Can the Born Rule Be Derived Within the Everett Interpretation?

[DrChinese]

Yes, that's because you insisted that the optical fiber was wound up and that the two detectors were essentially in the same place. There is only a possible ambiguity in time ordering when the two events are spacelike separated. When two events are timelike connected (as I understood it was) then there is no ambiguity.

That is exactly what I was intending to specify, that all measurements are local and in the same frame - your closestuff1/2...

Thanks for clarifying that point.

In your opinion, is this application of branching exactly the same as how the Born rule would be applied in oQM?

 

[vanesch]

In your opinion, is this application of branching exactly the same as how the Born rule would be applied in oQM?

Hehe, you'll get different replies to this one

As experimentally, when we say that "QM is confirmed by experiment", we ALWAYS use the Born rule, then if that branching has to have the slightest bit of chance to survive it *better* behave exactly as how the Born rule would be applied of course.

But let us remember what are the two main problems with the Born rule in oQM: 1) we don't have a physical mechanism for it (all physical mechanisms are described by unitary operators which cannot lead to a projection)
2) the technique is bluntly non-local (even though the *results* are not signal-non-local even though Bell non-local).

So how is this branching *supposed* to work ? Well, there is something "irreversible" in the projection postulate of course, and that "irreversibility" is established by entanglement with the environment. This is not mathematically irreversible of course (it happens by a unitary operator, and that one is of course reversible), but is "irreversible FAPP". So this is what separates practically "for good" the different terms which have classically different outcomes (pointerstates).
The discussion that remains (witness the different contributions here from players in the field!) is about how probabilities emerge in that context. The "most natural" probability rule would of course be that if you "happen to be" in one of those branches, well, you could just be in *any* of them, so give them all the same probability. (that's my famous :-) APP)
Trouble is, one has to twist oneself in a lot of strange positions to get the Born rule out that way!
The other (probably less severe) problem is: how do we know that the resulting terms which are now irreversibly entangled, correspond to the classical worlds we would like to get out ? Decoherence gives a hint at a solution there.

Now, I would like to re-iterate my point of view on all these matters: they are a picture of *current* quantum theory, as we know it today. But clearly it doesn't make sense to talk about macroscopic superpositions of systems without taking into account the *gravitational* effect (because macroscopically different states will clearly have a slightly different mass-energy distribution, and as such, correspond to slightly different classical spacetimes, and as such to different derivatives wrt time (what time ? Of which term ? In what spacetime ?)). As we have, as of now, not yet one single clue of how quantum theory will get married with gravity (no matter the hype in certain circles), it is difficult to say whether the MWI picture will make sense once one has found the riddle.

 

[straycat]

Zurek's derivation of the Born rule

Hey all,

How many here are familiar with Zurek's derivation of the Born rule? (See, eg, [1].) I know Howard is, having written a paper [2] on it. I just this evening watched an online lecture [3] by Zurek about his derivation, and skimmed the other papers listed below. It appears to me that Zurek's work assumes Patrick's "alternate projection postulate" (= outcome counting [Weissman] = the "equal probability postulate" [me]). Cool! (If Zurek gives his version of the APP a name, I haven't encountered it yet.) Actually, Zurek does not *assume* the APP - rather, he attempts iiuc to *derive* it, based on an assumption termed "envariance." From envariance, Zurek gets (again, iiuc) the APP. And from there, Zurek gets the Born rule -- although I'm not sure how exactly. Does the Born rule emerge because Zurek assumes a Hilbert space formalism, so that Gleason's theorem can be plugged in? Not sure -- I still need to look at Zurek's papers more in depth.

Here's another question: does Zurek's derivation of the APP from envariance make sense? I tend to agree with Schlosshauer and Fine [4] that it does not, ie that the APP stands as an independent probability assumption: "We cannot derive probabilities from a theory that does not already contain some probabilistic concept; at some stage, we need to 'put probabilities into get probabilities out'." I think Patrick would see it the same way.

Patrick, I think you definitely need to talk about Zurek a lot in your revised paper. How's it comin', by the way?

David

(PS I owe thanks to Simon Yu and Andy Sessler at Lawrence Berkeley for getting me interested in Zurek.)

[1]
Probabilities from Entanglement, Born's Rule from Envariance
Authors: W. H. Zurek
http://xxx.lanl.gov/abs/quant-ph?papernum=0405161

[2]
No-signalling-based version of Zurek's derivation of quantum
probabilities: A note on "Environment-assisted invariance,
entanglement, and probabilities in quantum physics"
Authors: Howard Barnum
http://xxx.lanl.gov/abs/quant-ph?papernum=0312150

[3]
http://www.physics.berkeley.edu/colloquia%20archive/5-9-05.html

[4]
On Zurek's derivation of the Born rule
Authors: Maximilian Schlosshauer, Arthur Fine
http://xxx.lanl.gov/abs/quant-ph?papernum=0312058

 

[vanesch]

And from there, Zurek gets the Born rule -- although I'm not sure how exactly. Does the Born rule emerge because Zurek assumes a Hilbert space formalism, so that Gleason's theorem can be plugged in? Not sure -- I still need to look at Zurek's papers more in depth.

The trick resides, I think, above equation 7b. There, it is assumed that if we do a fine-grained measurement corresponding to the mutually exclusive outcomes sk1...skn, that we get probability n/N (this is correct) ; however, one CANNOT conclude from this, that if we were only to perform the coarse-grained measurement testing the EIGENSPACE corresponding to sk1,...skn, it would STILL have the same probability.

In the entire discussion above that point, it was ASSUMED that our observable was going to be an entirely exhaustive measurement (a different outcome for each different |sk>). But here (as did, in fact, Deutch do in a very similar way !), we are going to introduce the probabilities for measurements with an outcome PER EIGENSPACE assuming that it equal the sum of the measurements per individual eigenvector, and then SUMMING OVER THE probabilities per eigenvector to restore the outcome of the eigenspace. BUT THAT IS NOTHING ELSE BUT NON-CONTEXTUALITY. It is always the same trick (equation 9a).

