component to go on, and hence her projection, along her axis, will depend on what projection bob did on his side.

2) in MWI: by making the bob-state interact with the wavefunction of the particles, and this INTERACTION (bob-particle_at_bob) is specified by the measurement angle at bob. The same on Alice's side: Alice will INTERACT with the wavefunction (unaltered) of her particle, and this interaction (alice-particle_at_alice) is specified by the measurement angle at alice.
So what we have now is not so much that the wavefunction of the particles has altered, but rather that bob interacted in a specific way with HIS particle, which put him in different states, and alice interacted with HER particle, which put HER in different states, and when these alice states finally come together with the bob states, and interact by exchanging information, then AT THAT MOMENT, Alice/bob paired-up states appear with different amplitudes, which are of course a function of the relative interactions of alice and bob with their particles, as this determined in what kind of states they happened to arrive. It is the fact that all these interactions can be considered as local that makes me say that the theory can still be considered local.

Still each term is a mathematical consequence of current physical states, including the current physical state of the angle of the measurement device. This is where I see the wavefunction is "anchored" (for lack of an established term).

In a single world interpretation, or MWI with global splits (splitting the whole universe at once), the result is explained by a mathematical relationship of current physical states.

However with local, not yet paired-up splits, this result is postponed to the "meeting point".

Yes. Like a photon, which "splits" over two slits, and then pairs up with itself again, to make an interference pattern. It is only at that place that an interference pattern is made, and the slit system could be destroyed by then.

And now it is difficult to see the mathematical relationship as being anchored in a current state. Where has the measurement angle gone? The device didn't come along, and also may have changed its angle meanwhile.

Yes, but the slits might have been blown up before the interference pattern was shown too.

That is, the "email", as it travels through the internet, needs additional physical states that anchor the wavefunction so that it can carry the additional information (possibly huge), as the wavefunction itself is just an abstract formulation of the potentials of making a measurement. Hilbert space is a space in which this mathematical function is calculated, not a space in which physical states exist, AFAIK.

Ah, that's then the difference. The way I look upon things (others have of course different views) is: in quantum theory, the REAL physical universe is the Hilbert space in MWI. We only get an impression of a kind of 3-dim space with objects in it, simply because these are the kinds of "classical states" we can experience. It is the property of locality which contributes strongly to that impression.

But again, note that I don't really think that "the universe is a vector in hilbert space", because I don't really think that quantum theory is the ultimate final theory. It's probably much more sophisticated than this. I put myself in this "frame of thinking" for the ease of understanding how quantum theory works.

I'm not sure whether you haven't thought about all this, because you just see it as a calculation method, or whether you are still trying to leave it out of the discussion for simplicity, in spite of my many questions, or whether you just don't have an answer.

However I now really see this as a crucial question, if MWI really wants to claim that there is a local *physical* theory, rather than merely a mathematical or information-theoretical statement of illustrating the categorical difference between "correlation" and "signal sending".

Again, I'm unable to make the distinction between a mathematical theory that has the ambition to be a physical theory, and a "real physical theory". If you have a mathematical theory that is saying something about physics, and you use it, then you ASSUME that its objects are "physical" of course.

Like Newton ASSUMED that there was a 3-dim euclidean space outside, simply because he used this in his theory to position his essential objects, which are point particles (mappings from the real axis into 3-dim space).

In terms of a *physical* theory, I still see this only as the claim that a theory should be possible, and it would remain to be seen if an actual, physically valid, "implementation" would require compromises, as I would expect based on the need to carry potentially large amounts of information, which would negate the possibility to store this information in a way that the theory could still be called local, as all the information needs to be available at the very edge of a light cone, so to speak.

I don't understand this. If you can define the necessary mathematical objects, then you simply DECLARE them to be physical, no ?

 

[colorSpace]

This is already a strange statement. Imagine I send an electron through a 2-slit setup, and then let it evolve afterwards for about a year. In the mean time, I blow up my setup and so on. Nevertheless, very very far from here, someone might do an experiment on the electron, and its wavefunction will determine the outcome there. The outcome there is not "anchored" in the state of my slit setup, which is blown up in the mean time: the electron wavefunction had to "carry that information" with it.

Now, you don't have much difficulties imagining this, because the electron is a simple point particle, and the "superposition of states" is in this case equivalent to a kind of "classical wave in space". But, as I pointed out, this only works for single point particles, and is in fact very unfortunate. People tend not to understand the REAL meaning of the superposition principle, because they confuse "superposition of classical location states" and "classical waves".

So your electron, which is, classically speaking a POINT particle, has to carry with it the memory of the state of your slit system when it passed through it, a year ago.

Well, no, the electron (or whichever particle) isn't in any sense just a particle.

But I guess we are again getting a small step closer.

The particle doesn't have, for example, a specific position, but a more-or-less spread out probability (uncertainty) to be at various positions. (Of course this statement is just an attempted approximation). This spread-out probability-of-positions evolves over time, influenced over time by different physical states at different locations (also, in Single World Interpretations, non-locally) at each time.

The specific wavefunction which describes the particle itself (so to speak, in isolation), will need to reflect only the information that is needed to describe the current probability-distribution of the particle. It is anchored in this distribution, not anymore in the splits that it may have went through. Depending on the distribution, it may look complicated, or simple. In its mathematical description, many mathematical terms may just fall away.

