Will quantum computers ever be possible?

[colorSpace]

[This message doesn't replace the two previous one's, it is in addition to them.]

Now, let's say that Bob does his measurement (along the original z axis). we now have:

|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>

At this moment, bob's brain inherits also the two labels A and B, from the v-states. Note that algebraically, A and B simply mean: first term and second term (from Bob's PoV).

Right, now Alice is going to send an email to bob with her results. First her keyboard gets hits from her fingers:

[This message doesn't replace the two previous one's, it is in addition to them.]

Your state description of "Bob's brain" already includes states from Alice, even before she sends the email. But as far as Bob is concerned, Alice might not even have performed the experiment, or with different measurement angles, or she might have gone to sleep and be dreaming, and/or sold her computer, or who knows what.

Therefore it is a non-local state description, and your use of non-local state descriptions may be part of the reason why you don't see the problem yet. You look only at these state descriptions, but they are non-local, and therefore they don't show the problem.

 

[vanesch]

[This message doesn't replace the two previous one's, it is in addition to them.]



[This message doesn't replace the two previous one's, it is in addition to them.]

Your state description of "Bob's brain" already includes states from Alice, even before she sends the email. But as far as Bob is concerned, Alice might not even have performed the experiment, or with different measurement angles, or she might have gone to sleep and be dreaming, and/or sold her computer, or who knows what.

Therefore it is a non-local state description, and your use of non-local state descriptions may be part of the reason why you don't see the problem yet. You look only at these state descriptions, but they are non-local, and therefore they don't show the problem.

No, look carefully to the state you quoted:

|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>


At this point, bob's brain is in 2 different states:

brainbob+A and brainbob-B.

It got the labels A and B from the particle v, which was transmitted by local interactions (v was the particle Bob did a measurement on). As such, it doesn't "include states from alice".

Alice's states are
brainalice+AC
brainalice-AD
brainalice+BE
brainalice-BF

The label A and B, she got from the particle u, and the labels C,D E and F she also got from the particle u, after it got written in another basis (angle of her analyser).

So alice's states get their labels ALSO from purely local interactions at Alice's place.

As I said before, the labels are unnecessary when writing down the wavefunction, because they simply "indicate the algebraic tree structure" of it. But the labels can be useful if you write things just as lists. Mind you, next to the labels, you need then also to keep track of the complex amplitudes.

So we can redo the story but this time without wavefunction.

There are two entangled particles u and v:

we have:
u+A and u-B for u
and
v+A and v-B for v

and with labels A and B goes a phase factor +1 and -1 respectively.

All the other states of systems in our thing are in a product state wrt this, so we don't write them down yet.

Next, Alice does her measurement along an axis on u.

This means that she measures on all the states of u, in the new basis.

So u+A is to be written:
|u+A> = x |uu+A> + y |uu-A>

and Alice's brain is going to entangle with it, so we have to write two new labels:
u+A> = x |uu+AC> + y |uu-AD>

where label C carries "amplitude x" and D amplitude "y".

and we have now:

brainalice+ AC
brainalice- AD

when alice interacted with u+ (that's locally, at her place)

but she also interacts with u-B:
u-B> = -y |uu+B> + x |uu-B>

and again we'll need two new labels because there's going to be an entanglement:
u-B> = -y |uu+BE> + x |uu-BF>

these labels E and F carry amplitudes -y and +x

and alice interacts with it
brainalice+ BE
brainalice- BF

Note that alice's states only inherited labels by local interactions! It came from the u-particle. She doesn't know what bob is going to do here.

On Bob's side, bob received a particle v which was in two states:

v+A and v-B

upon measuring along the z-axis, bob's brain entangled with them:

bobbrain+A
bobbrain-B

He doesn't know at all what alice did. He got his labels only locally.