The extra hypothesis is again, that we can construct an eigenspace of sufficient dimensionality which corresponds to the ONE outcome of the original eigenvector, so that we can make them all equal, and sum over the fine-grained probability outcomes (which are equal, through a symmetry argument), to obtain the original coarse-grained probability. But again, this assumption of the behaviour of probabilities is nothing else but the assumption of non-contextuality (and then, through Gleason, we already knew that we had the Born rule).

Zurek's derivation is here VERY VERY close to Deutsch's derivation. The language is different, but the statements are very close. In 7b and in 9a, he effectively eliminates the APP. As usual...

cheers,
Patrick.

 

[straycat]

The trick resides, I think, above equation 7b. There, it is assumed that if we do a fine-grained measurement corresponding to the mutually exclusive outcomes sk1...skn, that we get probability n/N (this is correct) ; however, one CANNOT conclude from this, that if we were only to perform the coarse-grained measurement testing the EIGENSPACE corresponding to sk1,...skn, it would STILL have the same probability.

This is a valid issue to raise. But my reading of the paper is that the "coarse-grained" measurement (yielding the value of k) should be reconceptualized as, in fact, being a "fine-grained" measurement (yielding the value of n, with n > k) in disguise.

Suppose the measurement is of a spin 1/2 particle, with premeasurement probabilities of k=up and k=down being 9/10 and 1/10, respectively. My reading of Zurek is that when we measure spin, we are doing more than measure the value of k; we are, in fact, measuring n, with n = 1, 2, 3, ..., 10; and we further assume that "n = 10" implies "k = down," and "n = anything between 1 and 9" implies "k = up." For this scheme to be compatible with the APP, we must assume that the spin measurement must give us the exact value of n. If the measurement gives only the binary result: "n = 10" versus "n somewhere between 1 and 9," then your criticism applies.

So does Zurek say somewhere that the measurement does not give us the exact value of n? I still am struggling through his paper, so it is possible that I've missed it if he did say such a thing. I would like to think that his scheme works the way I mentioned above, and hence evades your criticism, because that would mean that this part of Zurek's argument exactly matches the beginning of my own argument (up to Figure 1 B of my paper).

David

 

[straycat]

Zurek's derivation is here VERY VERY close to Deutsch's derivation. ...

OK, I have finally read the whole paper once through (excluding appendices). I note that Zurek agrees with us regarding Deutsch/Wallace decision theory -- ie, he thinks that it employs circular reasoning in the derivation of the Born rule:

"Reliance on the (classical) decision theory makes the arguments of [24] and [36] very much dependent on decoherence as Wallace often emphasizes. But as we have noted repeatedly, decoherence cannot be practiced without an independent prior derivation of Born's rule. Thus, Wallace's arguments (as well as similar 'operational aproach' of Saunders [52]) appears to be circular." (page 27, left column [arXived version])

Zurek states repeatedly in his paper that he has taken great care not to assume the Born rule in his derivation. So at the very least, he is aware of this danger!

David

 

[vanesch]

This is a valid issue to raise. But my reading of the paper is that the "coarse-grained" measurement (yielding the value of k) should be reconceptualized as, in fact, being a "fine-grained" measurement (yielding the value of n, with n > k) in disguise.
Suppose the measurement is of a spin 1/2 particle, with premeasurement probabilities of k=up and k=down being 9/10 and 1/10, respectively. My reading of Zurek is that when we measure spin, we are doing more than measure the value of k; we are, in fact, measuring n, with n = 1, 2, 3, ..., 10; and we further assume that "n = 10" implies "k = down," and "n = anything between 1 and 9" implies "k = up." For this scheme to be compatible with the APP, we must assume that the spin measurement must give us the exact value of n. If the measurement gives only the binary result: "n = 10" versus "n somewhere between 1 and 9," then your criticism applies.

The problem is that in his derivation of the probability of 9/10, he needs an extra space (which he can always find in the environment) with enough dimensional liberty to *imagine* that to the 9/10, he can use 9 dimensions, and for the remaining 1/10, he can have a 10th dimension, so that he can include this in an *imagined* finegrained measurement where all events are now equi-probable and have identical hilbert norms. As he argued before, from pure symmetry arguments, he can then derive that the probabilities of all of these outcomes are equal, and hence the probability of the "coarse grained event" is the sum of the respective probabilities of the fine-grained events. Now, admit that the way Zurek does it, is very artificial. There's no good reason why there should be exactly 9 extra dimensions, with equal lengths, in the environment corresponding to the "spin up" case, and 1 corresponding to the "spin down" case! He just gives this case, because then all fine-grained probabilities are equal because of a symmetry argument. But there's no reason why, in a real interaction, this should be the case, and it is certainly not argued that way. He only needs an artificial finegrained case which is exactly of the right composition so that his argument can work. Now, his argument works of course, because it is always *thinkable* that the fine-grained (but not too finegrained!) measurement works exactly that way on the environment ; meaning that we measure exactly SUCH an extra quantity of the environment that his scheme works. (If we measure too well the environment, it might not work - we may have too many or too few components for each term). So we can accept that SOME relatively finegrained measurement exists so that his scheme of things works out.

But this is implicitly assuming that the probability of the coarse grained event, when calculated from the probabilities of the fine-grained events, is the same probability as if we were going to perform only a coarsegrained measurement directly, without first fine-graining, and then not considering the information. As I tried to point out in my paper, *these are physically different measurements*. But it is very natural to assume that the two probabilities are equal. This is assuming that the probability of some coarse-grained event is NOT DEPENDING ON THE DEGREE OF EXTRA USELESS FINEGRAINING that is present in the measurement - and that is nothing else but postulating non-contextuality. Non-contextuality is exactly that: given the state and the eigenspace one wants to consider (the coarse-grained event), the probability can only depend upon the state and the eigenspace, and not upon the slicing up or not of that eigenspace and the complementary eigenspace. But that assumption is sufficient to derive Gleason's theorem!

