Will quantum computers ever be possible?

[colorSpace]

Adding quote, more specifically:

And if he waits until "Alice" comes along, he will meet with the particular Alice that was in his branch of course.

 

[vanesch]

Ok, I worked out the thing in all generality. I hope I didn't make to many
errors.

We start with a singlet state produced in the source.

|u+A> |v-A> - |u-B>|v+B>

bob measures under angle alpha1 (cos = x and sin = y):

u+ = x |u++> + y |u-->
u- = -y |u++> + x|u-->

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

Bob will be in two possible states, bob+ and bob-. However, his bob+ state will carry
2 labels, namely AC and BE (with amplitudes respectively x and +y), and his
bob- state will carry 2 labels, AD and BF with amplitudes respectively y and -x.

Alice measures under angle alpha2 (cos = r and sin = s)

v+ = r |v++> + s |v-->
v- = - s |v++> + r |v-->

So this becomes:
(x |bobAC+> |u++AC> + y |bobAD->|u--AD> )
(-s |aliceAG+>|v++AG) + r |aliceAH->|v--AH>)
- (-y |bobBE+> |u++BE> + x |bobBF-> |u--BF>)
(r |aliceBI+>|v++BI> + s |aliceBJ->|v--BJ>)

Alice is in two states, |alice+> and |alice->. alice+ has two labels attached to
it, namely AG and BI, with amplitudes -s and +r respectively. alice- has two labels
attached to it, namely AH and BJ, with amplitudes r and s respectively.

Imagine now that bob and alice meet somewhere, but imagine first that they do not
yet interact.

bobAC+ will then meet the alices with an A-label, namely aliceAG+ and aliceAH-.

bobAD- will then meet also aliceAH- and aliceAG+

bobBE+ will meet aliceBI+ and aliceBJ-

bobBF- will meet aliceBI+ and aliceBJ-


Now, they interact (exchange their findings). We seem to have 8 different couples now:

bobAC+ and AliceAG+ which find both the ++ result --> |bobalice++ ACG>
bobAC+ and AliceAH- which find both the +- result --> |bobalice+- ACH>
bobAD- and AliceAG+ which find both the -+ result --> |bobalice-+ ADG>
bobAD- and AliceAH- which find both the -- result --> |bobalice-- ADH>

bobBE+ and aliceBI+ which find both the ++ result --> |bobalice++ BEI>
bobBE+ and aliceBJ- which find both the +- result --> |bobalice+- BEJ>
bobBF- and aliceBI+ which find both the -+ result --> |bobalice-+ BFI>
bobBF- and aliceBJ- which find both the -- result --> |bobalice-- BFJ>

right. This is at first sight a strange situation, because it would seem that
there is a split into 8 different end states according to the labels, and in fact,
there are only 4 distinct states.

Indeed, the alice-bob couple can be in the bobalice++ state, but this state carries
two labelsets: ACG and BEI. That's the same as when the bob+ state also had 2 labels,
namely AC and BE. In fact, it is the "continuation" of this double label.

This double label finds its origin in the indeterminacy of the notation of the original
pair (the singlet state). Indeed, the singlet state |+>|-> - |->|+> has many different
algebraically identical representations:

|+z>|-z> - |-z>|+z> = |+x>|-x> - |-x>|+x>, where z and x are two different axes.
They represent the same state in hilbert space, but they are written in two different
entangled ways. It is because we "picked out the wrong entanglement representation" that
we now have to carry these double labels around. When we split the labels in A and B,
this was not the "right" way to split them. If we would have picked the right
representation (the axis that bob was going to choose), which, let's remind us,
wouldn't change a thing to the quantum state (the vector in hilbert space), then
the situation would have been like it was explained before.

Most entanglements do not allow for such a "representational degeneracy". It is
a property of the spherical symmetry of the s=0 state that results in this.

 

[vanesch]

According to everything I can tell, Bob doesn't know it, but each particular Bob is in a branch with the corresponding particular Alice. You just wrote yourself that the pairing-up happens at the source: "Indeed, it all depends on the initial pairing up... which happened in the source of the particles. Sorry for not having made that clear."

Yes, I was a bit sloppy about this. You are right that in general, a particular bob cannot pair up with any labeled alice, but only with an alice which shares the same label (is in the same branch) as himself. Only, there can be several Alices in his branch and they can cover all the possible alice states!

For instance:

|keyboardAlice0>|computerAlice0> ((x |uu+AC>|brainalice+AC> + y |uu-AD>|brainalice-AD>)|v+A>|brainbob+A> - (-y |uu+BE>|brainalice+BE> + x |uu-BF>|brainalice-BF> )|v-B>|brainbob-B>) |thunderbird_bob0>


brainbob+A can still pair up OR with brainalice+AC OR with brainalice-AD.

But that's "label talk". In "state talk" we have that a brainbob+ state can still pair up with a brainalice+ state OR with a brainalice- state.