Alice does now what she wants, like sending an e-mail (after having interacted with her fingers, keyboard, air, ocean, moon, the sun, Jupiter, ... who all get sooner or later entangled with her, directly or indirectly, through successive local interactions and will inherit the labels AC, AD, BE and BF).

So at a certain moment, we have:
half-of-the-universeAlice AC, containing the information that alice had a +
half-of-the-universeAlice AD, containing the informaton that alice had a -
half-of-the-universeAlice BE, containing the information that alice had a +
half-of-the-universeAlice BF, containing the information that alice had a -

We can tell the same story of Bob: he will interact with his environment, with the dinosaurs who still live on his planet, which will interact with the air, the water, etc.. on his remote planet, Bob will maybe die, do whatever, so 10 years later, we also have:

half-of-the-universeBob A containing the information that bob had +
half-of-the-universeBob B containing the information that bob had -

In the common part of these "half-of-the-universeAlice" and "half-of-the-universeBob", say, on a planet midway between both, there will have been an interaction with the AC part and with the A part, which will be recognized being the same label A.

So in the AC part in this common part, the information will be available that alice had + and bob had - ; in the AD part, it will be that alice had - and bob had +, in the BE part it will be that alice had + and bob had - and in the BF part, it will be that alice had - and bob had -.

"information available" will simply say, there has been a chain of local interactions through which it is *in principle* possible to find out. But that can be of course in an "obvious" way, such as an e-mail, or an old book in which, Alice wrote her result 100 years ago, or in an undisentangible configuration of air molecules or something. If Alice and bob did record their measurements in a readable way, say a book and a hard disk, then Joe, which will be in 4 different "universes" 100 year later, will be in 4 different states:
joeAC with a bookAC and a disk AC
joeAD with a bookAD and a disk AD
joeBE with a book BE and a disk BE
joeBF with a bookBF and a diskBF

On it, he will find the necessary information (book for alice's result and disk for bob's result).

 

[colorSpace]

No, look carefully to the state you quoted:

|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>At this point, bob's brain is in 2 different states:

brainbob+A and brainbob-B.

It got the labels A and B from the particle v, which was transmitted by local interactions (v was the particle Bob did a measurement on). As such, it doesn't "include states from alice".

Alice's states are
brainalice+AC
brainalice-AD
brainalice+BE
brainalice-BF

The label A and B, she got from the particle u, and the labels C,D E and F she also got from the particle u, after it got written in another basis (angle of her analyser).

So alice's states get their labels ALSO from purely local interactions at Alice's place.

As I said before, the labels are unnecessary when writing down the wavefunction, because they simply "indicate the algebraic tree structure" of it. But the labels can be useful if you write things just as lists. Mind you, next to the labels, you need then also to keep track of the complex amplitudes.

The complex amplitudes? I guess I forgot about those. Never heard about them!

On looking carefully once more, I only find confirmed what I said: You have associated each of Bob's states with two of Alice's state, and they even already share the same labels, A and B.

How is that not non-local?

So it appears that in the "back of your mind", so to speak, you are performing a non-local split of states right in the beginning. The "pairing-up" is there right in the beginning, non-locally.

So we can redo the story but this time without wavefunction.

We can redo the story as often as you want. I don't see you getting even one inch closer to answering my points.

There are two entangled particles u and v:

we have:
u+A and u-B for u
and
v+A and v-B for v

and with labels A and B goes a phase factor +1 and -1 respectively.

All the other states of systems in our thing are in a product state wrt this, so we don't write them down yet.

Next, Alice does her measurement along an axis on u.

This means that she measures on all the states of u, in the new basis.