Now, what's wrong with that ? Nothing of course, except that in order to be even able to _state_ that property of the probabilities that you would like to extract from the state and a set of eigenspaces, that you are going to HAVE TO STATE THAT PROBABILITIES EXIST IN THE FIRST PLACE. And if you state that, you already left the purely unitary part of QM. You already assumed that somehow, probabilities should emerge and have a certain property. So you are NOT deriving any probabilistic framework purely from the unitary machinery. Now, even Zurek himself seems to be aware of the non-triviality of the statement of additivity, because he addresses it (badly) in section V. I didn't see a convincing argument *without* invoking probabilities in section V.

I have to say that it is exactly in situations such as Zurek's paper that I think that my little paper is useful: take the APP, and see where it fails. THAT is the place where an extra (non-unitary QM) postulate has been sneaked in!

So does Zurek say somewhere that the measurement does not give us the exact value of n? I still am struggling through his paper, so it is possible that I've missed it if he did say such a thing.

He's making up the extra hilbert space of states in order to have equal-length components so that you can make orthogonal sums of them that come close to the hilbert norms of the original coefficients. He argues that in the big extra space of states of the environment, you will always find enough room to consider such an extra space. It is exactly the same scheme as is used by Deutsch to go from symmetrical states with equal probabilities to states with arbitrary coefficients.

 

[vanesch]

Zurek states repeatedly in his paper that he has taken great care not to assume the Born rule in his derivation. So at the very least, he is aware of this danger!

Well, he doesn't make that error, indeed. He makes the error of assuming non-contextuality, which he introduces by assuming the additivity of probabilities. He even seems to be aware of the danger (he refers to it, and a discussion in section V, which is however, deceiving).

From the moment you make ONE assumption about probabilities generated by states, apart from respecting the symmetries of the state, you're done!

 

[straycat]

Well, he doesn't make that error, indeed. He makes the error of assuming non-contextuality, which he introduces by assuming the additivity of probabilities.

Hmm. It seems to me that assuming additivity of probabilities is fine, if you assume that (to use my example above) the spin measurement is in fact the more fine-grained measurement of the exact value of n. I suppose n could be called a "hidden variable," and when we think we are only measuring spin, we are in fact measuring this hidden variable -- we just haven't been smart enough to figure it out yet!

I'll admit that he does not provide an explanation -- not that I see, at least -- for where n comes from, what it represents, what it means physically, what these "extra dimensions" are, why n turns out to be just the right amount of "fine-grained-ness" that we need to recover the Born rule, etc. (The reason I wrote my paper is to answer precisely these questions!) But that is a separate objection from the one you make. The way I see it, Zurek has taken a tiny baby step, and there are lots of questions (what is n and why does it have the properties Zurek postulates) that are left unanswered. But what's wrong with baby steps?

David

 

[straycat]

But this is implicitly assuming that the probability of the coarse grained event, when calculated from the probabilities of the fine-grained events, is the same probability as if we were going to perform only a coarsegrained measurement directly, without first fine-graining, and then not considering the information.

Where does Zurek make this assumption -- implicitly or otherwise?

 

[vanesch]

Where does Zurek make this assumption -- implicitly or otherwise?

He does it implicitly, in two places. He first does it when he introduces the states |C_k> in equation 8b, and his hilbert space HC of sufficient dimensionality in 9a. Clearly, he's now supposing a fine-grained measurement, where the c_j states are measured too, and from which are DERIVED afterwards the probabilities for the eventual coarse grained measurement. As such, he implicitly assumes that the the coarse grained measurement will give you the SAME probabilities as the sums of the probabilities of the fine grained measurement.

But he KNOWS that he's doing something fishy ! On p18, he writes (just under 1. Additivity...

In the axiomatic formulation ... as well as in the proof of the Born rule due to Gleason, additivity is an assumption motivated by the mathematical demand...

And he tries to weasel out with his Lemma 5 and his probabilities calculated from the state (27) "representing both fine-grained and coarse-grained records". However, he effectively only considers the probabilities of the fine-grained events.

Again, we will use our non non-contextual example to illustrate the flaw in his proof:

we consider |psi> = |x1>|y1> + |x1>|y2> + |x2>|y3>

As such, for the (finegrained) Y measurement, we have:
P_f(y1) = 1/3
P_f(y2) = 1/3
P_f(y3) = 1/3

and thus: P_f(x1) = 2/3 and P_f(x2) = 1/3

However, for the coarsegrained X measurement, we have:
P_c(x1) = 1/2
P_c(x2) = 1/2

AND IT MAKES NO SENSE TO TALK ABOUT THE PROBABILITY OF THE FINEGRAINED EVENTS. If I were to talk about the probabilities of y1, y2 and y3 for the probability measure P_c, I would get nonsense of course.

From the moment you ASSIGN a probability to the finegrained events, of course from the Kolmogorov axioms, additivity is implicitly incorporated.

Only, Zurek uses only ONE probability function, p(). As he is considering probabilities of fine-grained events in his subtraction procedure, the p() is the finegrained probability measure (P_f in my example). There of course, additivity is correct.

He's assuming that the probability function is the SAME ONE for fine grained and coarse grained measurements, and that is nothing else but the (rightly identified) extra assumption of Gleason of non-contextuality. But he's making the same assumption in his Lemma 5!

 

[vanesch]

Hmm. It seems to me that assuming additivity of probabilities is fine, if you assume that (to use my example above) the spin measurement is in fact the more fine-grained measurement of the exact value of n. I suppose n could be called a "hidden variable," and when we think we are only measuring spin, we are in fact measuring this hidden variable -- we just haven't been smart enough to figure it out yet!

Ok, but in that way, I can produce you ANY probability measure that is compatible with unitary dynamics: the APP, the Born rule, any other function that does the trick. If I'm allowed to say that the measurement of an observable O1 is in fact the measurement of the observable O1 x O2, where O2 works onto a yet to be specified Hilbert space with a yet to be established number of degrees of freedom and a yet to be established dynamics (interacting with O1) so that I get out the right number of "different" outcomes, I can provide you with just ANY probability rule.

