Will quantum computers ever be possible?

[vanesch]

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

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

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

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

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

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

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

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

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

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

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

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

 

[colorSpace]

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

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

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

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

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

 

[vanesch]

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

The (or one of the) GHZ states is:

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

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

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

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

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

which becomes, after working out:

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

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

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

 

[colorSpace]

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

 

[colorSpace]

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

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

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

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

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

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

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

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

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

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

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

 

[Enigma007]

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

Enigma Valdez

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

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

 

[vanesch]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[colorSpace]

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

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

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

However:

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

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

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

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

 

[vanesch]

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

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

However:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[colorSpace]

But all this doesn't matter.

(Excerpt from the last message illustrating MWi flavors.)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[colorSpace]

I think you may get the answer from the textbook

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

See any help.

hehe ^_^

Congratulations to your first post!

"Hehe".

 

[vanesch]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[colorSpace]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[...]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[vanesch]

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

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

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

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

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

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

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

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

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

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

The measurement basis of our analyser under 45 degrees:

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

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

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

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

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

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

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

The LOCAL particle, yes.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[colorSpace]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The measurement basis of our analyser under 45 degrees:

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

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

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

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

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

...

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

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

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

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

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

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

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

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

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

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

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

...

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[Mentz114]

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

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

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

 

[colorSpace]

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

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

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

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

 

[colorSpace]

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

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

0.5 |bob+> + 0.5 |bob->

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

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

 

[colorSpace]

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

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

 

[vanesch]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[vanesch]

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

You are right that the "global" way of dealing with it (namely, doing usual wave function dynamics) is more elegant, and as you correctly point out, the only thing we need is to show that it is *possible*. I pointed you to a paper (Rubin) where this has been worked out in all detail in the Heisenberg representation, and I simply took a version of the Schroedinger representation here to ILLUSTRATE the mechanism.

So the proof exists, and is given in Rubin's paper. I only illustrated it, in the case of the Schroedinger representation, for some specific examples.

Now, why is this important ? Many people claim, because of Bell's theorem, that quantum mechanics is non-local, but that's cutting corners a lot.
The thing that Bell's theorem shows, is this:

"there doesn't exist a local hidden variable theory that can produce the same correlations as those given by quantum theory in the analysis of entangled states, if we drop superdeterminism".

Right. Now, this doesn't mean that quantum mechanics itself is non-local. It simply means that there is not going to be found a local Newton-like gears-and-wheels deterministic (or even stochastic) theory which will reproduce the statistical results predicted by quantum theory.

Now, depending on how you look upon quantum theory, this can mean or not that quantum theory itself is non-local. In order to even be able to say whether a theory is local or not, there needs to be the hypothesis of causal links. Things that happen here and now are "dependent" (deterministically or stochastically) on "things that happened there and then". Locality means that the "there and then" must coincide with the "here and now", up to small differences. That is, if the "entire physical state" is given in the neighbourhood of "here and now", then the entire causal influence on the here and now is fully determined, and doesn't depend anymore ON TOP OF THIS on "things happening there and then".

But this already supposes that we have a picture of reality including causality ! It is impossible to talk about locality without having given a meaning to causality. Also, we need to have a picture of reality which has a localised physical state.

Now, if you see quantum theory only as a mathematical trick to help you calculate outcomes of experiments in a single world view, between a setup and a measurement, and refusing to consider that there are "physical states" in between, then the concept of causality, nor the concept of locality, make any sense. You cannot say that this is local or non-local. You've just a mathematical trick to do calculations, and you have outcomes. In this case, the only thing that Bell tells you, is that you WON'T BE ABLE TO REPLACE it by a Newton-like, causal and local theory.

Many people take this stance, and I can understand them. The thing that bothers me with that view, is that you have no "physical picture" and hence that you cannot gain any intuition for what "goes on".

If you insist that the wavefunction "exists", but that projections "really happen", then clearly, you HAVE a physical state, there IS causality (the dynamics of the wavefunction and the collapse), and there is a non-local effect. But upon analysing this in more detail, you see that the ONLY non-local effect occurs upon the moments of PROJECTION (collapse) and NOT during unitary quantum dynamics.

Finally, if you insist that the wavefunction "exists", and that you follow all the time the quantum dynamics, you:
1) have MWI
2) you have a dynamics that is local (in the sense I tried to explain in this thread).

So, contrary to what is often claimed, quantum theory by itself is not local or non-local. it depends on the interpretation you give to the elements of the theory to conclude this or that way. What is true however, thanks to Bell, is that we are not going to find a single-world, mechanistic causal theory that is going to be equivalent to quantum theory.

We can find mechanistic/causal theories which are non-local (Bohm, and "wavefunction is real and projection") ;

we can refuse to say anything about a physical reality, and as such the notion of local or non-local doesn't mean anything ;

we can stick with unitary quantum dynamics, and show that it is causal/local. The price to pay is MWI.

Note that the RESULTS of quantum dynamics are in the "twilight zone" between "obviously local" and "obviously non-local".

The "obviously local" would be a theory which satisfies Bell's theorem.

The obviously non-local would be a theory that allows immediate SIGNALLING across finite distances.

Well, quantum results are in between. You cannot SIGNAL immediately, but it doesn't satisfy Bell's requirements either.

 

[vanesch]

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

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

0.5 |bob+> + 0.5 |bob->

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

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

I suppose you start from this state:
(x |bobAC+> |u++AC> + y |bobAD->|u--AD> ) |v-A>
- (-y |bobBE+> |u++BE> + x |bobBF-> |u--BF>) |v+B>

The reason why you cannot "add up" the different bob states, is that they carry labels.

There are two different bob states here, |bob+> and |bob->. However, bob+ appears with two different labels, namely AC and BE. Now, the label A was a pair with the v-system, and as this system didn't "come back" yet (it will when Alice will meet bob, and Alice will carry the complementary A-label), it means that we have to add the amplitudes squared. We can say the same for the BE label.

So the probability for bob+ (whether with label AC or BE) is given by |x|^2 + |y|^2, the x^2 from the AC contribution and the y^2 from the BE contribution.

In the same way the probability for bob- is |y|^2 + |x|^2, because of the same reason, the y^2 from the AD contribution and the x^2 from the BF contribution.