In other words, until they pair up, the bob who saw the outcome + (and hence is a brainbob+ state) CANNOT MAKE THE HYPOTHESIS THAT ALICE ALREADY HAS AN OUTCOME, because from his PoV, the TWO alice states can still pair up with him (brainalice+ as well as brainalice-). The decision can be postponed until they meet and interact.

It is THIS feature which makes this story different from a local hidden variable model. In a LHV model, there is assumed to be 1 alice outcome and 1 bob outcome. So bob can assume that alice has a specific outcome, and talk about the probability of that outcome and all that. This is impossible for "our" bob, because until he meets alice, it doesn't make any sense to say that alice had an outcome, and he can, as such, not talk about the probability that alice has an outcome as of yet. Both alice's (and both outcomes) still "exist" from his PoV.

 

[fermio]

strange you two writing. I don't know any a|0->|1->+b|1>|0>... What means - after 1 or 0? Can you write in 0 and 1 basis? Bell theorem probably is right, but experiments showing that quantum computer never will work.

 

[colorSpace]

It seems that the state descriptions in your first response (message #72) are now more what I would expect, especially since they are more symmetrical, and use measurement angles on both sides. I'm still having difficulties with the details of this notation, but I can now see that the amplitudes are used to address the probabilities resulting from the angles, since it is now done on both sides.

There seems to be some redundancy (8 couples) because now you have 3 splits. (Correct me if I'm wrong.) But that seems to be better than your earlier version in this thread, which had one split before the particles separate, and then one split later only on one side. You seem to return to the earlier version in message #73. Whereas in your discussion with 'nrqed', you seemed to have only one split (before the particles separate) (per entangled pair).

Optimally, you might need only 2 splits, but they would have to be both after the particles separate, on both sides symmetrically. But I don't know what that would look like in the notation you are using.

So from the versions you provided, I think I prefer the version in message #72, so let me focus on this one.

Now each Bob-state has two options of Alice-states for pairing-up, for example bobAC+ can meet either AliceAG+ or AliceAH-. And each Alice-state has two options of Bob-states, for example AliceAG+ can meet either bobAC+ or bobAD-.Now the big question:
------------------------

When they meet, how is it decided which Bob-state meets which Alice-state? I think you can't say 'the state description says which states go together', since the state description includes all Bob-states as well as Alice-states in a single description, so it is a global (non-local) state description.

In order to be a local theory, the theory would have to explain how each Bob-state is paired-up with each Alice-state, when they meet, based only on locally available physical states.

I think the theory doesn't yet address this question, and it would be difficult for it to do so.

I'm not sure whether you are going to understand my question. Please don't just try to answer the question, but also give me some indication of how you understand the question.

 

[vanesch]

It seems that the state descriptions in your first response (message #72) are now more what I would expect, especially since they are more symmetrical, and use measurement angles on both sides. I'm still having difficulties with the details of this notation, but I can now see that the amplitudes are used to address the probabilities resulting from the angles, since it is now done on both sides.

There seems to be some redundancy (8 couples) because now you have 3 splits. (Correct me if I'm wrong.) But that seems to be better than your earlier version in this thread, which had one split before the particles separate, and then one split later only on one side. You seem to return to the earlier version in message #73. Whereas in your discussion with 'nrqed', you seemed to have only one split (before the particles separate) (per entangled pair).

Optimally, you might need only 2 splits, but they would have to be both after the particles separate, on both sides symmetrically. But I don't know what that would look like in the notation you are using.

I tried to address this. The extra split comes from the fact that we have a notational redundancy in the way of writing the singlet state (the element in the hilbert space Hu x Hv) as a superposition of product states.

You see, the state |z+>|z-> - |z->|z+> is mathematically equal to the state |theta+>|theta-> - |theta->|theta+> for all theta.

Here, |theta+> is the state that corresponds to "spin up" along the axis with angle "theta". So we have, mathematically, that:

|theta+>|theta-> - |theta->|theta+> = |theta2+>|theta2-> - |theta2->|theta2+>

for any 2 angles. This is in general not the case, but for the singlet state, it is.

So my "labels A and B" could have been "for theta1+ and theta1-, or I could have given them to "theta2+" and "theta2-",...

When working with wavefunctions, this doesn't matter, and it is usual practice to pick, after the fact, that basis (remember, it is just a matter of WRITING one single element of Hilbert space into a specific product basis) which simplifies calculation. Usually, we take it to coincide with the basis imposed by one of the analyser axes - I picked Bob's axis before.

If I don't do that, then I "picked the wrong basis to write down the same vector", and hence my split in A and B is not going to be aligned with the split of bob.

Now each Bob-state has two options of Alice-states for pairing-up, for example bobAC+ can meet either AliceAG+ or AliceAH-. And each Alice-state has two options of Bob-states, for example AliceAG+ can meet either bobAC+ or bobAD-.

Indeed.