But even there, you have a problem when I change something. Suppose that I start from a state u1|a> + u2|b> and I do a binary measurement (a versus b). Now, you claim that there is some physics that will evolve:

|a> (|x1>+|x2> +...|xn>) + |b> (|y1> + ... ym>)

such that n is proportional to u1^2 and m is proportional to u2^2, and that my "binary measurement" is in fact a measurement of the x1... ym states. Ok.

But suppose now that I'm measuring not u1|a> + u2|b>, but rather u2 |a> + u1 |b>. If we have the same unitary evolution of the measurement, I would now measure in fact the x1... ym states in the state:

u2/u1 |a> (|x1>+|x2> +...|xn>) + u1/u2 |b> (|y1> + ... ym>)

right ?

But using the APP, I would find probability |u1|^2 for |a> and |u2|^2 for |b> and not the opposite, no ?

Why would the dimensionality of the x1...xn depend on the coefficient u1 of |a> in the original state ? This cannot be achieved with a unitary
operator which is TRANSPARENT to the coefficient.

Isn't this a fundamental problem to assuming a certain dimensionality of hidden variables in order to restore the Born rule ?

 

[straycat]

Ok, but in that way, I can produce you ANY probability measure that is compatible with unitary dynamics: the APP, the Born rule, any other function that does the trick. If I'm allowed to say that the measurement of an observable O1 is in fact the measurement of the observable O1 x O2, where O2 works onto a yet to be specified Hilbert space with a yet to be established number of degrees of freedom and a yet to be established dynamics (interacting with O1) so that I get out the right number of "different" outcomes, I can provide you with just ANY probability rule.

Yes, you are correct: You could in fact take Zurek's basic idea and come up with any probability rule! This is what I mean by "baby steps." I do believe that Zurek has avoided the circularity trap. What he has not done afaict is to demonstrate why the Born rule, and not some other rule, must emerge. But that is progress, no?

So now we turn to your next argument:

Why would the dimensionality of the x1...xn depend on the coefficient u1 of |a> in the original state ?

I believe that some additional rule or set of rules is necessary to answer this question. And the sole motivation for postulating the "Born constraints" in my draft manuscript is to provide an "existence proof" that it is possible to accomplish this.

This cannot be achieved with a unitary
operator which is TRANSPARENT to the coefficient.
Isn't this a fundamental problem to assuming a certain dimensionality of hidden variables in order to restore the Born rule ?

I'm not sure I entirely follow your argument that this cannot be achieved. I have a feeling, though, that the answer has something to do with the fact that you need to consider, not only the state of the system under observation, but also the state of the measurement apparatus. To use your example above,

|a> (|x1>+|x2> +...|xn>) + |b> (|y1> + ... ym>)

suppose the "binary measurement" is a spin measurement along the x-axis. We could suppose that the number of dimensions of the fine-grained measurement has something to do with the interaction between the particle and the SG apparatus. IOW, if the SG apparatus is oriented to measure along the x-axis, then the relevant "number of dimensions" is n and m (following your notation above). But if we rotate the SG apparatus so that it measures along some other axis, then the relevant number of dimensions becomes n' and m'. Of course, it still is necessary to explain WHY this should work out just right, so that the Born rule emerges. But the point I wish to make is that there is no reason to ASSUME that this CANNOT be done! Unless I have missed some element of your argument, which is why I am enjoying this discussion ...

David

 

[straycat]

... we consider |psi> = |x1>|y1> + |x1>|y2> + |x2>|y3>
As such, for the (finegrained) Y measurement, we have:

P_f(y1) = 1/3
P_f(y2) = 1/3
P_f(y3) = 1/3

and thus: P_f(x1) = 2/3 and P_f(x2) = 1/3

However, for the coarsegrained X measurement, we have:

P_c(x1) = 1/2
P_c(x2) = 1/2

AND IT MAKES NO SENSE TO TALK ABOUT THE PROBABILITY OF THE FINEGRAINED EVENTS. If I were to talk about the probabilities of y1, y2 and y3 for the probability measure P_c, I would get nonsense of course.

You raise the issue: given a measurement of the above system, which should we use: P_f or P_c? How do we justify using one and not the other?

Following the spirit of Everett's original proposal, I believe that the number of branches (ie, the "number of dimensions") associated with a given measurment must be reflected in the number of distinct physical states that the observer can evolve into, as a result of the measurement process. So if the interaction of the observer with the environment results in evolution of the observer from one to 3 different possible states, then we have 1/3 probability associated with each state. If two of these observer-states are associated with x1, and the third is associated with x2, then we get:

P_c(x1) = 2/3
P_c(x2) = 1/3

So the above result, yielding probabilities 2/3 and 1/3, depends upon the assertion that there are two mutually exclusive distinct physical observer-states associated with x1, but only one observer-state associated with x2. A fully developed underlying theory must give an exact prescription for this number of observer states, as well as tell us which states are associated with which observable (x1 or x2).

My point is that the choice between P_f and P_c is not arbitrary, but should be uniquely determined by the underlying theory, which must (if it is going to work) describe the evolution of the physical state of the observer. This underlying theory has not been found yet, but I think it will be!

David

 

[vanesch]

Yes, you are correct: You could in fact take Zurek's basic idea and come up with any probability rule! This is what I mean by "baby steps." I do believe that Zurek has avoided the circularity trap. What he has not done afaict is to demonstrate why the Born rule, and not some other rule, must emerge. But that is progress, no?

Eh ? What progress ? That we can have any probability rule ?

So now we turn to your next argument:
I believe that some additional rule or set of rules is necessary to answer this question.

That's what I'm claiming all along! Now why can that extra rule not simply be: "use the Born rule" ?

And the sole motivation for postulating the "Born constraints" in my draft manuscript is to provide an "existence proof" that it is possible to accomplish this.