However, when alice meets bob, the bob A-label finds its "partner A label" in alice back, and the same for all the other labels. At this point, we FIRST add the amplitudes and THEN take the square.

In global notation, this is understandable:
the vector a|u1> |v1> is orthogonal to the vector b|u1> |v2> (even though u1 is of course collinear with u1). So the length of its sum is sqrt(|a|^2 + |b|^2)

But the vector a|u1>|v1> is of course collinear with the vector b|u1>|v1>, so the length of its sum is |a+b|.

Remind you, the "label" stuff does the "vector algebra" while the kets are the states that undergo the interactions.

 

[colorSpace]

Unfortunately I currently don't have enough time to address all points in your recent messages, I hope I can catch up a little over the weekend (although I read everything at least once). Meanwhile, feel free to reiterate any relevant points.

So I've tried to wrap my mind around the concept of the wavefunction a little more.

Apparently the distinct (measurable) physical states of the system, which are in superposition, are called "pure states". And the wavefunction expresses the probability of encountering each pure state in measurement. As the system evolves, various events (for example, in this case, measurements on particles u and v) affect the system, and using complex vector algebra, the wavefunction allows calculation of the resulting pure states and their probabilities.

I'm sure this is at best an approximation, so which other factors are relevant in this context, if I may ask?

I suppose you start from this state:
(x |bobAC+> |u++AC> + y |bobAD->|u--AD> ) |v-A>
- (-y |bobBE+> |u++BE> + x |bobBF-> |u--BF>) |v+B>

The reason why you cannot "add up" the different bob states, is that they carry labels.

The state above has the measurement angles, which is good, but it still has the 'v' particle in it, so it appears to be from the time before the particles separate. How do you remove the 'v' particle from this state, so that it becomes local for the time after measurement (but before they meet), without loosing all references that allow association later on?

What are the labels exactly? So far I took them as just notational markers, but apparently they are some physical state, otherwise they wouldn't keep the terms from adding up. I haven't noticed "labels" yet in the literature I've been reading, except for the text that you referenced earlier.

 

[colorSpace]

And if they keep the terms from adding up, won't that mean that there have to be more physically distinct Bob-states than the two, bob+ and bob-, and that some of these Bob states will, according to the logic of this concept, then be "killed" when they meet with the email, when the measurement angles are the same?

 

[vanesch]

So I've tried to wrap my mind around the concept of the wavefunction a little more.

Apparently the distinct (measurable) physical states of the system, which are in superposition, are called "pure states".

Yes. Well. Of some pure states, we might not even know in practice how to measure them, but you're right that *in principle* it should be possible to find some kind of measurement (even though not practically feasible) that can measure it.

Now, you have to know - and you will see the "naturalness" of the appearance of a Hilbert space here - that "a complete measurement" can only determine CERTAIN pure states, and that most of the pure states are not measurable with a given setup, but will be "superpositions" of these measurable states. If we do ANOTHER kind of "complete measurement", well, we will find that we can now measure OTHER pure states, and that those that we could measure with the first setup, are now superpositions of these new pure states.

This means that with a *specific* measurement setup, that the pure states we can measure, FORM AN ALGEBRAIC BASIS of "all possible pure states". That is: take a specific measurement, with it correspond a set of pure states that we can measure, let's call them states |a1>, |a2>, |a3> ...
Well, ANY pure state can be expressed as a superposition of these |a1>, |a2>, |a3> ...

So an arbitrary pure state |X> = x1 |a1> + x2 |a2> + x3 |a3> + ...

x1, x2, ... are complex numbers which determine fully the state |X>.

If we have ANOTHER measurement setup, the pure states that are measurable with this new setup, will be different than those of the first: |b1>, |b2>, |b3> ...
These states will ALSO form a basis of all states.

Now, because, say, |b4> is also a pure state, we can write it as |X> in the "a" basis:

|b4> = (b41) |a1> + (b42) |a2> + ...

(b41) is the complex number x1 when |X> is the state |b4> ...

In the same way, we can write |b17> in the "a" basis:

|b17> = (b17,1) |a1> + (b17,2) |a2> + ...

etc...

So these numbers (bxxx,yyy) describe the BASIS TRANSFORMATION between the "measurement basis a" and the "measurement basis b". And lo and behold: it turns out to be an ORTHOGONAL (or, because we are with complex numbers, a UNITARY) transformation!

So the link between the measurable pure states of experimental setup A, and the measurable pure states of experimental setup B, is A UNITARY TRANSFORMATION.

It is then natural to postulate that the basis {|a1>, |a2> ,... } itself is AN ORTHOGONAL BASIS. As such, then all measurement bases will be orthogonal, as they are linked between them by unitary transformations.

What we have done (up to some analytical properties), is to have defined a HILBERT SPACE. Indeed, a space of vectors (that is, elements which can be combined in linear superpositions) in which one can define orthogonal bases, which are linked by unitary transformations, is a (pre-) hilbert space. It is possible to define inproduct and norm over it.

For instance, the in-product can be defined as follows:
if |X> = x1 |a1> + x2 |a2> + x3 |a3> + ...

then < a3 | X > = x3.

The complex linear combinations (required by the postulate of superposition) of the full set of measurable states of a specific experiment span a (pre-) hilbert space.

The only thing we need to make it into a genuine hilbert space is a mathematical curiosity, which is to require that a Cauchy series converges. In other words, we don't want "holes" in our space (like there are "holes" in the rational numbers). But that's just mathematics.

So, if the system happens to be in a state that corresponds to one of the basis vectors of the measurement basis, and we do the appropriate measurement, then the outcome will be with certainty the outcome associated to that basis vector.

What if the system happens to be in a superposition of measurement basis states when we do the measurement ? Well, the answer is that we will *observe* it to be in one of the basis states, with the corresponding outcome, with a probability equal to the square of the "coefficient" of expansion in that basis.

I guess this is what you mean when you say:

And the wavefunction expresses the probability of encountering each pure state in measurement.

Now, mind you, it is not because the system was in a state |X> before measurement (in basis A), and we found, say |a3>, that this means that the system "was actually" in state |a3>. This is the kind of error that is often committed (cfr Afshar). No, the state |X> DIDN'T MEAN to be OR state |a1> OR state |a2> OR state |a3> ... but we simply don't know it. State |X>, because of the superposition principle, is a distinct physical state, but it APPEARS to us as state |a1> or ... |a3> ... only if we do a measurement in basis A.