Now the big question:
------------------------

When they meet, how is it decided which Bob-state meets which Alice-state? I think you can't say 'the state description says which states go together', since the state description includes all Bob-states as well as Alice-states in a single description, so it is a global (non-local) state description.

In order to be a local theory, the theory would have to explain how each Bob-state is paired-up with each Alice-state, when they meet, based only on locally available physical states.

I think the theory doesn't yet address this question, and it would be difficult for it to do so.

I'm not sure whether you are going to understand my question. Please don't just try to answer the question, but also give me some indication of how you understand the question.

This is the "multi" part in the MWI !

bobAC+ will BOTH meet AliceAG+ and AliceAH- !

But for a "bob experience" living "bobAC+", he has now a chance, given by the born rule, to be in one "world" (that is, to experience bobACG) or to be in the other world (that is, to experience bobACH).

So out of one "bob" now come two bobs. One will meet AliceAC and the other one will meet AliceAH. If YOU are bob, you will end up being one or the other, with a probability given by the born rule (and there will be a twin around now, doing the other option, but which you'll never meet = "interact with").

 

[colorSpace]

This is the "multi" part in the MWI !

bobAC+ will BOTH meet AliceAG+ and AliceAH- !

But for a "bob experience" living "bobAC+", he has now a chance, given by the born rule, to be in one "world" (that is, to experience bobACG) or to be in the other world (that is, to experience bobACH).

So out of one "bob" now come two bobs. One will meet AliceAC and the other one will meet AliceAH. If YOU are bob, you will end up being one or the other, with a probability given by the born rule (and there will be a twin around now, doing the other option, but which you'll never meet = "interact with").

Obviously Bob is now two Bobs, and some conscious entity is either one or the other. That is the "Multi" part in MWI that is already understood (though weird).

But the the point of the question was this:
-----------------------------------------------

If their measurement angles have been the same, which if they chose their angles independently and after the particles separated, neither one could know, and which, if their states are defined locally, can't yet be reflected in their local states, then how will it be decided, when they meet, that bobAC+ can meet only AliceAH-, but not AliceAG+ ?

And do you understand why I ask this question?

[Edit:]
(Note that all three, bobAC+, AliceAH- and AliceAG+, are now in the A-branch)

 

[vanesch]

Obviously Bob is now two Bobs, and some conscious entity is either one or the other. That is the "Multi" part in MWI that is already understood (though weird).

But the the point of the question was this:
-----------------------------------------------

If their measurement angles have been the same, which if they chose their angles independently and after the particles separated, neither one could know, and which, if their states are defined locally, can't yet be reflected in their local states, then how will it be decided, when they meet, that bobAC+ can meet only AliceAH-, but not AliceAG+ ?

And do you understand why I ask this question?

[Edit:]
(Note that all three, bobAC+, AliceAH- and AliceAG+, are now in the A-branch)

Good question. This is due to quantum interference. In the thing I showed, I didn't carry explicitly the amplitudes with me, but they are "included" in the labels.

For instance, in:
bobAC+ and AliceAG+ which find both the ++ result --> |bobalice++ ACG>

we have that bobAC+ carries an amplitude x from the C and AliceAG+ carries an amplitude -s (from the G), so bobalice++ ACG will carry an amplitude -xs.

we also have that
bobBE+ and aliceBI+ which find both the ++ result --> |bobalice++ BEI>

bobBE carries an amplitude (-y) from E and aliceBI carries an amplitude r (from I), so
bobalice++ BEI carries an amplitude -(-yr). The extra minus sign comes from the common B which had a -1 amplitude.

Now, bobalice++ is the same state as bobalice++, so BOTH THESE TERMS INTERFERE.
That is, we sum their amplitudes.

So the amplitude of bobalice++ with the TWO labels (BEI and ACG) has an overall amplitude of (-xs +yr).

In the case of identical angles, we have that x = r and y = s, so this amplitude vanishes.
So in other words, there won't be an overall state bobalice++.

You will object, yes but what happens with the bobBE+ state then ? What with a bob who was in the bobBE+ state ?

The answer is that there was never a bob in the sole and unique bobBE+ state. There was a bob in the bob+ state, which carried TWO LABELS, namely BE and AC. There is no physical distinction between these LABELS. Only STATES can be different. A bob+ is not "either in AC or BE", but in a "superposition" of both, but this is not a physical superposition, it is an "algebraic" superposition, of two times the same state.

This is like writing a vector |a> = 0.3|a> + 0.7|a>. Is something that is in state a now in the "first term" or in the "second term" ?

 

[colorSpace]

Good question. This is due to quantum interference. In the thing I showed, I didn't carry explicitly the amplitudes with me, but they are "included" in the labels.

That still appears to be a non-local explanation. I'm afraid you still don't understand the question, since I have to keep re-phrasing the same question, but perhaps we are now coming closer in small steps. But let me answer the last part of your message first, and then come back to this first part.

You will object, yes but what happens with the bobBE+ state then ? What with a bob who was in the bobBE+ state ?

