Different interpretations? No, different theories

[stevendaryl]

Yes, I think that last part is correct. But now I think that my statement that "QM without the Born rule + tensor products → QM with the Born rule" isn't accurate enough. I think that the real problem with Zurek's derivation is that it relies not only on "QM without the Born rule + tensor products", but also on an assumption about how the probabilities assigned by the formula correspond to measurement results.

On odd days, I agree with you that the Born rule must be added as an additional hypothesis. However, the weird thing about the Born rule is that you can push its application off indefinitely. What I mean by that is this: Suppose you are interested in measuring the spin of an electron that is in a superposition of states $\vert \Psi \rangle = \alpha\ \vert +\frac{1}{2}\rangle + \beta\ \vert -\frac{1}{2}\rangle$. You could

1. Say that the spin-measuring apparatus has a probability $\vert \alpha \vert^2$ of measuring spin-up
2. Treat the apparatus quantum mechanically, so there is no definite result of the measurement until an experimenter comes along and observes the apparatus, in which case the human has a probability of itex]\vert \alpha \vert^2[/itex] of observing the apparatus to be in the state of having measured an electron in the state spin-up.
3. Treat the experimenter quantum mechanically, so there is no definite result for his observation until a different observer comes along and reads his lab write-up.
4. Treat the second experimenter quantum mechanically...
5. Etc.

There is no need for the Born rule until the last step of however many steps you want to include in the list. And the last step could be pushed off until the far future, where our great-great-great-great-grandchildren read about the whole history of the human race.

 

[stevendaryl]

Religious dogma is usually defined as something that is taken on faith without experimental evidence. Like the MWI. Sorry if i hurt religious feelings about the trillion worlds.

You're using words ("religion" and "dogma") in a way that conveys no information except your own feelings.

 

[stevendaryl]

Your commment makes no sense either, as I didn't say that the MWI was a classical explantion

You said:

...you [are] after a classical explanation of quantum behavior. Of course this can't be done and no such explanation exists except fancy, religious ideas like the MWI.

I interpreted "no such explanation exists" as "no classical explanation of quantum behavior exists", and I interpreted the word "except" to mean that MWI is an exception. Which would imply that "MWI is an exception to the claim that no classical explanation of quantum behavior exists".

I guess I shouldn't try to interpret your words as conveying meaning, as opposed to pure scorn, which is how they were intended.

 

[Maui]

You said:
I interpreted "no such explanation exists" as "no classical explanation of quantum behavior exists", and I interpreted the word "except" to mean that MWI is an exception. Which would imply that "MWI is an exception to the claim that no classical explanation of quantum behavior exists".

I will modify my initial statement so that no confusion arises, though from the context it seems obvious what my motivation was:

"you [are] after a classical explanation of quantum behavior. Of course this can't be done and no such explanation exists except fancy, religious attempts like the MWI""attempts" here does not equal classical explanation within physics, though it probably does within religion.

I guess I shouldn't try to interpret your words as conveying meaning, as opposed to pure scorn, which is how they were intended.

So if i react to scorn being thrown at the CI(a minimalist, no nonsense interpretation of experimental results) by showing the same amount of dismay at the religious proposition of MWI, it suddenly makes my words meaningless? I guess you haven't read my posts or your motivation is different from addressing the point being made but the author who made the point(basically an ad homimnem attack).

You're using words ("religion" and "dogma") in a way that conveys no information except your own feelings.

...towards beliefs that rest on no experiemental evidence whatsoever.

 

[stevendaryl]

So if i react to scorn being thrown at the CI(a minimalist, no nonsense interpretation of experimental results) by showing the same amount of dismay at the religious proposition of MWI, it suddenly makes my words meaningless?

I would say yes, your claims are pretty devoid of anything but scorn.

 

[Nugatory]

no such explanation exists except fancy, religious attempts like the MWI

The thread, taken as a whole, appears to my support my position (which I believe to be widely shared) that there are a number of interpretations that cannot be falsified experimentally. Therefore:
1) the idea suggested in the thread title can be rejected; interpretations are not theories.
2) there's not a lot of point in arguing about which interpretation is "right", nor whether an interpretation is being accepted for aesthetic or religious or other reasons. (Me, I choose mine based on aesthetics).

 

[Fredrik]

...towards beliefs that rest on no experiemental evidence whatsoever.

The MWI is essentially just the idea that QM is not just an assignment of probabilities to possible results of experiments, but also a description of what's actually happening. There's no evidence for that statement, but there's also no evidence for its negation. So I think it would be hard to justify the claim that the defining assumption of the MWI is religion and its negation is not.