Ok... but...

To use your example above,
|a> (|x1>+|x2> +...|xn>) + |b> (|y1> + ... ym>)
suppose the "binary measurement" is a spin measurement along the x-axis. We could suppose that the number of dimensions of the fine-grained measurement has something to do with the interaction between the particle and the SG apparatus. IOW, if the SG apparatus is oriented to measure along the x-axis, then the relevant "number of dimensions" is n and m (following your notation above). But if we rotate the SG apparatus so that it measures along some other axis, then the relevant number of dimensions becomes n' and m'.

The point is that we're not going to rotate the apparatus, but simply the initial state of the system to be measured. As such, the apparatus and environment and whatever that is going to do the measurement is IDENTICAL in the two cases. So if you have an argument of why we need n extra finegrained outcomes for |a> and m extra finegrained outcomes for |b> is to hold for the first case, it should also hold for the second case, because *the only thing that is changed is the to-be-measured state of the system, not the apparatus.
Whatever may be your reason to expect the n extra finegrained steps in the case we have |a> and the m extra finegrained steps in the case we have |b>, this entire measurement procedure will be resumed into A UNITARY INTERACTION OPERATOR that will split the relevant observer state into the n + m distinct states. A unitary interaction operator being a linear operator, it SHIFTS THROUGH the coefficients.

Let us take this again: let us assume that there are n+m distinct observer states that can result from the measurement of the "binary" measurement, namely the |x1> ...|yn> states (which now include the observer states which are to be distinct, and to which you can apply the APP). Of course, in his great naivity, the observer will lump together his n "x" states, and call it "a", and lump together his m "y" states, and call it "x" (post-coarse graining using the Kolmogorov additivity of probabilities).

But the evolution operator of the measurement apparatus + observer + environment and everything that could eventually matter (and that you will use for your argument of WHY there ought to be n |x_i> states and so on) is not supposed to DEPEND upon the incoming state. It is supposed to ACT upon the incoming state. If it were to DEPEND on it, and then ACT on it, it would be a non-linear operation ! Let us call it U.

So U(u1 |a> + u2|b>) results in the known state
|a> (|x1>+|x2> +...|xn>) + |b> (|y1> + ... ym>)

This means that U (|a>) needs to result in 1/u1 |a> (|x1>+|x2> +...|xn>)
and U (|b>) needs to result in 1/u2 |b> (|y1> + ... ym>)

(I took a shortcut here. In order to really prove it, one should first consider what U is supposed to do on |a> only, then on |b> only, and then on u1|a> + u2|b>, with the extra hypothesis that U(|a>) will not contain a component of |b> and vice versa - IOW that we have an ideal measurement)

This means that U(u2|a> + u1|b>) will result in what I said it would result in, namely:
u2/u1|a> (|x1>+|x2> +...|xn>) + u1/u2|b> (|y1> + ... ym>)

and this simply by the linearity of the U operator, which in turn is supposed to depend only on the measurement apparatus + environment + observer and NOT on the to-be-measured state. As this measurement environment is supposed to be indentical for both states, I don't see how you're going to wiggle out in this way (because from the moment you make it depend upon the to-be-measured state, you kill the unitarity (and even linearity) of the time evolution!)

The way to wiggle out is of course to say that the split does not occur in the measurement setup, but in the incoming state already! However, then you will meet another problem: namely decoherence. If you claim that the initial state is in fact, when we think that it is u1 |a> + u2|b>, in fact:
|a> (|x1>+|x2> +...|xn>) + |b> (|y1> + ... ym>)

then it is going to be difficult to show how we are going to obtain interference if ever we are now measureing |c> = |a> + |b> and |d> = |a> - |b>. Try it:
You'll find out that you will have to assume |x1> = |y1> etc... to avoid the in product being zero (decoherence!).

Now: (a|b) is supposed to be equal to u1 u2, or to: (sqrt(n)xsqrt(m))/(m+n)
But the in product of the two terms with the finegraining is going like m/(m+n) (if m is smaller than n). I don't see how you're going to get the right in products in all cases (all values of n and m) in this approach, unless I'm missing something.

 

[straycat]

Eh ? What progress ? That we can have any probability rule ? ... Now why can that extra rule not simply be: "use the Born rule" ?

Well that's all fine and good ... IF YOU'RE FEELING COMPLACENT

Here's the idea: start with the APP, and justify it via an argument-by-symmetry (or envariance). Then, to recover the Born rule, add some additional rule or set of rules so that the Born rule and not some other rule emerges. Then, try to derive these newly postulated rules from GR. (Or if not GR, then something like it, maybe some alternate field theory.) iow, "work backwards" in baby steps from the APP, to some other set of rules, back to whatever the underlying theory is. Then clean up all the intermediate steps to make sure they're rigorous, etc etc. If not, start all over again.

Surely you can see that this would be a big payoff? :!)

Robin, Mike, and I have each (independently) proposed just such an alternate set of "extra rules" from which the Born rule is argued to emerge. Someday someone's going to hit paydirt!

I'll address the issue of unitarity in my next post.

David

 

[straycat]

unitarity, linearity

Regarding the issue of unitarity: There are two possibilities: 1) the underlying theory is unitary; or, 2) the underlying theory is NOT unitary; and QM is unitary because it is an approximation of the underlying theory.

wrt my own scheme, I'm not entirely sure, although I'm leaning toward the latter case. Mike has argued that the underlying theory should be nonlinear. (which of course means not unitary.) In my own scheme, the fundamental entity is the state of the observer, which is specified in full via providing the metric g_{ij} and its power series at a point in spacetime. Since the power series has an infinite number of components, observer state space in my scheme is infinite dimensional (as might be expected/hoped). The operator that gives us the time-dependent evolution of the state of the observer is calculated by the parallel-transport law, ie by the geodesic equations:

<br /> \frac{d^{2}x^{k}}{dt^{2}} + <br /> \Gamma^{k}_{ij}<br /> \frac{dx^{i}}{dt}<br /> \frac{dx^{j}}{dt} = 0<br />

Assuming that spacetime is multiply connected, there will be more than one geodesic equation that is a solution to the above equations. Each separate geodesic is interpreted as an alternate possible evolution of the observer; ie, each separate geodesic exists in "superposition."