Specifically, there will be visible interference effects only (at least according to A.Zeilinger, as far as I understand) if the information about which slit in went through, is *lost*.

But this is a statement that I don't understand. To me the ONLY things that can have "physical meaning" are mathematical objects, in the frame of a physical theory. So, saying that it is "just a mathematical construction" to me, is the first prerequisite for something to be able to become charged with "physical meaning". I cannot personally conceive something that has "physical meaning" if it is not, in the first place, a mathematical object.

Strange, to me that seemed to be a rather simple statement. The mathematical terms depend on your mathematical model, and how you compute it, all of which is arbitrary. There are often different mathematical possibilities to compute and describe the same physical state. For the term (0.7 sin alpha - 0.3 cos alpha), nobody expects each of the two terms to have its own physical reality. Of course

? Before the experiment, the wavefunction DOESN'T depend on the measurement angle of course. The measurement angle enters the game only:
1) in CI: by PROJECTING the wavefunction on one of its components, when it is written in the basis corresponding to the angle at Bob, for instance.
Of course, once this is done in a strictly global way, Alice only has this component to go on, and hence her projection, along her axis, will depend on what projection bob did on his side.

2) in MWI: by making the bob-state interact with the wavefunction of the particles, and this INTERACTION (bob-particle_at_bob) is specified by the measurement angle at bob. The same on Alice's side: Alice will INTERACT with the wavefunction (unaltered) of her particle, and this interaction (alice-particle_at_alice) is specified by the measurement angle at alice.
So what we have now is not so much that the wavefunction of the particles has altered, but rather that bob interacted in a specific way with HIS particle, which put him in different states, and alice interacted with HER particle, which put HER in different states, and when these alice states finally come together with the bob states, and interact by exchanging information, then AT THAT MOMENT, Alice/bob paired-up states appear with different amplitudes, which are of course a function of the relative interactions of alice and bob with their particles, as this determined in what kind of states they happened to arrive. It is the fact that all these interactions can be considered as local that makes me say that the theory can still be considered local.

How is this supposed to be a response to my simple statement: "And when a measurement is made in an entanglement experiment, the wavefunction of course depends also on the measurement angle." ?

Whether you explain it locally or non-locally, the measurement angle will influence both the state of the particle, as well as the result that Alice or Bob see.

Yes. Like a photon, which "splits" over two slits, and then pairs up with itself again, to make an interference pattern. It is only at that place that an interference pattern is made, and the slit system could be destroyed by then.

[...]

Yes, but the slits might have been blown up before the interference pattern was shown too.

That's exactly my point. The wave function will then be anchored by the current state of the particle (its probability distribution) and the configuration it interacts with at that time. Only that you may compute the wave function more easily from the previous state of the wavefunction.

Ah, that's then the difference. The way I look upon things (others have of course different views) is: in quantum theory, the REAL physical universe is the Hilbert space in MWI. We only get an impression of a kind of 3-dim space with objects in it, simply because these are the kinds of "classical states" we can experience. It is the property of locality which contributes strongly to that impression.

But again, note that I don't really think that "the universe is a vector in hilbert space", because I don't really think that quantum theory is the ultimate final theory. It's probably much more sophisticated than this. I put myself in this "frame of thinking" for the ease of understanding how quantum theory works.

That may be the case in MWI, and I've heard hints of that before, that in MWI the wavefunction is considered to be real (though not that Hilbert space is real), but that is not self-evident and something you need to say. In Bohmian mechanics, for example, it is not the wavefunction, but the 'quantum potential', that is physically real (although for example I don't know in which space the quantum potential is meant to exist).

Saying that is part of what my "question" is about.

But then, you say you don't "really" think the universe is a vector in Hilbert space. I am glad you don't, since I still see Hilbert space as an arbitrary mathematical construct, but then where does that leave "reality" ?

Again, I'm unable to make the distinction between a mathematical theory that has the ambition to be a physical theory, and a "real physical theory". If you have a mathematical theory that is saying something about physics, and you use it, then you ASSUME that its objects are "physical" of course.

Like Newton ASSUMED that there was a 3-dim euclidean space outside, simply because he used this in his theory to position his essential objects, which are point particles (mappings from the real axis into 3-dim space).

But that distinction is very easy, once you get my simple point. Take the distinction between Boolean Algebra, and a physical description of the computer electronics.

A quantum computer may perform operations that for a classical machine would require a computer of a size larger than the universe, I've heard.

Yet quantum physics may have its own limitations for what it can do within the size of the universe.

I don't understand this. If you can define the necessary mathematical objects, then you simply DECLARE them to be physical, no ?

What you need to show is that it would be possible to reconstruct the angles, or a factor that pairs-up the angles, from a probability distribution that has evolved in the most complex ways since the measurement was done.

You haven't done that yet. Instead you are patiently teaching me quantum physics, Thank you, maybe eventually I will be able to answer the question I am asking you. But I still don't see that you have an answer, and that doesn't convince me that an actual physical implementation won't have to make compromises in this universe which will allow it to supply *and* *use* this possibly huge amount of information at the very edge of any light cone, or email, passing through fingers, keyboards, and the internet, etc., to find in it a factor that pairs up complex states according to an entangle particle and measurement angle that was applied long ago, at a different space and time.

Until you have that, your model is not only non-local, but also non-temporal (spooky effects at a different time).