The reason for that is simple. Imagine that the state was state |b2> of measurement basis B. Now, this means that if we apply measurement "B", that we will find b2 WITH CERTAINTY. We will NEVER find b1 or b3.

However, in basis A, state |b2> is a superposition: |b2> = (b2,1)|a1> + (b2,2) |a2> + (b2,3) |a3> ...

So we have a probability |(b2,1)|^2 to find a1, a probability |(b2,2)|^2 to find a2...
But if we say that this means that the state b2 was actually a statistical mixture of states |a1>, |a2>, ... |a3> ... (meaning, it is actually one of these states, but we simply ignore which one), and we would do a measurement B, then state |a1> (which is itself a superposition of |b1>, |b2> ...) would have a certain probability to find b1, a certain probability to find b2, b3 ...
And state |a2> also, would give a certain probability to find b1, to find b2 ...

In other words, if we think of state |b2> as a statistical mixture of states |a1>, |a2> ... before measurement, we wouldn't be able to explain how it comes that state |b2> ALWAYS gives rise to outcome b2 and never b1 if we do measurement B.

So, again, one cannot give a statistical interpretation to a superposition as long as one hasn't done any measurement. It is stated by the superposition principle too: a superposition is NOT a statistical mixture.

It is the basic fallacy in the (interpretation of) a lot of experiments, leading to a lot of (pseudo) paradoxes, and the Afshar experiment is a brilliant example of this.

As the system evolves, various events (for example, in this case, measurements on particles u and v) affect the system, and using complex vector algebra, the wavefunction allows calculation of the resulting pure states and their probabilities.

Well, as time progresses, the state of a system changes. That is, it "wanders" in Hilbert space. And this is the second fundamental axiom of quantum theory:

Time evolution is given by a unitary operator U(t).

In the same way as we had a unitary matrix which linked one measurement basis to another (at a given moment), we also have a unitary operator (as a function of time) which describes the time evolution of any quantum state. The whole thing of quantum dynamics is to find out WHAT is the form of this unitary time evolution operator. It turns out that it is intimately linked to the energy of a system, and for classically-looking systems, we know how to build the unitary time evolution operator starting from the classical dynamics.

What are the labels exactly? So far I took them as just notational markers, but apparently they are some physical state, otherwise they wouldn't keep the terms from adding up. I haven't noticed "labels" yet in the literature I've been reading, except for the text that you referenced earlier.

The labels (and the way I did it, I invented it on the fly) are nothing else but encodings of the coordinates of the state vector in a specific basis. What I tried to do is to scatter these coordinates over different localizable entities, instead of having them in a big, global container (the wavefunction).

 

[colorSpace]

The labels (and the way I did it, I invented it on the fly) are nothing else but encodings of the coordinates of the state vector in a specific basis. What I tried to do is to scatter these coordinates over different localizable entities, instead of having them in a big, global container (the wavefunction).

I don't see the connection between your response and the intent of my question.

 

[vanesch]

I don't see the connection between your response and the intent of my question.

I could answer: I don't see how you can miss the connection between the intent of your question, and my response and we can go on for ever like this.

But I'll try again. In as much as the superposition principle tells us something "physical" (that we can take any two physical states, and combine them with complex numbers, to have a new physical state), it is clear that these complex numbers are somehow "physical", no ? So the complex numbers that describe the *superpositions* of states, or in another way, that make up the algebraic expression of the wavefunction in Hilbertspace in a certain basis, must have somehow something physical to them... Well, these "physical" properties are in the bucket. They are of course NOT in the "physical states" (the basis vectors) themselves, as they tell us HOW to put those in superposition.
So these "physically meaningful numbers" will not be found in the basis states (classical states). They must be "somewhere else".

 

[colorSpace]

I could answer: I don't see how you can miss the connection between the intent of your question, and my response and we can go on for ever like this.

Well, and I could answer that we can still go on forever, as your response below still doesn't answer my question. The fact that you 'locate' the labels on the 'superposition side' doesn't say whether they are physical properties or notational markers. Given your previous elaborations, it is not clear to me whether you try to take the position that there is (almost) no difference, and try to treat them as notational in one context, and as physical in another context.

But I'll try again. In as much as the superposition principle tells us something "physical" (that we can take any two physical states, and combine them with complex numbers, to have a new physical state), it is clear that these complex numbers are somehow "physical", no ? So the complex numbers that describe the *superpositions* of states, or in another way, that make up the algebraic expression of the wavefunction in Hilbertspace in a certain basis, must have somehow something physical to them... Well, these "physical" properties are in the bucket. They are of course NOT in the "physical states" (the basis vectors) themselves, as they tell us HOW to put those in superposition.

Is your term "basis states" somehow different that 'pure states' ? So far it seemed 'pure states' is the common term. Maybe better to clarify, than not.

So my limited understanding of superpositions, and the mathematical terms describing them, is that they are a combination of (often non-local) pure states. That is, a soon as one replaces all the variables in a wavefuntion with actual values, one has something that boils down to, for example:

50% purestate1, 50% purestate2.

Where "50%" is something more than a classical probability since it can describe interference, and it can have a phase or such, being a complex number.

If I am missing something else here, this is the right time to tell me.

So the "the complex numbers that describe the *superpositions* of states" are *in effect* a single complex number in front of each pure state. "In effect", meaning, once the variables are replaced with their actual values, which is a mathematical operation, not a physical one, correct me if I'm wrong.

The very trivial thing to remain clear about is that (0.2+0.8) oranges is the same as (0.3+0.7) oranges, and the same as 1.0 oranges. Once these terms are all in one bucket, these three cases can only be differentiated if there are physically-distinct orange-slices, rather than a single orange. And of course, in the case of entanglement, there needs to be a physical difference, since in the end there will be either 2 or 4 physically distinct Bob-states.

Consider all this to be followed by a big, now very familiar, question mark.

So these "physically meaningful numbers" will not be found in the basis states (classical states). They must be "somewhere else".

In all non-local interpretations, the pure states are often non-local, so the superposition itself does not have a single physical location that can be measured in meters or miles.

The superposition has either a non-local "existence" or is an abstraction that needs to be "deconstructed" into multiple independent "superpositions" before it could be assumed to have a local existence. Your state descriptions seem to be (mostly) non-local, including the last one, and the ones in message #72.