My point is that the above equation is nonlinear, so I'm thinking that my underlying scheme is fundamentally nonlinear, as Mike would argue it should be. The Born rule emerges in my scheme only if you make a particular approximation, in which a big chunk of the possible evolutions are approximated as being not there. (This is justified because this particular chunk has a very low probability, via the APP.)

But I'd like to understand your objection better -- it should help me understand why I should be so happy that my scheme is nonlinear.

The point is that we're not going to rotate the apparatus, but simply the initial state of the system to be measured. As such, the apparatus and environment and whatever that is going to do the measurement is IDENTICAL in the two cases.

I don't think I follow you here. If you are going to "rotate the initial state of the system to be measured," then there are several ways to do this, conceptually:

1) replace the initial system with a DIFFERENT system in a DIFFERENT (rotated) state; or

2) rotate the system (the particle), ie, PHYSICALLY; or

3) keep the particle in place, but rotate the SG apparatus, ie, by PHYSICALLY actually rotating the magnets.

However you do it, you are making a physical change of something: either the particle, or the apparatus. So you're not looking at the same overall (system+apparatus) state; you're looking at a DIFFERENT state. So why should you expect to go from one to the other via a unitary operator? I may just being dunce here, so keep hammering away ...

David

 

[vanesch]

Well that's all fine and good ... IF YOU'RE FEELING COMPLACENT
Here's the idea: start with the APP, and justify it via an argument-by-symmetry (or envariance).

Just as a note: in the case of a symmetry argument, such as envariance, ALL probability rules have to give equal probabilities. This is, however, the ONLY case where we can deduce something about a hypothetical probability rule without postulating something about it.

Then, to recover the Born rule, add some additional rule or set of rules so that the Born rule and not some other rule emerges. Then, try to derive these newly postulated rules from GR. (Or if not GR, then something like it, maybe some alternate field theory.) iow, "work backwards" in baby steps from the APP, to some other set of rules, back to whatever the underlying theory is. Then clean up all the intermediate steps to make sure they're rigorous, etc etc. If not, start all over again.
Surely you can see that this would be a big payoff? :!)

I'm not sure it can even work in principle. Of course, for SOME situations, one can derive the Born rule in this (artificial) way, but I think that you cannot build it up as a general way ; as I tried to show with the in product example (although I just typed it like that ; I might have messed up, I agree that it is for the moment still "just intuition"). And if it succeeds, you need to postulate A LOT of unexplained physics!

Robin, Mike, and I have each (independently) proposed just such an alternate set of "extra rules" from which the Born rule is argued to emerge. Someday someone's going to hit paydirt!
I'll address the issue of unitarity in my next post.
David

I have all respect for the different attempts. As I think I proved, you do not need much as an extra hypothesis to derive the Born rule. I think that non-contextuality is a fair hypothesis ; I find the additivity of probabilities also a fair hypothesis (these are in fact very very close!). But they ALL need you to already say that SOME probability rule should emerge, and then you find WHICH ONE can emerge, satisfying the desired property.
So it is logically equivalent to say: "the probabilities that should emerge must follow the Born rule", and "the probabilities that should emerge must do so in a non-contextual (or additive) way". These are logically equivalent statements.

I find this fine. I don't see why one needs to go through this "equal probability stuff" (which is NOT non-contextual!) by postulating extra physics (unknown degrees of freedom) so that we miraculously obtain the Born rule, just to avoid to say that, well, the probability to be in a certain state is given by the Born rule, and not by equal probability. Why should the probabilities of the different worlds be equal for you to be in ?

 

[straycat]

And if it succeeds, you need to postulate A LOT of unexplained physics!

True ... but that is (hopefully) only a temporary state of affairs. The ultimate goal is that the "lot of unexplained physics" can be replaced by a very simple physical postulate -- like Einstein's equation, something like that.

But they ALL need you to already say that SOME probability rule should emerge, and then you find WHICH ONE can emerge, satisfying the desired property.

So it is logically equivalent to say: "the probabilities that should emerge must follow the Born rule", and "the probabilities that should emerge must do so in a non-contextual (or additive) way". These are logically equivalent statements.

Again, the ultimate goal is that the assumption: "the probabilities that should emerge must follow the Born rule" is only a provisional one that is characteristic of a theory-in-progress. Once the "extra physics" mentioned above gets figured out, then the Born rule becomes derived, so need not be postulated.

I find this fine. I don't see why one needs to go through this "equal probability stuff" (which is NOT non-contextual!) by postulating extra physics (unknown degrees of freedom) so that we miraculously obtain the Born rule, just to avoid to say that, well, the probability to be in a certain state is given by the Born rule, and not by equal probability. Why should the probabilities of the different worlds be equal for you to be in ?

Well you know the arguments at least as well as I do! We want the final finished product to avoid the charge of non-contextuality, to rely fundamentally on the APP. And we want to say that the Born rule, ie quantum mechanics, emerges from some underlying field theory, like GR -- and the ONLY probabililty assumption that we need to plug in is the APP, which is justified by a symmetry argument. The Born rule then gets "demoted," in a manner of thinking, because the APP is more fundamental to it.

I agree this is a difficult endeavor to undertake. But let's just suppose that it will work.How hard could it be? There are currently, what, hundreds of really smart mathematicians working on quantum gravity (string, loop, etc etc) -- and NONE of them afaik have incorporated the beautiful symmetry of the APP. Instead, they all ASSUME the (not symmetric, therefore ugly) Born rule to be the FUNDAMENTAL probability rule. (Smolin of course argues that string theory, in addition, makes the non-symmetric, therefore ugly, assumption of background dependence ... an argument to which I am sympathetic, although that is a whole differenet can of worms!) Perhaps the reason that all of these brilliant folks have not succeeded in figuring out quantum gravity is that their programmes are all tainted by one or both of the aforementioned assumptions. Surely the APP (like background independence) is a worthy symmetry principle into which some effort should be invested.