So u+A is to be written:
|u+A> = x |uu+A> + y |uu-A>

and Alice's brain is going to entangle with it, so we have to write two new labels:
u+A> = x |uu+AC> + y |uu-AD>

where label C carries "amplitude x" and D amplitude "y".

and we have now:

brainalice+ AC
brainalice- AD

when alice interacted with u+ (that's locally, at her place)

but she also interacts with u-B:
u-B> = -y |uu+B> + x |uu-B>

and again we'll need two new labels because there's going to be an entanglement:
u-B> = -y |uu+BE> + x |uu-BF>

these labels E and F carry amplitudes -y and +x

and alice interacts with it
brainalice+ BE
brainalice- BF

Note that alice's states only inherited labels by local interactions! It came from the u-particle. She doesn't know what bob is going to do here.

On Bob's side, bob received a particle v which was in two states:

v+A and v-B

upon measuring along the z-axis, bob's brain entangled with them:

bobbrain+A
bobbrain-B

He doesn't know at all what alice did. He got his labels only locally.

Alice does now what she wants, like sending an e-mail (after having interacted with her fingers, keyboard, air, ocean, moon, the sun, Jupiter, ... who all get sooner or later entangled with her, directly or indirectly, through successive local interactions and will inherit the labels AC, AD, BE and BF).

So at a certain moment, we have:
half-of-the-universeAlice AC, containing the information that alice had a +
half-of-the-universeAlice AD, containing the informaton that alice had a -
half-of-the-universeAlice BE, containing the information that alice had a +
half-of-the-universeAlice BF, containing the information that alice had a -

Maybe Bob got his labels locally, but Alice already got Bob's labels inherited (A and B).

Again, how is that not non-local?

Bob's result should depend on his measurement angles which he chooses after Alice has received her particle already, and Alice's states should be free of any reference to Bob's.

According to what I can tell, you haven't understood my objections at all.

We can tell the same story of Bob: he will interact with his environment, with the dinosaurs who still live on his planet, which will interact with the air, the water, etc.. on his remote planet, Bob will maybe die, do whatever, so 10 years later, we also have:

half-of-the-universeBob A containing the information that bob had +
half-of-the-universeBob B containing the information that bob had -

In the common part of these "half-of-the-universeAlice" and "half-of-the-universeBob", say, on a planet midway between both, there will have been an interaction with the AC part and with the A part, which will be recognized being the same label A.

That's trivial since you shared the label A already at the beginning of your story.

So in the AC part in this common part, the information will be available that alice had + and bob had - ; in the AD part, it will be that alice had - and bob had +, in the BE part it will be that alice had + and bob had - and in the BF part, it will be that alice had - and bob had -.

"information available" will simply say, there has been a chain of local interactions through which it is *in principle* possible to find out. But that can be of course in an "obvious" way, such as an e-mail, or an old book in which, Alice wrote her result 100 years ago, or in an undisentangible configuration of air molecules or something. If Alice and bob did record their measurements in a readable way, say a book and a hard disk, then Joe, which will be in 4 different "universes" 100 year later, will be in 4 different states:
joeAC with a bookAC and a disk AC
joeAD with a bookAD and a disk AD
joeBE with a book BE and a disk BE
joeBF with a bookBF and a diskBF

On it, he will find the necessary information (book for alice's result and disk for bob's result).

I didn't get the slightest clue where or how this information is supposed to be available. It all appears to depend on the initial pairing-up of states and labels, which you preform at the beginning when everything is supposed to be spacelike separated.

In my book, your story is a non-local story.

 

[colorSpace]

[Continued from the previous message]

Are you perhaps assuming hidden variables?

If so, it would appear that your theory is quite different than the one in the link, and the one I discussed before.

In that case, I would assume that arguments related to local-hidden-variable theories will apply here as well.

[Again, continued from the previous message]

 

[colorSpace]

Actually, it looks almost like in your scenario, the splitting happens before the entangled particles separate...

 

[colorSpace]

Yes, Vanesh, it was difficult for me to tell, since you seemed to describe everything 'subjectively' from Bob's point of view, which would make the situation naturally asymmetrical.