 

[vanesch]

Well, and I could answer that we can still go on forever, as your response below still doesn't answer my question. The fact that you 'locate' the labels on the 'superposition side' doesn't say whether they are physical properties or notational markers. Given your previous elaborations, it is not clear to me whether you try to take the position that there is (almost) no difference, and try to treat them as notational in one context, and as physical in another context.

To me (not to everybody) superpositions have a physical meaning. That means that the complex numbers that enter in the superpositions also have a physical meaning. As I said, not everybody takes that position, but in MWI, we do. That is what it means to "give physical meaning to the statevector". As I said, some people don't ascribe any physical meaning to the statevector, and hence not to the concept of superposition as a physical phenomenon, but just as a calculational trick to help us find out which things "happen".

To me, the physical concept of superposition is the same as the concept of a statevector in hilbert space and is the same as giving physical meaning to the coefficients that build up this statevector from basis vectors.

Now, we can have different WAYS of WRITING DOWN the same abstract concept. We can think of a global statevector in hilbert space, or we can think of statevectors in smaller hilbert spaces, attached to localisable entities, and with "extra notation" to specify how these sub states "fit together" in the bigger one. All this are different MATHEMATICAL (notational ?) representations of the same abstract concept.

A bit in the same way as the electric and magnetic field vectors, or the electromagnetic 4-potential, or the F-tensor, are different representations of the same abstract concept, which is the EM field.

So the "physical reality" can be represented as well by a global state vector in a global hilbert space, or by "substates + extra notation" or even by other mathematical constructs. They are, again, different mathematical representations, notations, to represent the same abstract concept (which I identify with the "real" physical state).

Is your term "basis states" somehow different that 'pure states' ? So far it seemed 'pure
states' is the common term. Maybe better to clarify, than not.

basis states are a special, orthogonal, selection of pure states, and all pure states can be written as superpositions of basis states.

The whole point was to show that we can have basis states which are combinations of localisable sub-states of the composing systems, and that the coefficients of superposition can "walk with them".

So my limited understanding of superpositions, and the mathematical terms describing them, is that they are a combination of (often non-local) pure states. That is, a soon as one replaces all the variables in a wavefuntion with actual values, one has something that boils down to, for example:

I don't understand a word of what it means "replacing the variables in a wavefunction with actual values"...

Note that the word "wavefunction" is very badly chosen, and comes from the position representation of single-particle states. But it stuck. "wavefunction" means "vector in hilbert space". It doesn't have any "variables". It is a POINT in a big space.

If you insist on using a genuine "wavefunction", then the "variables" are the different classical positions, and the value is the coefficient of superposition of this classical position in the statevector.

So the "the complex numbers that describe the *superpositions* of states" are *in effect* a single complex number in front of each pure state. "In effect", meaning, once the variables are replaced with their actual values, which is a mathematical operation, not a physical one, correct me if I'm wrong.

The superposition of states means indeed, a complex coefficient in front of each BASIS STATE. This superposition itself is, however, itself a single pure state. So the coefficients depend on what set of basis states we have chosen. And I take as basis states, the combination of localisable classical states of each individual subsystem.

The very trivial thing to remain clear about is that (0.2+0.8) oranges is the same as (0.3+0.7) oranges, and the same as 1.0 oranges. Once these terms are all in one bucket, these three cases can only be differentiated if there are physically-distinct orange-slices, rather than a single orange. And of course, in the case of entanglement, there needs to be a physical difference, since in the end there will be either 2 or 4 physically distinct Bob-states.

Well, consider the 0.2 oranges that will pair up with the potatoes, and 0.8 oranges that will pair up with the apples. So yes, there is a difference between this situation, and the one with 1.0 oranges period. But the difference doesn't reside in the oranges. It are the same oranges, but the superposition principle requires us to make a distinction between the two situations. You cannot find any classical analogue of this, because the superposition principle is exactly what distinguishes quantum theory from classical theory.

In all non-local interpretations, the pure states are often non-local, so the superposition itself does not have a single physical location that can be measured in meters or miles.

The superposition has either a non-local "existence" or is an abstraction that needs to be "deconstructed" into multiple independent "superpositions" before it could be assumed to have a local existence. Your state descriptions seem to be (mostly) non-local, including the last one, and the ones in message #72.

Well, both are possible representations of the same abstract concept of course. But the fact that one CAN think of a local version, means that the abstract concept has the property of locality. Because that is what it means: CAN be represented by something that is localized.

 

[colorSpace]

To me (not to everybody) superpositions have a physical meaning. That means that the complex numbers that enter in the superpositions also have a physical meaning. As I said, not everybody takes that position, but in MWI, we do. That is what it means to "give physical meaning to the statevector". As I said, some people don't ascribe any physical meaning to the statevector, and hence not to the concept of superposition as a physical phenomenon, but just as a calculational trick to help us find out which things "happen".

To me, the physical concept of superposition is the same as the concept of a statevector in hilbert space and is the same as giving physical meaning to the coefficients that build up this statevector from basis vectors.

I think to some extent there is an obvious physical meaning since only the description of the superposition says which state will be more likely to be measured (as a combination of specific sub-states). Just the description of the basis state wouldn't indicate how likely it is to measure this state.

I'd guess it is just that some see wavefunctions as an abstraction for which the physical "implementation" is unknown, but they would probably agree that the raw information must be present in physical states somehow.

But that doesn't tell me whether 'labels' are just a notational difference or also a physical difference (and what kind of difference). You seemed to describe labels as just a notational marker.

Now, we can have different WAYS of WRITING DOWN the same abstract concept. We can think of a global statevector in hilbert space, or we can think of statevectors in smaller hilbert spaces, attached to localisable entities, and with "extra notation" to specify how these sub states "fit together" in the bigger one. All this are different MATHEMATICAL (notational ?) representations of the same abstract concept.

A bit in the same way as the electric and magnetic field vectors, or the electromagnetic 4-potential, or the F-tensor, are different representations of the same abstract concept, which is the EM field.

So the "physical reality" can be represented as well by a global state vector in a global hilbert space, or by "substates + extra notation" or even by other mathematical constructs. They are, again, different mathematical representations, notations, to represent the same abstract concept (which I identify with the "real" physical state).