Don't forget that seemingly innocuous and simple symmetry principles have a long history of unexpectedly big payoffs. Einstein's principle of relativity being one such example!

David

 

[hbarnum]

don't need no reeducation pleeez!

And in which context do you have a problem with it Howard? (...we are watching you... )

Hi Tez! Logic, me boy, logic... there's no logical implication that I have a problem with it in any context... really I don't, really... I didn't mean to imply I did... Please don't think I did! I don't need to be reeducated, really I don't!


Actually, I just meant to imply I'm not committed to probability *always* meaning subjective probability... in mathematics, it just refers to some things satisfying some formal axiomatic structure (s) ... so there, one is not committed to any view about its use in the world, or even whether it has any use... as far as probability in science, though, I'm a straight subjectivist a la Savage, even though I didn't bother to argue with the post about "unobservable preference orderings" vs. "measurable experimental outcomes"... yet.

Cheers,

H

 

[vanesch]

Surely the APP (like background independence) is a worthy symmetry principle into which some effort should be invested.

I fail to see what's so "symmetrical" about it:

You have the state a|u> + b|v>. If a and b are not equal, there is no symmetry between u and v, so I do not see why |u> should be equally probable to |v>. The "APP" arizes in those cases where a symmetry operator can swap the states ; but that's not the APP, it is *in the case of a symmetric state* that ALL probability rules need to assign equal probabilities to that particular case.
But I don't see why it is forbidden to say that the probability to observe |u> is not allowed to be a function of its hilbert coefficient ?

 

[straycat]

I fail to see what's so "symmetrical" about it:
You have the state a|u> + b|v>. If a and b are not equal, there is no symmetry between u and v, so I do not see why |u> should be equally probable to |v>. The "APP" arizes in those cases where a symmetry operator can swap the states ; but that's not the APP, it is *in the case of a symmetric state* that ALL probability rules need to assign equal probabilities to that particular case.

Well in the above, you have assumed standard QM, including the Born rule and the Hilbert space formalism. So of course you are correct that there is no symmetry. That's the point: the Born rule asserts asymmetry (except in special cases) wrt probabilities. The APP, otoh, is symmetrical wrt probabilities.

But I don't see why it is forbidden to say that the probability to observe |u> is not allowed to be a function of its hilbert coefficient ?

I do not mean to imply that we are forced to accept the APP over the Born rule. ie, a symmetry argument is insufficient to forbid any non-symmetric alternative.

Take Einstein's principle of relativity. This is a symmetry principle, stating that the laws of physics are the same in all frames of reference. Does symmetry in and of itself mean we are "forbidden" to postulate frame-dependent laws? Well, no. (I could replace GR with Lorentzian relativity, for example.) All it means is that if I am trying to come up with new physics, and I have the choice between a frame-dependent and a frame-independent postulate, and I haven't yet worked out all the math so I don't know yet which one will work out, then as a betting man, I would put my money on the frame-independent one, all other considerations being equal (no pun intended ).

David

 

[straycat]

You have the state a|u> + b|v>.

There is another hidden assumption you are making here: that the number of "branches" associated with at measurement is in one-to-one correspondence with the different possible states of the observed system. In keeping with the spirit of Everett, the former should be equated with the number of physically distinct states into which the observer may evolve as a result of a measurement. The latter is equated with the number of physically distinct state of the observed system. In the standard treatment of the MWI, these two numbers are assumed to be the same. But why? It is conceivable that these may be different. So we have yet another independent postulate that is implicit to the standard MWI.

David

 

[vanesch]

Take Einstein's principle of relativity. This is a symmetry principle, stating that the laws of physics are the same in all frames of reference. Does symmetry in and of itself mean we are "forbidden" to postulate frame-dependent laws? Well, no.

Eh, yes ! That's exactly the content of the principle of relativity: we are forbidden to postulate frame-dependent laws!

 

[vanesch]

Well in the above, you have assumed standard QM, including the Born rule and the Hilbert space formalism. So of course you are correct that there is no symmetry. That's the point: the Born rule asserts asymmetry (except in special cases) wrt probabilities. The APP, otoh, is symmetrical wrt probabilities.

No, I didn't assume the Born rule. I just got out of unitary QM (on which we agree) that the state of the entire system (including myself) is a |u> + b|v>. One cannot say that the state is invariant under a swap of |u> and |v> which would be the statement of symmetry. In this case, there is no symmetry between the |u> and the |v> state, and I simply claimed that I don't see how a "symmetry" principle can assign now equal probabilities to |u> and to |v>. We can do so, by postulate, but it doesn't follow from any symmetry consideration. Given that doing so attracts us a lot of trouble, why do we do so ? And given that by assuming the Born rule we don't have that trouble, why not do so ?