I think of interpretations the way I think of Venn diagrams for set theory. They aren't part of the theory. They are tools that can help us develop some intuition about the theory.

 

[JK423]

You've got me really confused!
Demystifier, indeed the OP seems to be impossible even in principle, i have to think about this a little bit more, and i will post in the future if i think of something.
Right now i am quite "obsessed" with the simplicity of MWI and the non-circular derivations of Born's rule. So, i would like to ask you all a question.

Say, that, tomorrow a paper appears on arXiv where Born's rule has been derived non-circularly and without inserting probabilities "by hand" in any way; assume that everything comes out naturally. What will be the meaning of this result? Will it mean that MWI is correct and infinite copies of the world exist simultaneously? I don't know why, but i have a feeling that the statement "simultaneous existence" involves hidden assumptions that require more than just deriving Born's rule.

 

[Fredrik]

Right now i am quite "obsessed" with the simplicity of MWI and the non-circular derivations of Born's rule.

You may want to take a look at Gleason's theorem then. I think it's a much better way to obtain the probability formula than what Zurek did. It even provides the motivation for the definition of state operators in QM.

Say, that, tomorrow a paper appears on arXiv where Born's rule has been derived non-circularly and without inserting probabilities "by hand" in any way; assume that everything comes out naturally. What will be the meaning of this result? Will it mean that MWI is correct and infinite copies of the world exist simultaneously?

Even without that result, the MWI can (probably) be viewed as a plausible description of what is happening to the universe. With that result, it can also be viewed as an explanation of why QM is a good theory.

 

[Demystifier]

You've got me really confused!
Demystifier, indeed the OP seems to be impossible even in principle, i have to think about this a little bit more, and i will post in the future if i think of something.
Right now i am quite "obsessed" with the simplicity of MWI and the non-circular derivations of Born's rule. So, i would like to ask you all a question.

Say, that, tomorrow a paper appears on arXiv where Born's rule has been derived non-circularly and without inserting probabilities "by hand" in any way; assume that everything comes out naturally. What will be the meaning of this result? Will it mean that MWI is correct and infinite copies of the world exist simultaneously? I don't know why, but i have a feeling that the statement "simultaneous existence" involves hidden assumptions that require more than just deriving Born's rule.

If there was a way to derive Born's rule non-circularly, I would still not be satisfied with MWI due to the following objection:
https://www.physicsforums.com/blog.php?b=4289

 

[JK423]

If there was a way to derive Born's rule non-circularly, I would still not be satisfied with MWI due to the following objection:
https://www.physicsforums.com/blog.php?b=4289

I need to study the paper, but my immediate response at this point is:
Aren't the interactions taking care of defining the subsystems? Can you give me an example of the ambiguity you mention?

 

[Fredrik]

If there was a way to derive Born's rule non-circularly, I would still not be satisfied with MWI due to the following objection:
https://www.physicsforums.com/blog.php?b=4289

I consider that a good reason to reject the idea that a preferred basis identifies the worlds, but not a good reason to reject the idea of many worlds. See e.g. my post #33.

Aren't the interactions taking care of defining the subsystems? Can you give me an example of the ambiguity you mention?

The universe can be decomposed into "you + everything else" or "you and the chair you're sitting on + everything else".

 

[Demystifier]

I need to study the paper, but my immediate response at this point is:
Aren't the interactions taking care of defining the subsystems? Can you give me an example of the ambiguity you mention?

Defining subsystems in terms of interactions looks like another circularity. Namely, a priori, all you have is a total Hamiltonian, not a split of the Hamiltonian into the "free" and "interacting" part. If you choose some splitting of the whole system into subsystems then you can also (at least to some extent) see what is the interacting part of the Hamiltonian. But then it is circular to use the interacting part to identify the subsystems again.

The explicit examples of ambiguity are presented in the paper.

 

[Demystifier]

I consider that a good reason to reject the idea that a preferred basis identifies the worlds, but not a good reason to reject the idea of many worlds. See e.g. my post #33.

Fine. But then, as the author says in the Conclusions, we deal with
"... Many Many Worlds Interpretation (because each of the arbitrary more complicated factorizations tells a different story about Many Worlds [7])."

 

[Fredrik]

Fine. But then, as the author says in the Conclusions, we deal with
"... Many Many Worlds Interpretation (because each of the arbitrary more complicated factorizations tells a different story about Many Worlds [7])."