That sounds like we start to get on the same page. Would you agree that in order to indicate two local states instead of one global non-local state, one needs to be able to have two (or more) separate and independent wavefunctions, each of which includes only basis states consisting only of *local* physical sub-states?

basis states are a special, orthogonal, selection of pure states, and all pure states can be written as superpositions of basis states.

Sorry for the confusion, I meant what you call "basis states", specifically, rather than pure states in general. I wouldn't even have mentioned the term "pure states", so far.

The whole point was to show that we can have basis states which are combinations of localisable sub-states of the composing systems, and that the coefficients of superposition can "walk with them".

What do you mean with "walk with them"? That seems like an important part of your concept.

Are you trying to use a single global wavefunction with localizable subsections within this single global wavefunction? If so, then I don't yet see how this would be a truly local concept.

It would seem to me that after measurement, and before meeting, the Bob system and the Alice system would have to be describable each by its own wavefunction, which uses only basis states consisting of local sub-states. That is, the description of the Bob system cannot use any "v particle" sub-state, nor any Alice-related sub-state.

I don't understand a word of what it means "replacing the variables in a wavefunction with actual values"...

Just very trivially that more complicated forms of a wavefuntions are due to mathematical variables having a more complicated relationship. For example the term a^2 + b^2 looks more complicated than 0.5 mathematically, but in a physical context, it may be the same, depending on the values of a and b. There is no additional physical information in a and b.

Note that the word "wavefunction" is very badly chosen, and comes from the position representation of single-particle states. But it stuck. "wavefunction" means "vector in hilbert space". It doesn't have any "variables". It is a POINT in a big space.

If you insist on using a genuine "wavefunction", then the "variables" are the different classical positions, and the value is the coefficient of superposition of this classical position in the statevector.

This sounds like a misunderstanding, but contains a point that I need to clarify.

This is Wikipedia's definition of 'wavefunction':

"A wave function is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a space that consists of the possible states of the system into the complex numbers. The laws of quantum mechanics (i.e. the Schrödinger equation) describe how the wave function evolves over time. The values of the wave function are probability amplitudes — complex numbers — the squares of the absolute values of which, give the probability distribution that the system will be in any of the possible states."

To me this would mean that if you have two separate (local) systems, then each has its own independent wavefunction, and each wavefunction is a function in a space over the possible states of each respective system.

Forgive my ignorance if this is off.

The superposition of states means indeed, a complex coefficient in front of each BASIS STATE. This superposition itself is, however, itself a single pure state. So the coefficients depend on what set of basis states we have chosen. And I take as basis states, the combination of localisable classical states of each individual subsystem.

This sounds about right, but seems to contradict that all major state descriptions you've given me so far, appear to include sub-states from both locations.

Well, consider the 0.2 oranges that will pair up with the potatoes, and 0.8 oranges that will pair up with the apples. So yes, there is a difference between this situation, and the one with 1.0 oranges period. But the difference doesn't reside in the oranges. It are the same oranges, but the superposition principle requires us to make a distinction between the two situations. You cannot find any classical analogue of this, because the superposition principle is exactly what distinguishes quantum theory from classical theory.

That's exactly what I'm trying to find out: how exactly you are trying to make that work. You are using labels, but they seems to be just notational.

The oranges can't know if they are (0.2+0.8) or (0.3+0.7). You say the superposition knows. But how, if it only consists of one complex number in front of each physically distinct basis state?

Well, both are possible representations of the same abstract concept of course. But the fact that one CAN think of a local version, means that the abstract concept has the property of locality. Because that is what it means: CAN be represented by something that is localized.

The thing is: I don't see local versions (yet) for the time after measurement and before meeting. I wonder how those maintain the necessary information. I think I would require them to be two completely separate and independent local state descriptions. That is what I am trying to get at.

 

[vanesch]

Ok, let's get a bit more technical then. Consider a universe in which there exist exactly 3 "particles" as subsystems. Or, let us first consider a universe in which there is only ONE particle.

A basis of states of a single particle is of course the "position basis". That is, with each individual, precisely located position in 3-d space P corresponds a quantum state, which we label |P>, and the set of all these |P> (there are as many of these basis states as there are points in 3-D space) forms a BASIS of the hilbert space of states of this single particle, H_1. So an arbitrary pure quantum state of a single particle is a superposition of all these basis states:
|X> = a1 |P1> + a2 |P2> + ... an |Pn> + ...

So to each basis state, there corresponds a complex number a in the particular state |X>. So to describe a single pure state, |X> in casu, we need to know all the complex numbers a1, a2, ... an, ... in this "position basis".

As, in this particular case, there is a 1-1 mapping between points in 3-d space, and basis vectors in the P-basis, we can also label the complex numbers a1, a2, ... .by their "point in space" which corresponds to the position basis state. This is what is the "wavefunction": the mapping from 3-D space (points P) onto the complex number (amplitude) - the a_i that goes with the basis state |P>.

But, but... the basis states |P> are not classical states. Indeed, they don't correspond to a classical dynamical situation of a point particle. One cannot associate any momentum to such a state (indeed, a single |P> state contains ALL "momentum states").

So a more classically-looking state is more a state "in between" a position state and a momentum state. For instance, "a gaussian wave packet" as wavefunction. So we can now have a basis of "classically looking states" which are "localised" (not much spread out) in space, and which also have a rather well-defined momentum. THESE are the basis states that we are going to use, because they are the quantum equivalent of the classical particle states (a position, and a velocity). We write them in ket notation:
|particle-at-joe-going-left> or something. We use them as basis states here. It is going to be difficult to define a "wavefunction" in this basis.

So let us accept that the hilbertspace of pure states H_1 is spanned by a basis of "classically-looking" states of the particle.

Over time, a classically-looking state will evolve in another state, thanks to the time evolution operator U_1 acting upon H_1. It isn't necessary that a classical state always evolves into another classical state. It can evolve into another pure state. That depends upon the specific dynamics of U_1.

Next, consider a universe of 3 particles. The hilbertspace of states of this universe consists of the tensor product of the 3 hilbertspaces of each individual particle:

H_tot = H_1 x H_2 x H_3

Now, a property of the tensor product of hilbert spaces is that it is spanned by a basis which is the set product of the bases of the subspaces.
So a basis of H_tot can be made up of all thinkable basis states:
|basisstate of particle 1> |basisstate of particle 2> |basisstate of particle 3> and hence the total hilbertspace is made up by superpositions of these basis states.