I know of course where this desire comes from, and it is the plague of most MWI versions: one DOESN'T WANT TO SAY ANYTHING ABOUT PROBABILITIES. As such one seems to be locked up in the situation where we have a lot of independent terms (worlds) and we have to explain somehow how it comes that we only perceive ONE of them, given that we have "some state" in each of them. It seems indeed the most innocent to say that all these independent worlds are 'equally probable to be in', but that is nevertheless a statement about probability. Because without such a statement, we should be aware of ALL of our states, not just of one. So YOU CANNOT AVOID postulating somewhere a probability. My point is: pick the one that works ! Because there is no symmetry between those different worlds, we are NOT OBLIGED to say that they are equally probable. There is no physical symmetry between the low-hilbert norm states and the high hilbert norm states, in the sense that the state is not an eigenstate of any swap operator. But it is indeed tempting to state that all worlds are "equally probable" because that's how we've been raised into probabilities. We've been raised into counting the number of possible outcomes, and then counting the number of "successes", and taking the ratio. This is usually because we had genuinly symmetrical situations (like a dice) and in this case of course a symmetry argument implies that the probabilities are to be equal. So we do this also to the different terms in the wavefunction (although, I repeat, there is no a priori reason to take THIS distribution over another one, given that there is no symmetry in the situation). And so one has a *lot* of terms with small hilbert norm, and relatively few terms with high hilbert norm, and we can only find correspondence between the (observed) Born rule behaviour by "giving more weight" (1) to the few terms with high hilbert norm, or by "eliminating" (2) the vast lot of terms with small hilbert norm. Hence a lot of brain activity to find a mechanism to do so. You do (1) by trying to find extra physical degrees of freedom which we ignore, but split the few high-hilbert norm terms into a lot of small ones, so that their population is finally much larger than the initial small-hilbert norm states. Hanson does (2) by taking that worlds with very small hilbert norm are "mangled" (continue to suffer interactions with big terms), so that they somehow "don't count" in the bookkeeping.

And at the end of the day, you want to find the Born rule. Why not say so from the start ?

 

[straycat]

Eh, yes ! That's exactly the content of the principle of relativity: we are forbidden to postulate frame-dependent laws!

I think you and I are merely speaking past each other a bit here. What I mean to say is that there is nothing that forbids us to say: "here is a symmetry principle, and despite its beauty, it's wrong." So, if we assume the principle of relativity is right, then we are forbidden from violating it; but we could, alternatively (and hypothetically), recognize the principle of relativity as being beautiful, but then turn around and not assume it. (Some people actually do this! ie proponents of Lorentzian relativity.)

My whole purpose in bringing up the principle of relativity is to compare it to the APP. Both are symmetry principles. With GR, of course, we assume the principle of relativity to be true. The question still remains, however, whether we should likewise adopt the APP as being true. I think that if the APP enables a succint derivation of quantum laws from deeper principles, in a manner that minimizes the total number of assumptions that must enter into the overall scheme, then the answer becomes "yes."

David

 

[straycat]

And at the end of the day, you want to find the Born rule. Why not say so from the start ?

I think this whole discussion boils down to the fact that we see the APP differently. To you, the APP is no "better" than the Born rule, in the sense that we still need to postulate it. I can respect this PoV. However, there is a part of me that feels that the APP is sufficiently "natural" that it does not require an independent postulate.

I realize I sort of equivocate on this issue. Perhaps I should be more forceful in saying that the APP must be true. According to this view, the assumption of the Born rule in the standard formalism is a "band-aid" that we use because it "works," but which is ultimately unsatisfactory. Hence the need to replace it with the APP and some more physics.

David

 

[straycat]

No ... I just got out of unitary QM (on which we agree) ...

Another instance of me equivocating ... I'm not sure I agree that the underlying theory must be characterized by unitary (= linear) evolution! See a few messages back.

 

[vanesch]

So, if we assume the principle of relativity is right, then we are forbidden from violating it; but we could, alternatively (and hypothetically), recognize the principle of relativity as being beautiful, but then turn around and not assume it.

Ah, ok! Yes, a wrong principle, no matter how beautiful, must not be adhered to

Both are symmetry principles. With GR, of course, we assume the principle of relativity to be true. The question still remains, however, whether we should likewise adopt the APP as being true. I think that if the APP enables a succint derivation of quantum laws from deeper principles, in a manner that minimizes the total number of assumptions that must enter into the overall scheme, then the answer becomes "yes."

I agree with that, IF you have a (totally different!) theory, in which for some or other reason, there is a symmetry between the states, then you are right. But I fail to see how *the APP* is a "symmetry principle" in quantum theory. A symmetry principle should apply to the mathematical structure that is supposed to describe nature ; that is: it is an operator acting upon whatever set is supposed to be the set of possible states of nature (in QM, it is the rays of hilbert space ; in classical physics, it is the phase space, in GR, it is the set of all 4-dimensional manifolds that respect certain properties...), and that transforms it into the same state, or a state which has identical meaning (in the case of redundancy in the state space, like is the case with gauge symmetries).

I fail to see what operator can correspond to something which makes all states equivalent in a random state. The *envariance* symmetry of Zurek, on the other hand, IS a genuine symmetry (which is based upon the arbitrary phases in the tensor product of two subsystems).
Probabilities (just as any other observable phenomenon) that are to be derived from a state respecting a certain symmetry, should also obey the symmetry that is implemented. As such, I can understand that the probabilities, in the case of equal hilbert coefficients, must be equal (no matter what rule; a rule that does not obey it will run into troubles).

Of course, if we would now have some "APP quantum theory" in which we ALWAYS have, in each relevant basis, the same hilbert coefficients, then of course you are right. If you trim the hilbert space by a superselection rule that only allows for states in which the components are all equally long, so that all allowed-for states are "swappable", then of course ANY probability rule will be equivalent to the APP, and so will the Born rule. But I wonder how you're going to implement this ! This is very far from the original idea of quantum theory, and its superposition principle.

 

[straycat]

Of course, if we would now have some "APP quantum theory" in which we ALWAYS have, in each relevant basis, the same hilbert coefficients, then of course you are right. If you trim the hilbert space by a superselection rule that only allows for states in which the components are all equally long, so that all allowed-for states are "swappable", ...

To be honest I do not fully understand how Zurek can define "swappability" without letting some piece of QM -- and hence the Born rule! -- "sneak" in. iiuc, two states are swappable iff they have the same Hilbert space coefficients. Why couldn't we assume that they are swappable even if they have different coefficients? Because that would mean that they have different physical properties. So at the very least, Zurek is assuming that states must be elements of a Hilbert space, and that the Hilbert space coefficient is some sort of property characteristic of that state. Well if we are going to assume all that, we may as well just plug in Gleason's theorem, right? Or am I missing something?

But I wonder how you're going to implement this !

Well you'll just have to read my paper! Which, btw, I am finally biting the bullet and submitting, today!

David