Ooh, now I think I have to read the conclusions, and check out reference 7.

 

[Fredrik]

I had a quick look. What he called "many many worlds" is the idea that decompositions are arbitrary, but then there's a preferred bases selected by decoherence or something. I guess what I'm advocating would be "many many many worlds" then, because I'm suggesting that decompositions are arbitrary, and that the basis is arbitrary. (The basis selected by decoherence is still "preferred", but not in the sense that it identifies the worlds; it just identifies worlds that are more interesting than most).

 

[JK423]

I read the paper and it, indeed, presents a serious problem. This problem seems to be relevant to classical physics as well, not just quantum.

Consider the Earth and Sun, where the Sun stands still and the Earth goes around it in circles. The Hamiltonian of the system is
${H_1} = {H_{{r_{Earth}}}} + {H_{{r_{sun}}}} + {H_{{\mathop{\rm int}} }},$
and the equations of motion show that an object (the earth) is moving in circles.
As we know, a change of coordinates to center of mass R and relative position r uncouples the system,
${H_2} = {H_R} + {H_r}$,
and the new equations of motion gives two static objects (or at least one of them -R- moving in constant motion).
If this system was all there is in the universe, then we would not be able to tell which description is the "real one" because they are mathematically equivalent. This holds for classical physics as well.
However, in practice when we (the observer) look at the system we see the first case -the Earth going around in circles- and not the second -two objects standing still. That's because the observer interacts specifically with the r_Earth and r_Sun, and not with R and r. The paper suggests now, that if we put the observer in the description and consider the system "earth+sun+observer" all there is in the universe, then the Hamiltonian describing the whole system can be expressed in various different bases and there is ambiguity in which interpretation is "the real one".

My objection:
I agree that only the postulate of the wavefunction seems not to be enough, it leads to the aforementioned 'paradoxa'. My first thought is to postulate spacetime; if we do that, the previous ackward situation disappears. In the postulated spacetime basis, all the interactions (in the global Hamiltonian) take their well-known form (which is also postulated) and this solves (?) the subsystem ambiguity. Now, Schwindt -in that paper- presents this idea of postulating spacetime to solve the problem (in page 8), but he argues that it's not enough, but to be honest i cannot understand why. He defines a new space in order to make his argument, but this new space is not related to the previous (postulated one) via a Lorentz transformation as it should, so his argument seems really weird. If you understand it, please explain.

In conclusion, i think that postulating
1) a spacetime basis (plus Lorentz transformations)
2) the interactions (which take the particular known local form in the postulated spacetime basis plus they are invariant under Lorentz transformations),
solves the problem.
For example, in the particular situation with Earth+Sun the postulated space basis involves r_Earth and r_Sun, and the transformed coordinates R and r that we later get are not related with a Lorentz transformation to the postulated ones.

Now, one may ask; how do you know which to postulate, r_Earth & r_Sun, or R & r? Hmmm. This system came about from an initial quantum state. If we also postulate an initial quantum state of the universe, together with the spacetime basis and interactions, then the later formation of Earth+Sun would (probably) involve r_Earth & r_Sun.

 

[JK423]

You may want to take a look at Gleason's theorem then. I think it's a much better way to obtain the probability formula than what Zurek did. It even provides the motivation for the definition of state operators in QM.

Hmm, i don't think so.. Gleason's theorem involves projection operators that act on states. You cannot use the projection operator formalism in the case of the quantum state of the universe, because you will need an extra system outside of the universe to do the job.

 

[Fredrik]

Hmm, i don't think so.. Gleason's theorem involves projection operators that act on states. You cannot use the projection operator formalism in the case of the quantum state of the universe, because you will need an extra system outside of the universe to do the job.

I assume that you mean that it's impossible to do an experiment in which the entire universe is being "measured". This is of course true. But this also makes the Born rule irrelevant to such experiments.

Gleason's theorem is relevant to all situations where the Born rule is relevant.

 

[JK423]

Agreed, i just want to point out that you cannot use Gleason's theorem to prove the probability formula in the context of MWI. For example, if the state of the universe is
$\left| {{\Psi _{{\rm{universe}}}}} \right\rangle = a\left| + \right\rangle \otimes \left| \uparrow \right\rangle + b\left| - \right\rangle \otimes \left| \downarrow \right\rangle$. How would you make sense of a and b? Gleason's theorem, in this case, uses projection operators on $\left| {{\Psi _{{\rm{universe}}}}} \right\rangle$ to deduce the probability formula. In MWI this cannot be, since nothing else exists to measure this state. Zurek's derivation is based on symmetry considerations and not on projection operators, so it's applicable in MWI. What I'm trying to say is, that, we cannot treat Zurek's work and Gleason's theorem on the same footing regarding this issue.