The answer is that there was never a bob in the sole and unique bobBE+ state. There was a bob in the bob+ state, which carried TWO LABELS, namely BE and AC. There is no physical distinction between these LABELS. Only STATES can be different. A bob+ is not "either in AC or BE", but in a "superposition" of both, but this is not a physical superposition, it is an "algebraic" superposition, of two times the same state.

This is like writing a vector |a> = 0.3|a> + 0.7|a>. Is something that is in state a now in the "first term" or in the "second term" ?

bobBE+ (or rather bob+ in general) doesn't seem to be a problem because he can meet with aliceBJ- (or rather alice- in general). (However there will be a problem of this kind in the triangular situation of GHZ entanglement with 3 or more particles).

I think the "interference" or superposition matters specifically in regard to your distinction between the labels A and B. You don't really need to split A and B, since there is no measurement at the source. A and B remain in superposition until the particles are measured at Alice and Bob. Then you need to physically split.

So you need only two physical splits, when Alice and Bob make their measurements, but to be correct, in the beginning the particles should remain in superposition, so that they can show interference. However I think that isn't all too relevant for the question I'm trying to get at. On the other hand, it might be an important detail for understanding the situation.

But I understand MWI to say, once there is a real split, then it is always a real physical split, and the real splits don't interfere with each other anymore, correct?

So label A and B don't indicate a split, they indicate a superposition. But other labels indicate a split, as they are related to states, correct?

For instance, in:
bobAC+ and AliceAG+ which find both the ++ result --> |bobalice++ ACG>

we have that bobAC+ carries an amplitude x from the C and AliceAG+ carries an amplitude -s (from the G), so bobalice++ ACG will carry an amplitude -xs.

we also have that
bobBE+ and aliceBI+ which find both the ++ result --> |bobalice++ BEI>

bobBE carries an amplitude (-y) from E and aliceBI carries an amplitude r (from I), so
bobalice++ BEI carries an amplitude -(-yr). The extra minus sign comes from the common B which had a -1 amplitude.

Now, bobalice++ is the same state as bobalice++, so BOTH THESE TERMS INTERFERE.
That is, we sum their amplitudes.

So the amplitude of bobalice++ with the TWO labels (BEI and ACG) has an overall amplitude of (-xs +yr).

In the case of identical angles, we have that x = r and y = s, so this amplitude vanishes.
So in other words, there won't be an overall state bobalice++.

This is back at the beginning of your message.

I understand very well that a non-local theory could explain this, and I kind of get the idea how the amplitudes would be used to do so. So the term that explains that Bob+ and Alice+ won't meet is (-xs +yr). But that appears to be a non-local term. x and y are Bob's, r and s are Alice'. In the time until they actually meet (and they could just send emails to a midpoint), their states will develop in complicated and different ways. It is not like two identical sine waves meeting, with opposite amplitudes, that could easily cancel each other out. How could this possibly be resolved when they meet, with local states (and local state descriptions) only?

This is the question.

And even if there were an answer: In the triangular case with three locations A, B and C, when there is a meeting at midpoint AB, the information (including the amplitudes) from C won't be locally available, so combinations of AB will 'survive', which suddenly will become impossible when they meet C at the midpoint ABC afterwards. [Edit: I expect they may become impossible, that depends on details of GHZ entanglement which I'm not too sure about.]

 

[vanesch]

But I understand MWI to say, once there is a real split, then it is always a real physical split, and the real splits don't interfere with each other anymore, correct?

So label A and B don't indicate a split, they indicate a superposition. But other labels indicate a split, as they are related to states, correct?

No, not at all. That's then your misunderstanding. A split is not necessarily definitive. A split only becomes practically definitive if decoherence sets in, that is, entanglement with so many degrees of freedom in such a complicated way that there's no way we will ever have identical states add together. That's what happens when there is a macroscopic measurement: you entangle air molecules, light photons, molecular vibrations, chemical states in your brain ... and there's no chance that you will do this in identical ways for different outcomes. Hence, at no point anymore you will add two collinear vectors in hilbert space, they will always be orthogonal.

I understand very well that a non-local theory could explain this, and I kind of get the idea how the amplitudes would be used to do so. So the term that explains that Bob+ and Alice+ won't meet is (-xs +yr). But that appears to be a non-local term. x and y are Bob's, r and s are Alice'. In the time until they actually meet (and they could just send emails to a midpoint), their states will develop in complicated and different ways. It is not like two identical sine waves meeting, with opposite amplitudes, that could easily cancel each other out. How could this possibly be resolved when they meet, with local states (and local state descriptions) only?

Their states WON'T devellop in "different" ways because they evolve from the same state. There are no *different* outcomes which have been macroscopically recorded which would make them evolve differently.
A bob+ state will evolve in exactly the same way as a bob+ state (unitary evolution is deterministic) and will keep exactly the same amplitude and phase information wrt another branch (also a property of unitary evolution). In other words:

U( blabla + u |bob+> + v |bob-> )

will result in:

blablabis + u (U|bob+>) + v (U|bob-> )

and the u and v come out of the terms.