Now, if the 3 particles don't interact, then they EACH have their own dynamics U_1, U_2 and U_3, which acts each upon the relevant vector.

So if, for non-interacting particles, we have a state:

|p1> |p2> |p3>, then we can have it that time evolution acts over this global vector in H_tot, but that this can be seen as an individual evolution within each subspace:

|p1> in H_1, will evolve under U_1 into |q1>
|p2> in H_2, will evolve under U_2 into |q2>
|p3> in H_3, will evolve under U_3 into |q3>

So the total state will evolve from |p1>|p2>|p3> into |q1>|q2>|q3> under the global time evolution operator U = U_1 x U_2 x U_3.

I think it is clear that this global notation is just a mimicking of 3 independent local evolutions.

Now, consider a superposition in H:

a |p1> |p2> |p3> + b |s1> |s2> |s3>

Under the time evolution U, this evolves into:

a |q1> |q2> |q3> + b |t1> |t2> |t3>

But it is clear that each evolution happened independently in each sub-hilbert space independently: p1 evolved into q1 under U_1, and s1 evolved into t1 under U_1 also in H1.

In the same way, p2 evolved into q2 under U_2 and s2 evolved into t2 under U_2 in H2.

And p3 evolved into q3 and s3 into t3 under U_3, in H3.

So although we write the global state as a superposition, you see that the individual evolutions happen in the subspaces, even if the global state is a superposition. The first term has nothing to do with the second, and WITHIN the first term, the factor belonging to H_1 has nothing to do with the one in H_2 etc...

But the 3 particles were not interacting here. Let us now consider interactions. Interaction means that the time evolution U doesn't act as a product upon the two subspaces. If particles 1 and 2 interact, then that means that there is a unitary time evolution operator which acts in a general way upon the product space of H_1 x H_2.

So a state |p1>|p2> can evolve into an entire superposition of states c1|q1>|q2> + c2|q1'>|q2'> ... It will be a general state in H_1 x H_2.

The thing I wanted to underline is that interactions in quantum theory happen to be such (it doesn't have to, but the U-operators are like this) that:
1) there is only an interaction upon |p1>|p2> if p1 and p2 are states which correspond to the same locality
2) the resulting superposition of states |q1>|q2> ... are ALSO only states which correspond to the same locality.

So let us imagine that p1 and p2 are states corresponding to a common location ("at joe's"), and that they interact:

|p1>|p2> will become c1 |q1>|q2> + c2 |q1'> |q2'>

The numbers c1 and c2 are only dependent on the fact that it was p1 and p2. And the states q1 and q1' are states of the first particle (in H_1), in the same neighbourhood (at joe's), in the same way as q2 and q2' are states in H_2 in the same neighbourhood, of the second particle.

We say that particles 1 and 2, in states p1 and p2, interacted at Joe's, to become now c1 |q1>|q2> + c2 |q1'> |q2'>.

If our initial global state was:
a |p1> |p2> |p3> + b |s1> |s2> |s3>

then this becomes now:

a (c1 |q1>|q2> + c2 |q1'> |q2'>) |p3> + b |t1> |t2> |t3>

Note that this evolution of the states |p1> |p2> into q1, q1', q2, q2' didn't have anything to do with the fact that there was also a state p3 of the third particle, nor with the things that happened to the states s1 or s2 of the same particles 1 and 2 (which were for instance not at the same location, and hence didn't interact but evolved independently into the t1 and t2 states).

So it is not inconceivable to consider the "walkings" of each localised state individually, but of course to keep track of the superpositions, we have to keep track then in a localised way of which amplitudes walk where, with which.

We see that state p1 met state p2 at Joe's, and evolved (at Joe's) into an entangled state c1|q1>|q2> + c2 |q1'>|q2'>. From this point on, we are going to be able to follow the states q1, q2, q1', and q2' independently. We can also follow the states s1, s2, and s3 individually, evolving into t1, t2 and t3.

So this whole game can be done by following just "localised states" evolving in their own subspace, and, during interactions, by considering the product space of the interacting subsystems, but then only for the localised states at their location of interaction.

Now, sometimes, an evolution of two states happens so that terms "get together" again:
imagine that |q1> |q2> evolves into |r1>|r2> + K|r1'>|r2'> and that |q1'>|q2'> evolves into |r1>|r2> - L |r1'>|r2'>, somewhere else.

Now, this means that
a (c1 |q1>|q2> + c2 |q1'> |q2'>) |p3> + b |t1> |t2> |t3>

evolves into:
a ( c1 (|r1>|r2> + K|r1'>|r2'>) + c2 (|r1>|r2> - L |r1'>|r2'>) ) |p3> + b |t1> |t2> |t3>

Clearly AT THE LOCATION where this happens, the first and second system are in the state (c1 + c2) |r1>|r2> + (c1 K - c2 L) |r1'> |r2'> in the first term, and remains in the |t1> |t2> state in the second term.

The taking together of the terms evolving out of q1q2 and q1'q2' follows algebraically if we do this as above, but if we insist on "taking our stuff with us" then we need of course to keep the entire bookkeeping of amplitudes and so on with each individual localised state of each individual system. That's not elegant, but it is possible.
Indeed, the states q1 and q2 need to know that they had a factor c1 in the superposition with the states q1' and q2'. But that's not difficult: this superposition occurred during the interaction that had p1 p2 evolve into c1 |q1>|q2> + c2 |q1'> |q2'> at Joe's. So the pointer that is going to go with |q1> simply has to remember that it was together with q2, and with a factor c1 in relation to the other state q1', which got a factor c2. This information is locally available at Joe's.

As such, it is possible to have "local entities" walking around all over 3-d space, following the successive localities where-ever the dynamics of the states they are associated with, leads them. Call it the "soul" of the different localised states, if you wish :-)

At no point, the "soul" of the state of particle 1 has to learn something from a far-away soul, which it couldn't have learned during a previous encounter.

So the unitary quantum dynamics of states in hilbert space can be described in a local way in this manner.

Mind you that in all of this, I'm not talking about probabilities at all. I'm simply talking about the dynamics in hilbert space, or an equivalent formulation, which is the walkings of several states in several subspaces H_1, H_2, H_3...