 

[Fredrik]

I don't follow your argument. Projection is a simple mathematical operation. I don't see how it has anything to do with the fact that there's no way to perform a measurement on the universe.

Also, since you agree that Gleason's theorem applies to all situations where the Born rule can be used, how can you say that Zurek's derivation is valid? These two statements look contradictory to me.

 

[Fredrik]

Fine. But then, as the author says in the Conclusions, we deal with
"... Many Many Worlds Interpretation (because each of the arbitrary more complicated factorizations tells a different story about Many Worlds [7])."

I have started reading the article. Maybe this will clear up when I read more of it, but it seems to me that he's saying that there's always a way to decompose the universe into non-interacting subsystems. Is this really what he means when he says that "nothing happens" in the universe? Isn't it still possible that subsystems of those subsystems are interacting with each other, and in that case, wouldn't it mean that something is happening?

 

[Demystifier]

I read the paper and it, indeed, presents a serious problem. This problem seems to be relevant to classical physics as well, not just quantum.

Consider the Earth and Sun, where the Sun stands still and the Earth goes around it in circles. The Hamiltonian of the system is
${H_1} = {H_{{r_{Earth}}}} + {H_{{r_{sun}}}} + {H_{{\mathop{\rm int}} }},$
and the equations of motion show that an object (the earth) is moving in circles.
As we know, a change of coordinates to center of mass R and relative position r uncouples the system,
${H_2} = {H_R} + {H_r}$,
and the new equations of motion gives two static objects (or at least one of them -R- moving in constant motion).
If this system was all there is in the universe, then we would not be able to tell which description is the "real one" because they are mathematically equivalent. This holds for classical physics as well.
However, in practice when we (the observer) look at the system we see the first case -the Earth going around in circles- and not the second -two objects standing still. That's because the observer interacts specifically with the r_Earth and r_Sun, and not with R and r. The paper suggests now, that if we put the observer in the description and consider the system "earth+sun+observer" all there is in the universe, then the Hamiltonian describing the whole system can be expressed in various different bases and there is ambiguity in which interpretation is "the real one".

Good point!

Now, Schwindt -in that paper- presents this idea of postulating spacetime to solve the problem (in page 8), but he argues that it's not enough, but to be honest i cannot understand why.

I can't see where exactly (in page 8) does he argue that. Can you quote the exact statement?

 

[bhobba]

Hmm, i don't think so.. Gleason's theorem involves projection operators that act on states.

Gleasons Theorem is prior to the definition and interpretation of states. It shows for an observable R E(R) = Tr(PR) where P is a positive operator of trace 1. P is by definition the state of the system. The only assumptions in the derivation is the eigenvalues of the Hermitian operator R are the possible outcomes of the observation and the eigenvectors are non contextual. The exact meaning of the eigenvectors as the state of the system after the observation is not required at this stage. After Gleasons Theorem is proved from the assumption of continuity in the change of system states after the observation (ie it will give the same result an infinitesimal instant later) you can show it must be in the state corresponding to the eigenvector associated with the outcome.

Thanks
Bill

 

[stevendaryl]

I have started reading the article. Maybe this will clear up when I read more of it, but it seems to me that he's saying that there's always a way to decompose the universe into non-interacting subsystems. Is this really what he means when he says that "nothing happens" in the universe? Isn't it still possible that subsystems of those subsystems are interacting with each other, and in that case, wouldn't it mean that something is happening?

I had a similar disconcerting feeling about a "pure wavefunction" interpretation of quantum mechanics, that went by a slightly different route: Any wave function (in nonrelativistic quantum mechanics, anyway) can be interpreted as a superposition of energy eigenstates. Within each such "branch" of the wavefunction, the universe is essentially unchanging (the time dependence $e^{-i \omega t}$ is trivial). Now, you can argue that splitting the wave function into energy eigenstates is the "wrong" decomposition into possible worlds, but it seems to me that the notion in which it is wrong has to go beyond just unitary evolution of states in Hilbert space. So you need something besides the wave function to get any notion of dynamics at all.

 

[JK423]

I can't see where exactly (in page 8) does he argue that. Can you quote the exact statement?

At page 8, last paragraph he says

In fact, there are infinitely many different spaces that can serve as a basis.

