Will quantum computers ever be possible?

[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.