 

[vanesch]

The particle doesn't have, for example, a specific position, but a more-or-less spread out probability (uncertainty) to be at various positions. (Of course this statement is just an attempted approximation). This spread-out probability-of-positions evolves over time, influenced over time by different physical states at different locations (also, in Single World Interpretations, non-locally) at each time.

Ah, but that's an error! The particle doesn't have a "probability to be at different positions" ; if you do that, you run into all kinds of paradoxes. The quantum state of the single point particle is a SUPERPOSITION of its possible positions, phase information included. You can have identical probability distributions, and different phase (complex number) relationships, and this will yield in entirely different results. You can even have "one branch" of the particle following a totally different way in space than the other branch, envellop a planet or more, and still have them interfere. This is btw what happens to the photons that come from distant galaxies and suffer gravitational lensing: the same photon went "left" and "right" of an entire galaxy, and then curved back to interfere with its "other half" on the photodetector of a telescope.

The specific wavefunction which describes the particle itself (so to speak, in isolation), will need to reflect only the information that is needed to describe the current probability-distribution of the particle. It is anchored in this distribution, not anymore in the splits that it may have went through. Depending on the distribution, it may look complicated, or simple. In its mathematical description, many mathematical terms may just fall away.

First of all, it is not sufficient to describe the "probability distribution", which doesn't make sense in between detections. A quantum state doesn't give you consistent probability distributions in between measurements. It is a common error, which leads to a lot of pseudoparadoxes. But putting that aside, the "simplicity" comes about because we are dealing here with a classical system which corresponds to a single point. So the superposition principle is limited to "superpositions of points in space" which we can mistake for classical fields. But this is not so anymore for, say a system with TWO points. The superpositions are now all thinkable complex superpositions of COUPLES of points in space. This is shown by the fact that the wavefunction is now psi(x1,y1,z1,x2,y2,z2) (which gives you the complex amplitude of the couple of points, at (x1,y1,z1) and (x2,y2,z2) ) and is not in general "splittable" in a "state of particle 1" and "a state of particle 2".
So the mathematical description is now giving you the probabilities of COUPLES OF POINTS when you measure.

Specifically, there will be visible interference effects only (at least according to A.Zeilinger, as far as I understand) if the information about which slit in went through, is *lost*.

This is silly blahblah. A quantum particle that went through the two slits simultaneously, never had any "information about through which the particle went". This is the kind of nonsense which leads also to Afshar's experiment and so on.
Again, in a 2-slit experiment, the particle was in a state which was a superposition of being at each slit individually. Now, you can entangle these states with, say, polarisation states of the particle (that's what Zeilinger does), and you will not get an interference pattern with these states (which is normal, they are entangled states), unless you "disentangle" them by doing a measurement of the polarization along an axis which is 45 degrees with the two original polarizations. Now, in the "which way" mumbo jumbo, you say that you have "erased" the information about the which way which was encoded in the polarization states. But what you actually did, was to re-arrange the terms in the wavefunction in such a way, that THOSE that give spin up in the +45 degree polarization, make one spatial interference pattern, and those that give spin down in the +45 degree polarization, make the opposite interference pattern.

So overall, you STILL don't have an interference pattern, but by doing coincidence counts with the outcome of the 45 degree polarizer, you can select one, or the other subset of data, which show complementary interference patterns.

But look at how nice this works out ( I skip all factors 1/sqrt(2) ):

The initial state is:
|slit1> |x> + |slit2> |y>

slit 1 and slit 2 give the position of the photon at slit 1 and slit 2 ;
x is the polarisation along x -axis
y is the polarisation along y-axis

The measurement basis of our analyser under 45 degrees:

|x> = |45> + |135>
|y> = |45> - |135>

Re-writing the original wavefunction:
|slit1> (|45> + |135>) + |slit2> (|45> - |135>)

= |45> (|slit1> + |slit2> ) + |135> (|slit1> - |slit2> )

So we see that for THOSE THAT SAW |45>, they will find the "position quantum state" to be |slit1> + |slit2> which gives you the usual interference pattern, and those that saw |135> will find pair up with:
|slit1> - |slit2> which gives you the interference pattern with a 180 degree phase shift, in other words, the complementary interference pattern.

Strange, to me that seemed to be a rather simple statement. The mathematical terms depend on your mathematical model, and how you compute it, all of which is arbitrary. There are often different mathematical possibilities to compute and describe the same physical state. For the term (0.7 sin alpha - 0.3 cos alpha), nobody expects each of the two terms to have its own physical reality. Of course

There's a difference between the abstract mathematical object, and then "tricks to compute". I was talking about the abstract mathematical objects.

Whether you explain it locally or non-locally, the measurement angle will influence both the state of the particle, as well as the result that Alice or Bob see.

The LOCAL particle, yes.

That's exactly my point. The wave function will then be anchored by the current state of the particle (its probability distribution) and the configuration it interacts with at that time.

You can twist and turn it as you want, from the probability distribution you cannot recover the wavefunction, not for a single particle, and even less so for couples, or triples of particles.

That may be the case in MWI, and I've heard hints of that before, that in MWI the wavefunction is considered to be real (though not that Hilbert space is real), but that is not self-evident and something you need to say.

Well, it is hard to conceive the wavefunction without the hilbertspace of which it is an element. That's a bit like saying that one considers the position of the moon to be real, but not the euclidean space in which this position is an element...