However, what you have done appears to be, more or less, that you made the first universe-split before the particles separate, and then in each universe you develop a local-hidden-variable based situation. Except that you complicate it with an additional split only on Alice's side regarding some "amplitudes" that I haven't heard of, neither when talking about polarized photons, nor when talking about electron spins, which are the common examples for entanglement. The second split appears somewhat bogus, in one of your earlier messages you even use the description "to show how for THAT SPECIFIC ALICE things LOOK AS IF a collapse took place".

So if one removes this complication on Alice's side, what's left is two universes with a local-hidden-variable situation in each, and the same arguments that apply to disproving a local-hidden-variable theory in a single universe, also apply here to each of the two universes, since the split happens before the particles are separated, and becomes irrelevant to the application of Bell's theorem.

 

[vanesch]

The complex amplitudes? I guess I forgot about those. Never heard about them!

On looking carefully once more, I only find confirmed what I said: You have associated each of Bob's states with two of Alice's state, and they even already share the same labels, A and B.

OF COURSE, but these labels do not come from "bob" or from "alice" but from the SOURCE OF THE ENTANGLED PARTICLES.

The source of the entangled particles (which is a *local* happening, right ?) makes two particles, u and v, in an entangled state:
|u+>|v+> - |u->|v->

It is *this entangled state* (the fact that there are two terms here) that makes up for the labels A and B. They are "produced" locally in the source of the entangled particles. You could even look at it this way. Imagine, in the source, that we have two particles "sitting there and waiting to be entangled": u and v.

They are as of yet "independent systems": u is in the state u0 and v is in the state v0.
As long as this remains so, we can treat u in its hilbertspace Hu and v in its hilbertspace Hv.
But now, inside the source, an interaction is provoked between the u-system and the v-system. Well, the superposition principle requires us now to use the tensor product state space Hu x Hv. What does this mean ? It means that different states from u can "couple" with different states from v. It is when this happens, that we have to introduce labels, if we want to keep track of which state of u goes with which state of v.

But, and this is the core of the locality in quantum theory: this coupling can only occur for states which share the same spacetime points. This is a property of the locality of the interactions. It didn't need to be, but that's the way it turns out to be. It is a property of the interaction hamiltonian.

Well, a state of particle u that "is in the neighbourhood of point P" can interact with a state of particle v that is "in the neighbourhood of point P", that is, they can change into an eventual superposition of states of u and v, all in the neighbourhood of point P.

Before the interaction, they were "independent", after the interaction, if there are multiple terms, they are "entangled".

We take it that our states u0 and v0 respectively are "in the neighbourhood of the source". So they can now interact, and form an entangled state:

a|u1>|v1> + b|u2>|v2> + c|u3> |v3>

where, remember, u1,u2, u3 and v1, v2, v3 are also states "in the neighbourhood of the source".

It is when THIS happens, that we can introduce LABELS (which do nothing else but codify for the algebraic expression above):

state "u0" "v0" evolved into a SUPERPOSITION OF:

state u1 with v1 with amplitude a, which we give LABEL A
state u2 with v2 with amplitude b, which we give LABEL B
state u3 with v3 with amplitude c, which we give LABEL C

the labels are nothing else but a way to say that it was state u1 that got with state v1, and not with state v2, that it was state u3 that got with state v3, and not v1.

When you write out the wavefunction algebraically, this is evident of course, but if you want to keep track "per system" of what happens, then you need the labels.

Remember that these labels come from two things:
1) the superposition principle applied to the union of two systems, which requires the existence of entangled states
2) the fact that a LOCAL interaction (at the source in this case) can make a product state (independent states for the two subsystems) evolve into such an entangled state (superposition of product states).

How is that not non-local?

Because it came forth of a past local interaction.

In our example, we used 3 terms in the entangled state (with labels A, B and C), but usually when using the spin states of photons or electrons, we only have two states available for each, so we only have two terms, and need two labels, A and B:

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

Remember that this entangled system is LOCALLY produced in the source.