In Eq. (8) he postulates a basis {|x>}, and then in Eq. (9) he postulates another space basis{|y>} and he concludes

The relations (9) and (10) define the new space. The two spaces are totally different.
You don’t get y-space by simply moving around some points of x-space. And yet we
can write |Ψ> as a wave function in x-space or in y-space, and there is no reason why
|Ψ> should look simpler in x-space than in y-space. (This is well known for x-space
versus k-space, the Fourier transformed space. This is just a reminder that there
is an infinity of such spaces.) The Hamilton operator will look very unpleasant in
y-space. The position operator X looks nice in x-space and unpleasant in y-space.
The position operator Y looks nice in y-space and unpleasant in x-space, etc.

...

Again, H may take on its simplest form if it is written in terms
of such an integral. But again the question is why an observer, arising as some part
of the global state vector, “sees” this particular space over which the integrals run.

To be honest, i am not sure what he is trying to prove.. To my mind, postulation of a space basis plus interactions expressed in that basis plus an initial quantum state to start with, solves the factorization problem. Ofcourse these are "handwaving" arguments, i cannot be sure. What do you think?

 

[Demystifier]

At page 8, last paragraph he says

In Eq. (8) he postulates a basis {|x>}, and then in Eq. (9) he postulates another space basis{|y>} and he concludes


To be honest, i am not sure what he is trying to prove.. To my mind, postulation of a space basis plus interactions expressed in that basis plus an initial quantum state to start with, solves the factorization problem. Ofcourse these are "handwaving" arguments, i cannot be sure. What do you think?

If one takes some basis {|x>} to be a PREFERRED basis, then you are right that the problem is essentially removed. But he does not take {|x>} to be a preferred basis. In his discussion, it is merely SOME basis, not better than any other.

 

[JK423]

I don't follow your argument. Projection is a simple mathematical operation. I don't see how it has anything to do with the fact that there's no way to perform a measurement on the universe.

Also, since you agree that Gleason's theorem applies to all situations where the Born rule can be used, how can you say that Zurek's derivation is valid? These two statements look contradictory to me.

I am quite confused.. You are right, Gleason's theorem applies to all those situation that Born's rule is relevant. I need to understand the differences between Gleason's and Zurek's work, so i cannot say anything else at the moment. Does Gleason's theorem assumes Born's rule for example?

 

[JK423]

If one takes some basis {|x>} to be a PREFERRED basis, then you are right that the problem is essentially removed. But he does not take {|x>} to be a preferred basis. In his discussion, it is merely SOME basis, not better than any other.

What do you mean "preferred" ? That the physical interactions take their well-known form in that particular basis?

 

[Demystifier]

What do you mean "preferred" ? That the physical interactions take their well-known form in that particular basis?

No. In classical physics, it would mean that positions x correspond to positions of some physical objects. In quantum physics, it depends on the interpretation. In particular, in MWI it means that the physical object is not an abstract vector psi in the Hilbert space, but the wave function psi(x) in one particular basis x.

 

[JK423]

No. In classical physics, it would mean that positions x correspond to positions of some physical objects. In quantum physics, it depends on the interpretation. In particular, in MWI it means that the physical object is not an abstract vector psi in the Hilbert space, but the wave function psi(x) in one particular basis x.

Let the physical object be an abstract vector psi in the Hilbert space. Choose a basis {|x>} via which you express your laws (interactions) in their well known form. I think this solves any ambiguities. No preferred basis in the sense that you define it.
Schmindt changes to another basis {|y>} which is not a Lorentz transform of the first {|x>}, hence he messes up the interactions. What does that prove?
Like in my example with earth+sun, when i move to center of mass R and relative position r, the transformation is not a Lorentz one, hence the R and r do not represent the position of the physical objects in space.

 

[Fredrik]

I am quite confused.. You are right, Gleason's theorem applies to all those situation that Born's rule is relevant. I need to understand the differences between Gleason's and Zurek's work, so i cannot say anything else at the moment. Does Gleason's theorem assumes Born's rule for example?

Gleason only assumes that we're dealing with a Hilbert space that's at least 3-dimensional.