In Bohmian mechanics, for example, it is not the wavefunction, but the 'quantum potential', that is physically real (although for example I don't know in which space the quantum potential is meant to exist).

Haha, in Bohmian mechanics, the quantum potential IS the wavefunction! And it doesn't live in 3-d space, but in ... Hilbert space, all the same. Bohmian mechanics has the entire "MWI dynamics" (for the quantum potential) PLUS extra dynamics for the particles. This allows Bohmians to have a purely deterministic theory, and the particle part looks strongly like Newtonian mechanics ; only, they need on top of that the entire quantum dynamics to have the quantum potential.

But then, you say you don't "really" think the universe is a vector in Hilbert space. I am glad you don't, since I still see Hilbert space as an arbitrary mathematical construct, but then where does that leave "reality" ?

For me, the concept of reality is a working hypothesis which helps us organize our perceptions. As we have different classes of perceptions, and not yet a coherent theory of everything, we need different, incompatible working hypotheses to make sense of different classes of perceptions. In daily life, we can usually do with a working hypothesis that we have a physical body, that there are objects around us, that we live in a kind of patch of 3-dim Euclidean-like space and so on. It's the hypothesis which makes most sense, and which helps us most make sense of our sensations, and it seems that this is the kind of working hypothesis our brain seems to be wired up for.
But when doing more sophisticated things, we know that this runs into trouble. So we switch to different working hypotheses - which might very well be incompatible with our "daily life" working hypothesis. And as we don't have any intuition here, but we DO have mathematical models, then I take as "ontological hypothesis" simply the Platonic existence of the mathematical models in question.

In relativity, "the world" is a 4-dim static blob of manifold. Nothing "moves" in it. In quantum physics, the world is a speck in hilbert space. In Newtonian physics, things are closer to our "daily life" model. However, in Hamiltonian mechanics, the world is a 6N dim manifold (or better, a spec in a 6N dim manifold, following the hamiltonian flow).

But that distinction is very easy, once you get my simple point. Take the distinction between Boolean Algebra, and a physical description of the computer electronics.

And what is that physical description ? The still classical description in 3dim euclidean space of the pieces of silicon, copper, PCB, etc... as in a Solid Works drawing ? But what do we do with the electrons in the silicon ? Wavefunctions ? Electric fields ? Drude model ?

So in order to describe a logical gate, you need quite a lot of mathematical objects! So you simplify... and in the end, you simplify to the point of just writing some VHDL, representing the boolean algebra of your device!

What you need to show is that it would be possible to reconstruct the angles, or a factor that pairs-up the angles, from a probability distribution that has evolved in the most complex ways since the measurement was done.

But that means that you take as basic physical objects, the probability distributions (in 3-space ?) of what ?

But this IS NOT GOING TO WORK, for 2 reasons:

first of all, the probability distributions (over configuration space, and certainly not its projections on 3-space) do not contain enough information to reconstruct the quantum state, as I said before.

But second, it is conceptually difficult to consider a PROBABILITY distribution to have some physical meaning. Probability is "lack of knowledge". You seem to associate "probability waves" with some kinds of "classical fields in 3-d space" and you seem to accept that as the "only containers of physical reality". Well, you're not alone, and you will run in A LOT OF SPOOKY PARADOXES if you cling onto that view, as do many others.

A quantum state is NOT a probability distribution and certainly not one in 3-dim space, from the moment that we have more than 1 point particle.

 

[colorSpace]

Ah, but that's an error! The particle doesn't have a "probability to be at different positions" ; if you do that, you run into all kinds of paradoxes. The quantum state of the single point particle is a SUPERPOSITION of its possible positions, phase information included. You can have identical probability distributions, and different phase (complex number) relationships, and this will yield in entirely different results. You can even have "one branch" of the particle following a totally different way in space than the other branch, envellop a planet or more, and still have them interfere. This is btw what happens to the photons that come from distant galaxies and suffer gravitational lensing: the same photon went "left" and "right" of an entire galaxy, and then curved back to interfere with its "other half" on the photodetector of a telescope.

You seem to be keen to detect errors in the writings of a non-physicist. :)

But I don't even see why a photon going left and right of an entire galaxy, and then interfering, would contradict a 'probability distribution', except that it has to be (as I already wrote elsewhere) a little more than a probability in the classical sense, in order to interfere with itself. That's just due to the shortness of expression. Otherwise your description doesn't seem to contradict my mental picture at all. And that's why I wrote:

" (Of course this statement is just an attempted approximation)"

First of all, it is not sufficient to describe the "probability distribution", which doesn't make sense in between detections. A quantum state doesn't give you consistent probability distributions in between measurements. It is a common error, which leads to a lot of pseudoparadoxes. But putting that aside, the "simplicity" comes about because we are dealing here with a classical system which corresponds to a single point. So the superposition principle is limited to "superpositions of points in space" which we can mistake for classical fields. But this is not so anymore for, say a system with TWO points. The superpositions are now all thinkable complex superpositions of COUPLES of points in space. This is shown by the fact that the wavefunction is now psi(x1,y1,z1,x2,y2,z2) (which gives you the complex amplitude of the couple of points, at (x1,y1,z1) and (x2,y2,z2) ) and is not in general "splittable" in a "state of particle 1" and "a state of particle 2".
So the mathematical description is now giving you the probabilities of COUPLES OF POINTS when you measure.