So the "complicated evolution" you're talking about is U|bob+>. But that doesn't change the factor u in front of it, nor does it change the factor in front of |bob-> (which is v), and their ratio (amplitude and phase relationship).

So whatever happens to |bob+> (his email, computer, etc...) will ALL get the exact factor u and whatever happens to bob- will all get this factor v.

This is the question.

And even if there were an answer: In the triangular case with three locations A, B and C, when there is a meeting at midpoint AB, the information (including the amplitudes) from C won't be locally available, so combinations of AB will 'survive', which suddenly will become impossible when they meet C at the midpoint ABC afterwards. [Edit: I expect they may become impossible, that depends on details of GHZ entanglement which I'm not too sure about.]

Well: work it out in detail and show me.

 

[vanesch]

I think the "interference" or superposition matters specifically in regard to your distinction between the labels A and B. You don't really need to split A and B, since there is no measurement at the source. A and B remain in superposition until the particles are measured at Alice and Bob. Then you need to physically split.

Oh but that's an error! The A and B labels are essential, although that there is no "physical" split as you say (I take it you mean an irreversible branching). The labels do not indicate decohered splits, they indicate ENTANGLEMENT. Entanglement can be reversible, but it is entangled nevertheless. We need the labels A and B to say WHICH state of system 1 goes with WHICH state of system 2.

If you think that you can only introduce labels when you have an irreversible split, or if you think that any entanglement is irreversible, then you could never obtain interference with entangled states, and there would be no difference between entangled states and statistical mixtures. The whole point of these experiments is that entangled states give rise to interference effects (that is, identical states with different amplitudes which followed "different paths", and then come together, so that we add their amplitudes).

If |u> |v> can evolve into |u1>|v1> + |u2>|v2> then this state can evolve back in |u>|v> too, but you will have |u1>|v1> evolve into 1/2|u>|v> + |something> and |u2>|v2> evolve into 1/2|u>|v> - |something>.

 

[colorSpace]

Oh but that's an error! The A and B labels are essential, although that there is no "physical" split as you say (I take it you mean an irreversible branching). The labels do not indicate decohered splits, they indicate ENTANGLEMENT. Entanglement can be reversible, but it is entangled nevertheless. We need the labels A and B to say WHICH state of system 1 goes with WHICH state of system 2.

Just to get this out of the way first, since it seems to be what your 2. message is all about: No, I'm not saying the labels wouldn't be necessary, just that they don't indicate a split (I guess what I mean with split is always what you call a "decohered split", as long as it is coherent I'd call it a "superposition").

However it means that looking more closely, there is only one Bob until Bob measures the particle. In your notation, you started to differentiate bob+ and bob- before his measurement, which makes things perhaps simpler to calculate, but isn't exactly correct. In the beginning, only the particles exist in a superposition of + and -, and they remain coherent at first.

As far as your first message is concerned, it seems a bit vague, I have to figure out if I can relate it to my question.

[Edit added:] Actually, in the beginning, I think, the particles exist in a superposition of all directions. + and - acquire meaning only in relation to a specific measurement angle, which comes later.

 

[colorSpace]

No, not at all. That's then your misunderstanding. A split is not necessarily definitive. A split only becomes practically definitive if decoherence sets in, that is, entanglement with so many degrees of freedom in such a complicated way that there's no way we will ever have identical states add together. That's what happens when there is a macroscopic measurement: you entangle air molecules, light photons, molecular vibrations, chemical states in your brain ... and there's no chance that you will do this in identical ways for different outcomes. Hence, at no point anymore you will add two collinear vectors in hilbert space, they will always be orthogonal.

As I said, with "split" I mean "decoherent split", otherwise I'd call it a superposition. Unless I'm missing some other cases right now..

Their states WON'T devellop in "different" ways because they evolve from the same state. There are no *different* outcomes which have been macroscopically recorded which would make them evolve differently.
A bob+ state will evolve in exactly the same way as a bob+ state (unitary evolution is deterministic) and will keep exactly the same amplitude and phase information wrt another branch (also a property of unitary evolution). In other words:

U( blabla + u |bob+> + v |bob-> )

will result in:

blablabis + u (U|bob+>) + v (U|bob-> )

and the u and v come out of the terms.

So the "complicated evolution" you're talking about is U|bob+>. But that doesn't change the factor u in front of it, nor does it change the factor in front of |bob-> (which is v), and their ratio (amplitude and phase relationship).

So whatever happens to |bob+> (his email, computer, etc...) will ALL get the exact factor u and whatever happens to bob- will all get this factor v.

Sorry, I can't understand what you mean with "factors" u and v. In message #72, you seemed to refer to u and v as the particles. How are u and v factors, and what does it mean for Bob to have u as a factor?

And what happened to my main point that the term (-xs +yr) is a non-local term? What, again, happened to my question?