 

[Rade]

Ok, let's get a bit more technical then. Consider a universe in which there exist exactly 3 "particles" as subsystems...The hilbertspace of states of this universe consists of the tensor product of the 3 hilbertspaces of each individual particle:
H_tot = H_1 x H_2 x H_3

I have a question that relates to what you explain here. Suppose a universe with only two particles, (1) a particle of antimatter with 2 mass units, think of deuteron [N^P^], where ^ = antimatter, and (2) a particle of matter with 3 mass units, think of He3[PNP]. So my questions, what is the hilbertspace equation if these two particles "interact" ? See that because the mass units are not identical we do not predict annihilation. What is the predicted result of this interaction ? Note: it may be important to consider in the solution that the [N^P^] antimatter is a spin 1 "vector" while the [PNP] matter is a spin 1/2 "spinor"--I do not know. In short, I am looking for a mathamatical explanation using QM that shows how these two particles can form superposition. Any help is appreciated.

 

[colorSpace]

Ok, let's get a bit more technical then. Consider a universe in which there exist exactly 3 "particles" as subsystems. Or, let us first consider a universe in which there is only ONE particle.

A basis of states of a single particle is of course the "position basis". That is, with each individual, precisely located position in 3-d space P corresponds a quantum state, which we label |P>, and the set of all these |P> (there are as many of these basis states as there are points in 3-D space) forms a BASIS of the hilbert space of states of this single particle, H_1. So an arbitrary pure quantum state of a single particle is a superposition of all these basis states:
|X> = a1 |P1> + a2 |P2> + ... an |Pn> + ...

[...snip...]

So the unitary quantum dynamics of states in hilbert space can be described in a local way in this manner.

Mind you that in all of this, I'm not talking about probabilities at all. I'm simply talking about the dynamics in hilbert space, or an equivalent formulation, which is the walkings of several states in several subspaces H_1, H_2, H_3...

Interesting. That seems a nice example of how a non-local superposition evolves when its basis states are subject to local evolutions and interactions.

Still, it seems, the highest level superposition continues as a non-local relationship as it is a relationship across multiple locations. Even if all particles meet again at the end.

 

[vanesch]

I have a question that relates to what you explain here. Suppose a universe with only two particles, (1) a particle of antimatter with 2 mass units, think of deuteron [N^P^], where ^ = antimatter, and (2) a particle of matter with 3 mass units, think of He3[PNP]. So my questions, what is the hilbertspace equation if these two particles "interact" ?

if only we knew ! The best answer I can give, is the one quantum field theory gives. Now, quantum field theory has a lot of mathematical difficulties describing the unitary evolution operator as a function of time, and the only thing we know more or less how to reasonably calculate, is the asymptotic value of the unitary evolution for t-> infinity. The trouble here is that we are in relativistic quantum theory, where we cannot consider a finite and fixed number of particles, and we have to switch to quantum fields as state descriptions.

So, in QFT, there is no "universe with just 3 particles", as the fields are present or are not present, and each field can represent as many particles as one desires (it are the different states of the field!). There is simply a universe with "only QCD fields" for instance, or with all the "standard model fields".

See that because the mass units are not identical we do not predict annihilation. What is the predicted result of this interaction ? Note: it may be important to consider in the solution that the [N^P^] antimatter is a spin 1 "vector" while the [PNP] matter is a spin 1/2 "spinor"--I do not know. In short, I am looking for a mathamatical explanation using QM that shows how these two particles can form superposition. Any help is appreciated.

We have to consider this as an anti-deuteron/helium interaction, but the bound state doesn't matter if we are at high energies. We then only need the "structure functions" of the anti-deuteron and of the helium, which give us the quark and gluon content of each hadronic particle, and (it's the only thing we know how to do in QFT) calculate the "free" interactions of the components of each particle (say, an anti-u quark from the anti-deuteron collides with a d-quark from the helium...).
The structure functions consider in fact the incoming hadrons as statistical mixtures of free quark and gluon states. It is an approximation which only works well at high energies.

 

[vanesch]

Interesting. That seems a nice example of how a non-local superposition evolves when its basis states are subject to local evolutions and interactions.

Still, it seems, the highest level superposition continues as a non-local relationship as it is a relationship across multiple locations. Even if all particles meet again at the end.

Well, what I tried to show in this thread is that all the needed information in any transformation can "travel" with each path, and that at no point, one needs "coefficients created at a distance" without there being a path to vehicle this information.

 

[colorSpace]

Well, what I tried to show in this thread is that all the needed information in any transformation can "travel" with each path, and that at no point, one needs "coefficients created at a distance" without there being a path to vehicle this information.

From my point of view, you have shown a path, but not a vehicle, and here many of my objections remain. In an abstract mathematical way, this succeeds in highlighting the difference between non-local 'correlation' and non-local 'signal sending', however without a physically viable theory of a vehicle, and some other points that need to be shown, it is not a physically viable local theory. I interpret the referenced texts by Richter as admitting that 'the vehicle' is at least an open problem.

Without that, one would have to say that other ideas, such as 'everything could be a computer simulation', or 'everything could be a dream', would have to be taken as objections to non-locality as well.

A universe which is in a superposition of all possible physical states, and splits off observer-specific universes whenever necessary, might even explain true FTL 'signal sending' in a 'local' fashion, so to speak on a gradient from objective reality to subjective reality.

 

[vanesch]

From my point of view, you have shown a path, but not a vehicle, and here many of my objections remain. In an abstract mathematical way, this succeeds in highlighting the difference between non-local 'correlation' and non-local 'signal sending', however without a physically viable theory of a vehicle, and some other points that need to be shown, it is not a physically viable local theory. I interpret the referenced texts by Richter as admitting that 'the vehicle' is at least an open problem.

Ok, and where is the "vehicle" or the "storage" in global quantum theory ?

Without that, one would have to say that other ideas, such as 'everything could be a computer simulation', or 'everything could be a dream', would have to be taken as objections to non-locality as well.

No, because a computer simulation can be "local" or not, depending on whether we can find an equivalent version of it that is given by mappings (vehicles!) in 3D or not.

A universe which is in a superposition of all possible physical states, and splits off observer-specific universes whenever necessary, might even explain true FTL 'signal sending' in a 'local' fashion, so to speak on a gradient from objective reality to subjective reality.