It seems with a superposition of two points, you are in this case talking about entanglement. (Otherwise what I wrote above applies.)

Yes, entanglement requires the assumption of additional physical states, certainly at least in a single world interpretation, it requires non-local physical states or connections. Wavefunctions might be the best way to describe entanglement states, but that doesn't mean that a wavefunction, which appears to be a mathematical construct of arbitrary formulation, has each of its terms anchored in its own physical reality.

One can point out that all our information comes from measurements in 3D space, and although I don't philosophically limit myself to 3D space at all, I haven't yet seen that physical reality needs more than that. Even string theorists seem to be happy with a limited number of 10 or 11, or so, dimensions. The rest appears to be mathematical convenience.

This is silly blahblah. A quantum particle that went through the two slits simultaneously, never had any "information about through which the particle went". This is the kind of nonsense which leads also to Afshar's experiment and so on.
Again, in a 2-slit experiment, the particle was in a state which was a superposition of being at each slit individually. Now, you can entangle these states with, say, polarisation states of the particle (that's what Zeilinger does), and you will not get an interference pattern with these states (which is normal, they are entangled states), unless you "disentangle" them by doing a measurement of the polarization along an axis which is 45 degrees with the two original polarizations. Now, in the "which way" mumbo jumbo, you say that you have "erased" the information about the which way which was encoded in the polarization states. But what you actually did, was to re-arrange the terms in the wavefunction in such a way, that THOSE that give spin up in the +45 degree polarization, make one spatial interference pattern, and those that give spin down in the +45 degree polarization, make the opposite interference pattern.

In this case, I was probably more thinking of delayed choice experiments, where the experimental configuration is such that the different "ways" are first separate for a while, and then re-join in a way that the which-way information is "lost", although "lost" isn't really the right word. My point here is that the state doesn't carry along information about its history, under such circumstances, even though the arrival of the particle depends on specific and separate conditions along each path, such as the presence of mirrors.

So overall, you STILL don't have an interference pattern, but by doing coincidence counts with the outcome of the 45 degree polarizer, you can select one, or the other subset of data, which show complementary interference patterns.

But look at how nice this works out ( I skip all factors 1/sqrt(2) ):

The initial state is:
|slit1> |x> + |slit2> |y>

slit 1 and slit 2 give the position of the photon at slit 1 and slit 2 ;
x is the polarisation along x -axis
y is the polarisation along y-axis

The measurement basis of our analyser under 45 degrees:

|x> = |45> + |135>
|y> = |45> - |135>

Re-writing the original wavefunction:
|slit1> (|45> + |135>) + |slit2> (|45> - |135>)

= |45> (|slit1> + |slit2> ) + |135> (|slit1> - |slit2> )

So we see that for THOSE THAT SAW |45>, they will find the "position quantum state" to be |slit1> + |slit2> which gives you the usual interference pattern, and those that saw |135> will find pair up with:
|slit1> - |slit2> which gives you the interference pattern with a 180 degree phase shift, in other words, the complementary interference pattern.

There's a difference between the abstract mathematical object, and then "tricks to compute". I was talking about the abstract mathematical objects.
The LOCAL particle, yes.

...

You can twist and turn it as you want, from the probability distribution you cannot recover the wavefunction, not for a single particle, and even less so for couples, or triples of particles.

It wasn't meant as an exhaustive physical model to explain all phenomena, just as an illustration of what I would consider a "physical state" as opposed to a mathematical function describing the mathematical relationship between physical states.

My simple point is that one cannot automatically expect each term of such a mathematical function to have a physical reality of its own.

Well, it is hard to conceive the wavefunction without the hilbertspace of which it is an element. That's a bit like saying that one considers the position of the moon to be real, but not the euclidean space in which this position is an element...

Well my personal opinion (not a physical one) is that space is a mental construct altogether, not an ultimate reality.

I wonder why you put the moon in "euclidian" space, though.

I think when you say that wavefunctions are not conceivable without Hilbert space, then you made yourself subject to a specific mathematical model that was invented for convenience. You cannot derive any conclusion about physical reality from a space that was invented for convenience of calculation. That's arguing backwards, it seems to me.

Haha, in Bohmian mechanics, the quantum potential IS the wavefunction! And it doesn't live in 3-d space, but in ... Hilbert space, all the same. Bohmian mechanics has the entire "MWI dynamics" (for the quantum potential) PLUS extra dynamics for the particles. This allows Bohmians to have a purely deterministic theory, and the particle part looks strongly like Newtonian mechanics ; only, they need on top of that the entire quantum dynamics to have the quantum potential.

I heard that the quantum potential was derived from the wavefunction, rather than identical with it. If that isn't correct, then I would first have to read more about that, in order to discuss it.

For me, the concept of reality is a working hypothesis which helps us organize our perceptions. As we have different classes of perceptions, and not yet a coherent theory of everything, we need different, incompatible working hypotheses to make sense of different classes of perceptions. In daily life, we can usually do with a working hypothesis that we have a physical body, that there are objects around us, that we live in a kind of patch of 3-dim Euclidean-like space and so on. It's the hypothesis which makes most sense, and which helps us most make sense of our sensations, and it seems that this is the kind of working hypothesis our brain seems to be wired up for.
But when doing more sophisticated things, we know that this runs into trouble. So we switch to different working hypotheses - which might very well be incompatible with our "daily life" working hypothesis. And as we don't have any intuition here, but we DO have mathematical models, then I take as "ontological hypothesis" simply the Platonic existence of the mathematical models in question.