Well: work it out in detail and show me.

I'll come back to that as soon as we are more clear about how "pairing-up" (as the text you referenced calls it) could be understood to be a 'local' concept.

 

[colorSpace]

And what do you mean with:

A bob+ state will evolve in exactly the same way as a bob+ state (unitary evolution is deterministic)

Is it a typo and means bob+ develops the same way as bob- ?
Or that all variations of bob+ (bobAC+ and bobBE+) develop the same?

 

[colorSpace]

After thinking some more about this, I'm getting the impression that this notation is a purely mathematical, non-local description.

So the two terms for bobalice++ which cancel each other out when they meet, do simply mathematically result in zero. There are no corresponding two waves, or such, which would cancel each other out. It is just mathematical terms that add up to zero.

So I am getting the impression that, in the first place, this notation is not able to give any physical explanation, let alone a local one.

 

[vanesch]

After thinking some more about this, I'm getting the impression that this notation is a purely mathematical, non-local description.

So the two terms for bobalice++ which cancel each other out when they meet, do simply mathematically result in zero. There are no corresponding two waves, or such, which would cancel each other out. It is just mathematical terms that add up to zero.

Of course there are no waves ! The point is that you can have mathematical entities attached to "locations in space" which can carry all the necessary information and which only exchange information with other entities at the same locations in space. That's what I'd call "local". The entities that walk around are then elements of a hilbertspace + a set of labels, which are themselves mathematical structures. Think of "objects" as in "object-oriented programming".

The set of labels is simply the equivalent of the algebraic expression of the wavefunction, but the fact that we CAN have these labels means that we CAN consider them, if we like so, to be "objects which wander through space and which only interact locally", the point which was to be demonstrated, and which isn't indeed evident when looking at the global wavefunction.

So in this whole story I'm trying to construct "localisable entities" which carry with them all the needed information that allows them to transform only by exchanges with other entities at the same location, and which continue to be represented globally by the wavefunction.

The localisable entities are "kets equipped with labels". The STATES are just the "ket" part, but the way they interact, combine, have probabilities etc... are determined by the kets AND the labels (where the label interaction is just the equivalent of the algebraic rules of manipulation of a global wavefunction of course).

This exercise illustrates then that it is *conceivable* to have localised entities which nevertheless only interact with "nearby" other entities, and nevertheless, are all the time equivalent to a global wavefunction.

I will now respond to different of your remarks:

However it means that looking more closely, there is only one Bob until Bob measures the particle. In your notation, you started to differentiate bob+ and bob- before his measurement, which makes things perhaps simpler to calculate, but isn't exactly correct. In the beginning, only the particles exist in a superposition of + and -, and they remain coherent at first.

I'm not supposed to. Can you show me where ?

Bob0 only became bob+ after interacting with the particle on his side. In:

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

I already presumed that bob did his measurement on u. I could have written the state before:

(|u+A> |v-A> - |u-B>|v+B>) |bob0> |alice0>

but that was trivial, no ? It is when Bob measures the u-system that he interacts with it. But this interaction is going to take place in bob's measurement basis, which is not |u+> and |u->, but rather |u++> and |u--> so I first have to write the |u+> and the |u-> in his measurement basis. This is what gives me the former expression.

Actually, in the beginning, I think, the particles exist in a superposition of all directions. + and - acquire meaning only in relation to a specific measurement angle, which comes later.

Yes, that's true, but |u+> is NOT in a superposition with |u->. It is |u+>|v-> that is in a superposition with |u->|v+>. This is very important, and it is to record this twinning, that we need the labels. It is a specificity of the singlet state that you can write this superposition in different ways, but nevertheless, each individual way of writing of the superposition will associate ONE specific u-state to ONE specific v-state.
You cannot consider them independently, and that is why the labels are essential here. They are in fact the core feature of the whole thing.

For instance, |u+>|v+> + |u->|v-> is NOT re-writable in a different direction. It is the only decomposition of this state that you can have.

Sorry, I can't understand what you mean with "factors" u and v. In message #72, you seemed to refer to u and v as the particles. How are u and v factors, and what does it mean for Bob to have u as a factor?

Sorry for the confusion in notation. u and v are here just complex numbers, I wasn't thinking of any particles here. Totally independent example.

The factor is the amplitude/phase information that is attached to the labelled state. It will indicate, in the case of a "mind state" what will be the probability to be experienced. In all other cases, it is just a piece of information to be carried along with the labels (as it is a part of the algebraic structure of the wavefunction).
The important part is to realize that if a unitary evolution is going to act locally on a state (that is, is going to transform a state into another one through interaction) that it cannot alter this complex factor. Simply because it is a linear operator. This is what allows one to make "locally abstraction of the rest of the wavefunction", and what allows us also to "recombine" afterwards identical states (add their amplitudes).