From a physicist's point of view, the reason we want a theorem like Gleason's is that subspaces of the Hilbert space can be thought of as mathematical representations of yes-no experiments. For example, let a,b be real numbers such that b>a, and suppose that we design a device that measures an observable A and returns the value "yes", if the result of the A-measurement is in the interval [a,b] and "no" otherwise. The system's state vector after the measurement will be in the subspace spanned by eigenvectors of A with eigenvalues in [a,b]. There's always a subspace associated with each yes-no experiment, so we need to be able to assign them probabilities in a way that makes sense. For example, the probability associated with the trivial subspace {0} must be 0, and the probability associated with the entire Hilbert space must be 1. Further, if E and F are orthogonal subspaces, then the probability associated with the smallest subspace that contains E and F must be the sum of the probability associated with E and the probability associated with F.

These rules are similar to, but not identical to, the rules that define a probability measure. The main difference is that the domain of a probability measure, as defined in measure theory, is a σ-algebra, and the set of subspaces of a Hilbert space isn't. However, it is a lattice, and σ-algebras can be thought of as a special kind of lattice. So what we need is a generalization of the term "probability measure" to lattices.

The generalization is straightforward. It involves a version of the rules I described above. With this definition in place, we can state the problem accurately: Find all probability measures on the lattice of subspaces of a Hilbert space. This is the problem Gleason solved. His answer goes like this:

For each probability measure P on the lattice L of subspaces, there's a unique state operator S such that $P(M)=Tr(P_M S)$ for all M in L, where $P_M$ is the projection operator associated with the subspace M. The map $P\mapsto S$ is a bijection from the set of probability measures onto the set of state operators.

In my opinion, it's far more natural to define a "state" as a probability measure than as a state operator, so to me, this theorem is the justification for why state operators can be thought of as states.

Now consider the special case of a pure state $S=|\psi\rangle\langle\psi|$ and let M be the eigenspace associated with some eigenvalue a of a self-adjoint operator A with non-degenerate spectrum (i.e. a 1-dimensional eigenspace for each eigenvalue). Then the formula above reduces to
$$P\left(|a\rangle \langle a|\right) =\operatorname{Tr} |\psi\rangle \langle\psi|a\rangle\langle a| =\sum_{a'} \langle a'|\psi\rangle \langle|\psi|a\rangle \langle a|a'\rangle =\left|\langle a|\psi\rangle\right|^2.$$ This right-hand side is of course the probability that the Born rule assigns to the result a when we're measuring A.

One final comment, when I say "subspace" in this post, I always mean a closed linear subspace. (A Hilbert space can have subsets that are inner product spaces but not Hilbert spaces, but we're not interested in those. By requiring that the subset isn't just a vector space, but also a closed set with respect to the norm topology, we ensure that the subset is also a Hilbert space).

 

[bhobba]

I am quite confused.. You are right, Gleason's theorem applies to all those situation that Born's rule is relevant. I need to understand the differences between Gleason's and Zurek's work, so i cannot say anything else at the moment. Does Gleason's theorem assumes Born's rule for example?

Gleason derives Born's rule with no assumption about it.

What it shows is given a resolution of the identity Ei, in a Hilbert space of dimension greater than 2, the only function f(Ei) 0<=f(Ei)<=1 such that f(Ei) sums to one is via the Born rule ie there exists a positive operator P of trace 1 such that f(Ei) = Tr(PEi).

You can find the details here:
http://kof.physto.se/theses/helena-master.pdf

The way probabilities enters into it is the interpretation of Ei as an observable whose outcome is one or zero so that the expected value of that observation is the probability of getting a one which is of course an f(Ei) so the theorem applies. Under this interpretation given any observable R = sum ri Ei, E(R) = sum ri Tr(PEi) = Tr(P sum ri Ei) = Tr (PR) and you have the Born rule.

If you have Balletines wonderful book on QM you see you have derived his second axiom from the first and the rest of QM follows from what he details in that book.

Note however there is no free lunch - there is an assumption being made in the proof - namely the f(Ei) does not depend on the elements of the resolution of the identity it is part of - this is called non contextuality. Mathematically it is a very clear assumption and necessary to make sense of the formalism. Physically however what it means is not quite so clear and you will find a fair amount of literature discussing it in particular the Kochen-Specker theorem - which follows fairly easily from Gleason - although the proof is usually presented in its own right.

Thanks
Bill

 

[bhobba]

In my opinion, it's far more natural to define a "state" as a probability measure than as a state operator, so to me, this theorem is the justification for why state operators can be thought of as states.

Definitely.

Thanks
Bill

 

[JK423]

Fredrik, very nice analysis, thanks. Peres' book is very descriptive on this issue as well, bhobba also thank you for that diploma thesis!
I still don't understand though why all the fuss with deriving Born's rule non-circularly, and what's the point of Zurek's work.