In relativity, "the world" is a 4-dim static blob of manifold. Nothing "moves" in it. In quantum physics, the world is a speck in hilbert space. In Newtonian physics, things are closer to our "daily life" model. However, in Hamiltonian mechanics, the world is a 6N dim manifold (or better, a spec in a 6N dim manifold, following the hamiltonian flow).

...

And what is that physical description ? The still classical description in 3dim euclidean space of the pieces of silicon, copper, PCB, etc... as in a Solid Works drawing ? But what do we do with the electrons in the silicon ? Wavefunctions ? Electric fields ? Drude model ?

So in order to describe a logical gate, you need quite a lot of mathematical objects! So you simplify... and in the end, you simplify to the point of just writing some VHDL, representing the boolean algebra of your device!

Well, the point is that Boolean Algebra makes sense without any physical computer, it is not just a high level description of a computer, but completely independent of computers, or any specific physical implementation. You could implement it with hydraulics, for example. Or not at all.

But that means that you take as basic physical objects, the probability distributions (in 3-space ?) of what ?

But this IS NOT GOING TO WORK, for 2 reasons:

first of all, the probability distributions (over configuration space, and certainly not its projections on 3-space) do not contain enough information to reconstruct the quantum state, as I said before.

But second, it is conceptually difficult to consider a PROBABILITY distribution to have some physical meaning. Probability is "lack of knowledge". You seem to associate "probability waves" with some kinds of "classical fields in 3-d space" and you seem to accept that as the "only containers of physical reality". Well, you're not alone, and you will run in A LOT OF SPOOKY PARADOXES if you cling onto that view, as do many others.

A quantum state is NOT a probability distribution and certainly not one in 3-dim space, from the moment that we have more than 1 point particle.

The real point is the difference between an abstract mathematical description, and a physical implementation.

If even that can't be addressed clearly, then we can't discuss.

As far as I am concerned, you could equally claim that classical waves can't be described without sin or cos, and that therefore the sin and cos functions must have physical reality. Which would be absurd. I hope. Not so sure anymore.

The thing seems to be that you carry over mathematical terms from an unfinished calculation, and then act as if physical reality could simply finish that calculation later on, where the calculation still looks like a birds-eye-view calculation that is performed on terms that have no individual physically meaningful reality.

[Edit:] And when physical states are added to represent them, it would seem that they need to represent so much independent information, that they would have their own requirements in terms of space and time. And that is just one of the challenges which remain unanswered, the best I can tell.

 

[Mentz114]

Having read the whole thread, I congratulate Vanesch on a well argued case. ColorSpace, you like a good argument, don't you ? But you are not seeing things like a practical physicist and it's hard to discern what your objections are.

The real point is the difference between an abstract mathematical description, and a physical implementation.

You must remember that the theories that survive are the ones that agree with experiment, and common observation. The quantities used in the calculations may have no physical reality, but as long as the equations help avoid crashes and collapsed bridges, why worry ? There is no physical thing that corresponds to a wave-function, and no physical thing that corresponds to space-time curvature. But they are really useful concepts. Physics is not mathematics, nor is playing with equations. There are no theorems in physics, only theories.

 

[colorSpace]

Having read the whole thread, I congratulate Vanesch on a well argued case. ColorSpace, you like a good argument, don't you ? But you are not seeing things like a practical physicist and it's hard to discern what your objections are. You must remember that the theories that survive are the ones that agree with experiment, and common observation. The quantities used in the calculations may have no physical reality, but as long as the equations help avoid crashes and collapsed bridges, why worry ? There is no physical thing that corresponds to a wave-function, and no physical thing that corresponds to space-time curvature. But they are really useful concepts. Physics is not mathematics, nor is playing with equations. There are no theorems in physics, only theories.

I'm not questioning the usefulness of wavefunctions, every interpretation uses them, AFAIK.

We are discussing whether, within MWI, a local interpretation makes sense, or not. Also I'm learning a lot about MWI this way. Vanesh hasn't made the point that the concept of local splits would be a practical simplification, on the contrary, it seems to make things more complicated. Perhaps he disagrees, I don't know, but the question seems to be whether it is possible, even if it makes things more complicated. To me it would seem that it would be easier to make global, non-local splits, but I wouldn't be sure since I know MWI very little. And outside MWI, this concept doesn't seem to work in any case, as far as I can tell.

Regarding the possibility, my main point is that the concept he presents doesn't seem to have been worked out to a point where the question can be answered. Specifically, I don't get any clarity about what kind of additional physical states would be required. [Edit:] Or not required... :)

 

[colorSpace]

I guess a simple question might bring a more productive point of view to the discussion (or not):

In a local view, what would keep the state description on Bob's side from mathematically adding up to:
[Edit: after measurement, as a result.]

0.5 |bob+> + 0.5 |bob->

if that is the correct notation to express that the only distinct possible physical states are bob+ and bob-, and that their probability is 50% each.

Doesn't a more complex state also require more distinct physical states, which haven't been specified yet?

 

[colorSpace]

The quantities used in the calculations may have no physical reality, but as long as the equations help avoid crashes and collapsed bridges, why worry ? There is no physical thing that corresponds to a wave-function, and no physical thing that corresponds to space-time curvature.