No, you won't find such a possibility ! That's exactly what I wanted to point out.
You won't find a potential (even "abstractly mathematical") pathway of information in 3D space that can mimick FTL signalling!

We have two DIFFERENT issues here:
- we have the local/global issue
- we have the issue of "physical storage" of the information in the wavefunction

You seem to object that I cannot point you to a (classical) local "storage" for the information included in the quantum mechanical superposition (the complex coefficients and pairing-ups and so on), and you seem to use this as an argument against LOCALITY.

But now I say: ok, let's say for argument's sake that I admit your objection. So the fact that there is not a LOCAL physical storage is admitted. So now my question to you: if I admit it to be GLOBALLY stored, where is the GLOBAL physical storage then ?
And be sure to show me that it has the physical properties you required the LOCAL storage to have ! Otherwise, your objection (lack of a storage) doesn't discriminate between local and global.

 

[colorSpace]

Ok, and where is the "vehicle" or the "storage" in global quantum theory ?

Since you were making a summarizing statement, rather than addressing my point about the highest level in your state description still being a non-local superposition, I had the impression that this was going to be the end of the discussion, and made my final statement. I find that you are generally quite reluctant to address my points directly.

To your point: Non-local quantum theory doesn't need to answer the question of "where" since it is non-local. By acknowledging that there are physical states outside of our common understanding of physical space, it has acquired the right of not having to answer that question.

However it describes that information precisely in the wavefunction, whereas you don't seem to answer my question of what exactly "labels" are, instead you just keep repeating that superpositions are part of physical reality, which of course they are in any case.

You act as if wavefunctions would accommodate labels naturally, as if that would be just a notational difference. However wavefunctions as they are defined in quantum mechanics appear not to have that capability, and my questions in this direction are answered only vaguely if at all.

No, because a computer simulation can be "local" or not, depending on whether we can find an equivalent version of it that is given by mappings (vehicles!) in 3D or not.

A computer simulation can simulate anything it wants, even if the computer itself functions "locally" (given enough computer power, of course, but hey, our resources are unlimited).

No, you won't find such a possibility ! That's exactly what I wanted to point out.
You won't find a potential (even "abstractly mathematical") pathway of information in 3D space that can mimick FTL signalling!

Sure, a universe which is a superposition of all possible physical states can mimmic anything a traveling observer might expect to see when checking whether any message already arrived. You just have to implement the logic in his perceptive apparatus which makes the universe pair-up the corresponding reality to this branch of the universe which this observer expects to find at any location of the travel. It can mimmic anything, just like a computer simulation.

Unless you provide a major new point, this will be last message in this sequence, for the time being. It was an interesting discussion, and I thank you for all general explanations regarding quantum physics! I'm sure we'll discuss again soon enough, however my time is limited and we seem to be going in circles lately.

My view remains as in my previous message.

 

[vanesch]

To your point: Non-local quantum theory doesn't need to answer the question of "where" since it is non-local. By acknowledging that there are physical states outside of our common understanding of physical space, it has acquired the right of not having to answer that question.

just to be clear: I didn't mean with "where" a point in 3-d space, but a "conceptual" where (like: in a computer's memory - you know, the computer that simulates the universe, or the mind of your favorite goddess, or a mathematical structure or so). As you seemed to insist upon the necessity of a "physical vehicle" for anything physical, I took it that you would apply the same requirement for a "physical" non-local state. But you seem to equate "non-local" with a lesser requirement of physicality than you seem to require of something "local". Well, I take up that right then also for a local theory ! The ONLY difference is that my "storage" is now "distributed" over 3-d space (that is, each of its components can be *put in relationship with a local environment in 3-d space*), while yours cannot be put in such a relationship with 3-d space.

Of course, in BOTH CASES, the "physical content" is outside of our common understanding of "physical" (3d) space. My "local" version of quantum theory is of course just as much outside of 3d space as the "global" version. I'm not disputing this. But "local" doesn't mean "is inside of our common understanding of physical 3d space". It only means that the thing that is "outside" *can be put in relationship* with physical 3d space.

However it describes that information precisely in the wavefunction, whereas you don't seem to answer my question of what exactly "labels" are, instead you just keep repeating that superpositions are part of physical reality, which of course they are in any case.

You act as if wavefunctions would accommodate labels naturally, as if that would be just a notational difference. However wavefunctions as they are defined in quantum mechanics appear not to have that capability, and my questions in this direction are answered only vaguely if at all.

Of course that wavefunctions would accommodate labels naturally ! That's what I did all the time (probably that's why I don't succeed in answering your questions... things that seem totally obvious to me seem to be not possible for you).

If I write the global wavefunction:

0.3|a+> |b+> + 0.8i |a-> |b->

or "I say that system a is in a state a+ which is entangled with a state b+ and a coefficient 0.3, and also in a state a- which is entangled with a state b- and a coefficient
i", don't you think that both statements vehicle exactly the same information ? Nevertheless, my verbal phrase is entirely local (a can carry it with itself, say, in a text file structure), while the formal expression can be said to be global as an expression in a global hilbert space.
The b-system can carry with itself ALSO its own text file (mind you, the text files are OUTSIDE of 3-d physical space, of course, but each of these text files can be PUT IN RELATIONSHIP with a small neighbourhood of space). So, consider that, in the same way as you can imagine "little arrows" attached to each point in space in classical electrodynamics, you can now imagine "text files" travelling, being attached from neighbourhood to neighbourhood in space. They contain the "verbatim" expressions of what I wrote above, and they travel along with the localised particle states as these move through space. They are the verbatim expressions of (parts of) the global wavefunction. Mind you, these are "mathematical" text files which are NOT implemented by "objects" in physical space, just as the "little arrows" of the E and B field are not "iron pieces of arrow" but are abstract mathematical concepts, attached to a point in space.

Sure, a universe which is a superposition of all possible physical states can mimmic anything a traveling observer might expect to see when checking whether any message already arrived. You just have to implement the logic in his perceptive apparatus which makes the universe pair-up the corresponding reality to this branch of the universe which this observer expects to find at any location of the travel. It can mimmic anything, just like a computer simulation.

Yes, but the square of the amplitudes has then still to come in agreement with his statistical observations ! I'm sure you understood that THAT was the interesting part: to get the AMPLITUDES right. It is the only link with observation.