I will try to give an example. If we have a system with a global wavefunction:

a |bob+_at_joe> |alice+_at_suze> + b |bob-_at_joe> |alice-_at_suze>

then there can happen a lot of things with bob at joe, which is described by a local interaction operator U_joe:

U_joe { a |bob+_at_joe> |alice+_at_suze> + b |bob-_at_joe> |alice-_at_suze>)

= a U_joe {|bob+_at_joe> } |alice+_at_suze> + b U_joe {|bob-_at_joe>} |alice-_at_suze>)

So what is going to happen to bob+at_joe (described by the operator U_joe) is independent of the phasefactor a and what's going to happen to bob-_at_joe is independent of the phasefactor b. These two phasefactors are determined by the superposition that happened somehow of alice and joe (which must be the result of a past interaction of alice and joe when they were together, or by interaction with something that was entangled or whatever). The interaction at joe can be as complicated as you want, we can bring in fred, the air at joe, the moonshine at joe's etc...
It won't alter the relative amplitudes and phases of these alice/joe pairs. In the same way, we could have introduced a local interaction operator U_alice. It could also do all kinds of complicated things to alice and her environment. But it won't alter anything to the a and b factors.

And what happened to my main point that the term (-xs +yr) is a non-local term? What, again, happened to my question?

I don't see why you insist on it being global, as all the constituents have been braught in locally, at the same location, to do the sum ??


I repeat what I wrote:

For instance, in:
bobAC+ and AliceAG+ which find both the ++ result --> |bobalice++ ACG>

we have that bobAC+ carries an amplitude x from the C and AliceAG+ carries an amplitude -s (from the G), so bobalice++ ACG will carry an amplitude -xs.

we also have that
bobBE+ and aliceBI+ which find both the ++ result --> |bobalice++ BEI>

bobBE carries an amplitude (-y) from E and aliceBI carries an amplitude r (from I), so
bobalice++ BEI carries an amplitude -(-yr). The extra minus sign comes from the common B which had a -1 amplitude.

Now, bobalice++ is the same state as bobalice++, so BOTH THESE TERMS INTERFERE.
That is, we sum their amplitudes.

So the amplitude of bobalice++ with the TWO labels (BEI and ACG) has an overall amplitude of (-xs +yr).

All the labels and kets have been carried to the place where alice and bob meet to make the state |alicebob++>, and the labels have carried with them the necessary amplitudes (r,s,x,y and 1,-1) in order to have locally all the necessary information to calculate (-xs + yr), so why do you insist on this being non-local ??

Is it a typo and means bob+ develops the same way as bob- ?
Or that all variations of bob+ (bobAC+ and bobBE+) develop the same?

No, it isn't a typo, it is clearly bob+ and bob+. As you say, one is bob+AC and the other is bob+BE, but they have identical kets, and will hence evolve identically under unitary evolution.
Not one single air molecule will be different under whatever evolution resulting from bob+AC and resulting from bob+BE.

The labels are only a way of showing that it is in principle possible to carry them with the states of the subsystems locally, and allow at any moment to reconstruct the global wavefunction. They have no dynamical significance apart from mimicking the algebra that happens to the global wavefunction. They are not necessarily connected to "individual branches", they are rather connected to "individual terms" in the wavefunction. There's a subtle difference between them, if not everything has irreversibly decohered.


This is a bit an analoguous situation as in classical hamiltonian mechanics. Imagine that we first discovered Hamiltonian mechanics, before Newton. We would then think that the universe is a phase space of 6N dimensions, over which a single dynamical law rules: the Hamiltonian flow (specified by the vector field given by the hamilton equations).
So "reality" is really 6N dimensional, and the "universe" is a single point in that 6N dimensional space. But then someone comes along and tries to ask whether this dynamics can be made "local in 3 dimensions". First of all, you say, how can this be so ? The universe is specified by a single point in 6N dim space ? How could you "re-map" this 6N-dim space on something like a 3-dim space ?
But then you look at the dynamics, and you start to realize that you can write all of the elements of the 6N-dim "universe state vector" as 3-tuples, as long as you pick very peculiar degrees of freedom for this state vector, namely those that correspond to "spacepoints in a 3-dim euclidean space".
And when you work this further out, you see that you can "lump" parts of the 6N-dimensional vector coordinates into pieces, which you can label "particle 1", "particle 2",... and you see that you can write the hamilton flow as following from individual, local interactions (collisions! No Newtonian gravity which is non-local of course) between these labeled parts. So you say that this apparently "global" dynamics given by the flow in the "real" 6N space, can be seen as "sub-interactions" of "labeled particles" in a 3-D space. That's sufficient to show that this system is "local", whether or not you add some belief to the real existence of a 3-dim space and those many particles (you've always believed that the universe was a single point in 6N dimensions). Is the apparent "3-d structure" of space "for real" or just a figement of the "locality" property of the 6N dim dynamics ?