I read this a third time. It sounds like you already thought about what I am thinking about right now !

 

[vanesch]

But I don't even see why a photon going left and right of an entire galaxy, and then interfering, would contradict a 'probability distribution', except that it has to be (as I already wrote elsewhere) a little more than a probability in the classical sense, in order to interfere with itself. That's just due to the shortness of expression. Otherwise your description doesn't seem to contradict my mental picture at all. And that's why I wrote:

" (Of course this statement is just an attempted approximation)"

The point I wanted to stress was this: if you say a *probability* distribution, then it means that the photon REALLY is somewhere specific, but that we DON'T KNOW (or can't know) exactly where. But it IS at a single point, and occupies one specific point in space (3D space). This is BTW what happens also if you give a probabistic interpretation of the wavefunction when it is not being measured. In other words, "probability" is not a physically meaningful concept that could have an ontological existence outside of our knowledge, it is a description of what we know, and what we don't know. So if you say that the photon is a point particle, with a spatial probability distribution, then you simply mean, with all you can know, the point particle can be here OR there OR there. But it is at ONE of these places.
Well, if you do that in quantum mechanics, you run into troubles each time there is what one calls "quantum interference". Our photon going left can then not interfere with itself going right, because it was OR left, OR right. It is the fundamental "mystery" of the two-slit experiment. But the *quantum state* (the state vector, the wavefunction, the point in hilbertspace) doesn't say that the particle is OR here, OR there. It tells you that it is in a superposition of being here AND there. So the particle, in this description, is "in both places at once".
Being ONLY here is a different physical state, and being ONLY there is still a different physical state, and the statistical mixture of 50% chance of being here and 50% chance of being there is a mixture of the last two physical states with uncertainty, and has nothing to do with the DIFFERENT state which says that the particle is BOTH here and there.

It seems with a superposition of two points, you are in this case talking about entanglement. (Otherwise what I wrote above applies.)

Yes, entanglement is the application of the superposition principle to more than one single subsystem. But you have to see that it is the same principle at work. We have a superposition of the different possible classical states of a "pair of points". It is not different from the assumption of superposition of the classical states of a single point. The only difference is that this time, no confusion is possible anymore between a classical wave in 3D and this superposition, which was, unfortunately the case for the single point particle case. (and hence all the wave/particle mumbo jumbo, which breaks down in any case in the multiparticle case).

Yes, entanglement requires the assumption of additional physical states, certainly at least in a single world interpretation, it requires non-local physical states or connections. Wavefunctions might be the best way to describe entanglement states, but that doesn't mean that a wavefunction, which appears to be a mathematical construct of arbitrary formulation, has each of its terms anchored in its own physical reality.

Again I don't see why you insist both on the "non-physicality" of these entangled states as described in hilbert space, as well as its non-locality.

One can point out that all our information comes from measurements in 3D space, and although I don't philosophically limit myself to 3D space at all, I haven't yet seen that physical reality needs more than that. Even string theorists seem to be happy with a limited number of 10 or 11, or so, dimensions. The rest appears to be mathematical convenience.

As I discussed before, I don't see why you make a distinction between the mathematical convenience of a 3-d Euclidean space (which is a pre-Newtonian invention, strongly supported by our intuition), and the mathematical convenience of Hilbert space. Both are mathematical constructions used in physical theories. What gives the 3-d space "more" right to an ontology than Hilbert space ?
You haven't yet seen a physical reality that "needs" more than that, nevertheless, you seem to run into troubles with the non-locality of entangled states nevertheless. So maybe it is time to reconsider this "no need" of more than a 3-d Euclidean space.

However, note that I don't abolish 3-d Euclidean space. It still has an important role as BASE SPACE. In fact, this is what locality is all about: does 3-d Euclidean space has a physical meaning ? Does it play any particular role ? In a local theory, clearly, the answer is yes. In a non-local theory, the answer is of course no. In a non-local theory, there is no need to split the state in "many states, each mapped from the 3-d space".

It seems that the only objection you have to my local version of MWI quantum theory, is the complexity and the arbitrariness of the local objects that travel around in 3-d space, but you don't seem to object that one CAN construct such objects.

In the same way as you seem to insist on the non-locality of an entangled state just for the reason that you can only conceive classical point particles and classical waves in 3-d space, and not more complicated objects in 3-d space.

But I don't see why the QUANTUM state |a+>|b+> + |a->|b->, where - I hope you agreed - such a state can only OCCUR after an interaction of a with b, or of an interaction of a and b with something that was already entangled (which shifts us then to the previous couple of entangled things), I don't see why you refuse to consider that the a+ state carries with it the information that it was entangled with the b+ state of the b-system, and that the b+ state carries with it that it was entangled with the a+ state. As I pointed out, you are not going to find a CLASSICAL bucket where you can store this information, because the classical bucket is already fully described by |a+> alone. It is not INSIDE |a+> that this information can reside. If we go back to our point particle, which is in a superposition of position states, it is not "inside the position state" (which is simply a point in 3-d space, so 3 real coordinates) that you are going to store its amplitude. The amplitude is not stored in the 3 real coordinates. The amplitude "goes with" the 3 real coordinates (is a *function* of these 3 real coordinates, it is not the 3 coordinates itself). And in this case, it is only one single amplitude, because the quantum system is that of a single point particle.