Next, u will evolve (in both its states + and -) towards Alice, and v will evolve towards Bob, but they ALREADY HAVE THEIR LABELS.

Note that this is interesting: it is not possible to "entangle at a distance". If u was independently at Alice, and v at Bob, it would not have been possible to create an entangled state. It is in the source, locally, that the states of u and v got entangled.

So it appears that in the "back of your mind", so to speak, you are performing a non-local split of states right in the beginning. The "pairing-up" is there right in the beginning, non-locally.

Nope, locally, in the source. But I'm sorry not having mentioned that, I thought that that was obvious.

Maybe Bob got his labels locally, but Alice already got Bob's labels inherited (A and B).

Again, how is that not non-local?

I hope that this is cleared up now. A and B come from the source of entangled particles.

Bob's result should depend on his measurement angles which he chooses after Alice has received her particle already, and Alice's states should be free of any reference to Bob's.

Consider that bob did his measurement first then. Given that they are at spacelike intervals, you can choose which one is "first" :-) You can always write the entangled state "in the basis of one of the measurers". True, you can work it out in more generality, it is just more typing, it comes down all the same.

I didn't get the slightest clue where or how this information is supposed to be available. It all appears to depend on the initial pairing-up of states and labels, which you preform at the beginning when everything is supposed to be spacelike separated.

Indeed, it all depends on the initial pairing up... which happened in the source of the particles. Sorry for not having made that clear.

Indeed, if you think that "particles just become entangled at a distance" then you are right that this would be a non-local phenomenon.

 

[colorSpace]

Nope, locally, in the source. But I'm sorry not having mentioned that, I thought that that was obvious.

Meanwhile I did figure this out, in my last two messages, where I wrote that you are splitting the universe before the particles are separated. Thank you for the confirmation.

After writing my last message, I also looked again more closely at the thread of your previous discussion with 'nrqed', and especially this quote (of you) was clarifying:

If you are now going to consider the case of A PARTICULAR BOB, then the rule is, that the corresponding state you have to pick out for THAT Bob is given by the Born rule. In this case, it is 50-50: so *a particular bob* will experience, with 50% chance, the first category of "bob states", and with 50% chance, the second category of "bob states".
And if he waits until "Alice" comes along, he will meet with the particular Alice that was in his branch of course.

(The thread is at https://www.physicsforums.com/showthread.php?t=114207 )

So you are keeping the corresponding "particular Bob" and "particular Alice" in the same branch from the beginning. Of course this doesn't make it difficult for them to find each other again. Actually, that is now trivial.

This is indeed a new situation for me, and I saw that it also took 'nrqed' quite a while to figure it out.

However before we go into the details, allow me to ask you whether you are familiar with the way in which local-hidden-variable theories have been disproved, and how you would (very shortly) describe this in your language. I am asking this question because of what I have written in my previous message.

 

[vanesch]

However before we go into the details, allow me to ask you whether you are familiar with the way in which local-hidden-variable theories have been disproved, and how you would (very shortly) describe this in your language.

Yes, I think I know Bell's theorem quite well.

I am asking this question because of what I have written in my previous message.

We are NOT in a local hidden variable theory here, simply because there has not been a deterministic outcome at the two measurements: the two possible outcomes exist.

I am btw not keeping a particular bob and a particular alice in the same branch. Bob "doesn't know" with which alice he's going to "pair up".

 

[colorSpace]

We are NOT in a local hidden variable theory here, simply because there has not been a deterministic outcome at the two measurements: the two possible outcomes exist.

The two possible outcomes appear to be distinguished (and labelled) at the beginning. From then on, each of the two branches behaves like a local-hidden-variable scenario, since all further outcomes are defined by the initial pairing-up.

I am btw not keeping a particular bob and a particular alice in the same branch. Bob "doesn't know" with which alice he's going to "pair up".

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."

This is what I finally figured out from our discussion, and saw confirmed in your discussion from which I quoted in my last message.

 

[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".