Well, I try to show something similar. In quantum mechanics, we believe that the universe is a vector in a hilbertspace. It gives us the impression sometimes that we have classical worlds in a 3-dim space. Is this for real, or is it a figement of some "locality" property of the dynamics in hilbert space ? I won't try to answer this, but I'm trying to show that it is *in principle possible* to construct a "localised version" of the evolution of the global wavefunction, by adding labels (mathematical structures) to "localised" states. If that works, then that's all I have to do to say that the global hilbert space dynamics can also be seen as local. In the same way as the 6N dim hamiltonian flow can also be seen as a local dynamics.

 

[thharrimw]

i think that as science avances it may be possable to have quantum computers and when you thik about it in the early 60's poeple didn't think that the internet was possable and look at the internet today

 

[colorSpace]

All the labels and kets have been carried to the place where alice and bob meet to make the state |alicebob++>, and the labels have carried with them the necessary amplitudes (r,s,x,y and 1,-1) in order to have locally all the necessary information to calculate (-xs + yr), so why do you insist on this being non-local ??

First of all, it is good that in your message you gave more background information about what you are trying to do.

So I may write another response after reading all the different points carefully, but I'm not sure that will be necessary.

You may remember that my objection was that a local explanation in MWI terms would require the local physical states to carry a lot of additional information. which implies, at the beginning, that I acknowledged the possibility of a local explanation *if* it were possible to add that information, *and* to have a physical process that allows using that information in order to do the "pairing up".

The reason I refer to (-xs + yr) as a non-local term is that so far, the variables x, y, s and r have been abstract mathematical values that refer to the measurement angles at Bob's and Alice' measurement locations which at the time of measurement have been at a large distance.

So now you are saying that all events influenced by the measurement results are affected physically not only by the measurement result in terms of being + or -, but also in a special physical state that corresponds to these variables. I was so far thinking of these variables to indicate something like a probability of the + or - state, rather than being a physical state of itself.

That is, Bob is not only in a special physical state that corresponds to + and - (with probabilities indicated by x and y), and but also in a special physical state that corresponds to x and y.

So this latter physical state is the "additional information" that I was talking about.

If they conduct many experiments at the same time and location, that is, if they exchange not only one bit of +/-, but millions, let's say 10 Mbit, then all their physical states, and all resulting physical states in the universe, have to carry 10Mbit in a way that corresponds to the 10 Million variables of the kind x, y, r, and s.

And then, there needs to be a physical process which, when they meet, makes this 10Mbit of information result in the "removal" of bobalice++, for example.

So one of the questions is: What kind of physical state could represent 10Mbit of information in each particle in the universe affected by the measurements?

[Edit:] Actually it is much more than 10MBit because x and y are two complex numbers, rather than just a bit. So the amount of information is 20 Million complex numbers. In each of all particles in the universe affected.

 

[colorSpace]

[Continued from the last message]

In addition, the x and y numbers need to be associated with a unique identifier of the entangled particle pair it relates to, otherwise the universe would mix-up states from different experiments, just based on the angles alone.

So any particle which is affected by 10 Million transmitted entangled particles (which can happen in less than a one second given today's transmission speeds) would have to carry information equivalent to 20 million complex numbers plus 10 Million unique id's.

[Edit:] (That is, in each "world", of course.)

[Edit] (In my last quote from you, you indicate that this is actually information that the "kets" need to "carry" with them. That's the only explanation I can discuss at this point.)

 

[vanesch]

You may remember that my objection was that a local explanation in MWI terms would require the local physical states to carry a lot of additional information. which implies, at the beginning, that I acknowledged the possibility of a local explanation *if* it were possible to add that information, *and* to have a physical process that allows using that information in order to do the "pairing up".

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.

The reason I refer to (-xs + yr) as a non-local term is that so far, the variables x, y, s and r have been abstract mathematical values that refer to the measurement angles at Bob's and Alice' measurement locations which at the time of measurement have been at a large distance.

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.

So now you are saying that all events influenced by the measurement results are affected physically not only by the measurement result in terms of being + or -, but also in a special physical state that corresponds to these variables. I was so far thinking of these variables to indicate something like a probability of the + or - state, rather than being a physical state of itself.

That is, Bob is not only in a special physical state that corresponds to + and - (with probabilities indicated by x and y), and but also in a special physical state that corresponds to x and y.

So this latter physical state is the "additional information" that I was talking about.

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.

If they conduct many experiments at the same time and location, that is, if they exchange not only one bit of +/-, but millions, let's say 10 Mbit, then all their physical states, and all resulting physical states in the universe, have to carry 10Mbit in a way that corresponds to the 10 Million variables of the kind x, y, r, and s.

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.

And then, there needs to be a physical process which, when they meet, makes this 10Mbit of information result in the "removal" of bobalice++, for example.

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

So one of the questions is: What kind of physical state could represent 10Mbit of information in each particle in the universe affected by the measurements?

[Edit:] Actually it is much more than 10MBit because x and y are two complex numbers, rather than just a bit. So the amount of information is 20 Million complex numbers. In each of all particles in the universe affected.

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".

 

[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.