I tried to show that it is possible (not that it is elegant or anything) to "attach" the informations needed to calculate all interactions, all amplitudes, and all probabilities of outcomes, to the "moving subsystems", and that in all these operations, you only NEED the as thus locally brought-in information to do the necessary transformations on them during interactions. This demonstration by itself is sufficient to show that quantum dynamics doesn't need any non-locality.

It wasn't meant as an exhaustive physical model to explain all phenomena, just as an illustration of what I would consider a "physical state" as opposed to a mathematical function describing the mathematical relationship between physical states.

My simple point is that one cannot automatically expect each term of such a mathematical function to have a physical reality of its own.

Not automatically, no. But it seems to me that you have peculiar, and rather arbitrary, criteria to decide what mathematical objects are to be "physical" and what not. Again, what makes the 3-d Euclidean space more eligible than Hilbert space ?

I think when you say that wavefunctions are not conceivable without Hilbert space, then you made yourself subject to a specific mathematical model that was invented for convenience. You cannot derive any conclusion about physical reality from a space that was invented for convenience of calculation. That's arguing backwards, it seems to me.

Well, I don't see why you think that hilbert space is a "convenience invented for calculations". You can't actually do much calculations in hilbertspace, you do them actually in C^n. Hilbert space is what results if you apply the superposition principle, which is the corner physical axiom of quantum theory (in the same way as the invariance of the speed of light is the corner axiom of special relativity).

If you claim, as a physical principle, that for each two physical states of a system A and B, you generate NEW distinct physical states for each pair of complex numbers c1 and c2 such that (c1,c2) is not equal to c(c1',c2') as superpositions of A and B with c1 attached to A and c2 attached to B, and you then say that each classical configuration of the system is an acceptable physical state, then you've automatically introduced a Hilbert space.
(ok, there are mathematical subtleties: actually a *projective* Hilbert space, which means that elements which are a complex multiple of one another are identified).

THIS is the corner physical axiom of quantum theory: the superposition principle. So if you accept the superposition principle as a fundamental physical axiom, then it is difficult not to attach some kind of physical meaning to the hilbert space of states, which is nothing else but the set of allowed physical states of a system, no ? A bit in the same way as one can give physical meaning to the concept of "velocity" or "momentum" in Newtonian physics, even though it is only introduced because Newton's equation of motion is a second-order differential equation. So you DO consider that, in Newtonian physics, a point particle doesn't only "have a position" in 3-d space, but also "carries the necessary initial conditions" with it in a small information bucket, namely its "momentum". Its momentum is NOT stored in the position, or in space, or nowhere else. If you don't carry it with a particle, then things become pretty non-local too.

I heard that the quantum potential was derived from the wavefunction, rather than identical with it. If that isn't correct, then I would first have to read more about that, in order to discuss it.

Yes, that's correct, the quantum potential is not identical to the wavefunction, but it is mathematically derived from it, and it needs the wavefunction to have its dynamics correct. That is, you need the wavefunction and its dynamics in an essential way to have something like the quantum potential. As such, if you want to give "physical meaning" to the quantum potential, I don't see how you will get away with that without giving also physical meaning to the wavefunction.

The real point is the difference between an abstract mathematical description, and a physical implementation.

If even that can't be addressed clearly, then we can't discuss.

As far as I am concerned, you could equally claim that classical waves can't be described without sin or cos, and that therefore the sin and cos functions must have physical reality. Which would be absurd. I hope. Not so sure anymore.

Indeed, not so clear! Is the spectral decomposition of light in different colors "physical" or not ?

The thing seems to be that you carry over mathematical terms from an unfinished calculation, and then act as if physical reality could simply finish that calculation later on, where the calculation still looks like a birds-eye-view calculation that is performed on terms that have no individual physically meaningful reality.

If you do a calculation of a trajectory in Newtonian physics, do you consider then that the intermediate positions (as a function of time) are also "unfinished calculations" and that the moon, between this morning and this evening, didn't "take these positions" as if "nature had to finish the integration" of Newton's equations which you were doing on your computer ?

The states I showed in the bob/alice examples where the quantum states at different moments in time, like, before and after interaction and so on. They were not "unfinished", but represented the quantum state of that moment, just as intermediate integration points along the orbit of the moon are not an "unfinished calculation", but represent the state of the moon at different moments in time.

[Edit:] And when physical states are added to represent them, it would seem that they need to represent so much independent information, that they would have their own requirements in terms of space and time. And that is just one of the challenges which remain unanswered, the best I can tell.

Again, "space and time" are classical concepts, which cannot contain the information needed to apply the superposition principle. That's also why you don't find hidden characters in the e-mails and so on which specify the "quantum state" of the e-mail. Obviously, and if you see that, it is something very nice, the superposition principle introduces "buckets of information" which have no classical representation. You won't find any classical (hence, directly observable) state which SHOWS you where the amplitudes introduced by the superposition principle "hide". It is IN ANY CASE something totally new.

So my surprise for you to insist that this information must be "global" and cannot be "distributed locally". Where do you think that the superposition principle "puts the information contained in the different amplitudes" ? If the superposition principle is a physical principle, clearly the amplitudes of the superpositions are physical quantities. Where do they hide ? Why is there no problem in having them "globally" but why is it unconceivable to have them "locally" ?