Can the Born Rule Be Derived Within the Everett Interpretation?

[vanesch]

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.

Well, Zurek accepts (of course) entirely the unitary part of QM, unaltered, and without "extra degrees of freedom". He introduces a unitary symmetry operator which "turns" randomly the phases of the basis vectors of system 1, and turns in opposite ways the phases of the basis vectors of system 2, and calls this an envariance symmetry operator. He argues that we can never measure the *phases* of the different basis vectors of system 1 (this comes from the redundancy in state description, namely the fact that a physical state corresponds to a RAY and not an element in hilbert space) or of system 2, and that, as such, his symmetry operator does not affect the physical state. He then goes on to enlarge the envariance symmetry operators, in which he swaps at the same time two states in the two hilbert spaces 1 and 2 (so that the overall effect of the Schmidt decomposition is simply to swap the hilbert coefficients), and notices that in the case of EQUAL COEFFICIENTS, this is a symmetry of the state.
He then introduces some assumptions (in that a unitary transformation of system 2 should not affect outcomes of system 1, including probabilities) and from some considerations arrives at showing that in such a case, all probabilities should be equal FOR THIS SPECIFIC STATE.

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.

Yes, in other words, he's accepting unitary quantum theory.

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?

In order to derive Gleason's theorem, you have to make an extra assumption related to probabilities, which is the non-contextuality ; in other words, to assume that the probability of an outcome ONLY depends upon the properties of the component of the state within the compatible eigenspace corresponding to the desired outcome, and NOT on other properties of the state or the observable, such as THE NUMBER of different eigenspaces, and the components in the OTHER eigenspaces (of other, potential, outcomes). (the other possible outcomes, and their relation to the state, are the *context* of the measurement). As you know, the APP NEEDS this information: it needs to know HOW MANY OTHER EIGENSPACES have a non-zero component of the state in them. The Born rule doesn't: the length of the component in the relevant eigenspace is sufficient. And Gleason proves that the Born rule is THE ONLY rule which satisfies this property.

Zurek does something else, which is assuming additivity of the probabilities of the "fine-grained" state components and then uses the specific case where there are exactly a sufficient number of fine-grained state components to arrive at the requested Born rule. The request that, for this specific fine-grained situation, the probability of the component in the relevant (coarse-grained) eigenspace is given ONLY by the sum of "component probabilities" within this eigenspace, is yet another form of requiring non-contextuality: namely that the probability is entirely determined by the component in the relevant eigenspace (by taking the sum), and NOT by the context, which is how the rest is sliced up, and how the components are distributed over the rest. So in an involved way, he also requires non-contextuality. And then, by Gleason, you find the Born rule.

 

[straycat]

In order to derive Gleason's theorem, you have to make an extra assumption related to probabilities, which is the non-contextuality ; in other words, to assume that the probability of an outcome ONLY depends upon the properties of the component of the state within the compatible eigenspace corresponding to the desired outcome, and NOT on other properties of the state or the observable, such as THE NUMBER of different eigenspaces, and the components in the OTHER eigenspaces (of other, potential, outcomes). (the other possible outcomes, and their relation to the state, are the *context* of the measurement). As you know, the APP NEEDS this information: it needs to know HOW MANY OTHER EIGENSPACES have a non-zero component of the state in them. The Born rule doesn't: the length of the component in the relevant eigenspace is sufficient.

True, the length of the component is not the same thing as the number of other eigenspaces. However, the length is a normalized length, right? and the normalization process injects, does it not, information regarding the other eigenspaces? ie, instead of counting the total number of eigenspaces, we are adding up the "measure" of all the eigenspaces when we normalize.

David

 

[vanesch]

True, the length of the component is not the same thing as the number of other eigenspaces. However, the length is a normalized length, right? and the normalization process injects, does it not, information regarding the other eigenspaces? ie, instead of counting the total number of eigenspaces, we are adding up the "measure" of all the eigenspaces when we normalize.
David

Eh, yes. But the eigenspace, and its complement, are of course representing "the same" outcome (A or NOT A). So, ok, if you want to allow for non-normalized states, you need to allow for the eigenspace and its complement - I was assuming that we could normalize the states (and so does Gleason). We all know that the same physical state is represented by a ray in Hilbert space, so a common coefficient has no meaning and may just as well be normalized out. In fact, if the initial state is normalized, unitary time evolution will preserve this normalization. What counts is that the way that the complementary eigenspace is eventually sliced up or not, should not influence the outcome A to have non-contextuality.

 

[straycat]

Eh, yes. But the eigenspace, and its complement, are of course representing "the same" outcome (A or NOT A). So, ok, if you want to allow for non-normalized states, you need to allow for the eigenspace and its complement - I was assuming that we could normalize the states (and so does Gleason). We all know that the same physical state is represented by a ray in Hilbert space, so a common coefficient has no meaning and may just as well be normalized out. In fact, if the initial state is normalized, unitary time evolution will preserve this normalization.

Although in the case of successive measurements (say A -> B -> C), you have to renormalize with each measurement result. So if we want to assert that normalization happens only once, at the beginning, then we are restricting ourselves to a "no-collapse" framework. Let's suppose we want to know the conditional probability: what is the probability of outcome b', given outcome a' ? To answer this, we need to "collapse" onto outcome a', which means we have to recalculate the wavefunction, which includes a renormalization procedure. So ok, we could calculate the probability of b' using only unitary evolution (allowing A to remain superpositioned), but NOT if we want CONDITIONAL probabilities based on the outcome of A.

 

[vanesch]

Although in the case of successive measurements (say A -> B -> C), you have to renormalize with each measurement result. So if we want to assert that normalization happens only once, at the beginning, then we are restricting ourselves to a "no-collapse" framework. Let's suppose we want to know the conditional probability: what is the probability of outcome b', given outcome a' ? To answer this, we need to "collapse" onto outcome a', which means we have to recalculate the wavefunction, which includes a renormalization procedure. So ok, we could calculate the probability of b' using only unitary evolution (allowing A to remain superpositioned), but NOT if we want CONDITIONAL probabilities based on the outcome of A.

Bzzzzt ! The conditional probability P(a|b) is completely known if we know P(a and b) and P(b), because it is equal (by definition) to P(a and b)/P(b).

Now, imagine that the initial state is |psi0>, and that this first evolves into u1|b> + u2|c>, where of course |u1|^2 + |u2|^2 = 1 if psi0 was normalized. This means that we had probability |u1|^2 to observe |b> (so P(b) will be equal to |u1|^2).
Now, imagine that this further evolves into:
u1 (v1|a> + v2 |d>) |b> + u2 |c'>

Clearly, |v1|^2 + |v2|^2 = 1 too if the entire state is to stay normalized, which it will, through unitary evolution. If, after having observed |b>, we now observe |a>, we are in fact in the branch |a>|b>, which has hilbert norm u1 v1 and thus probability |u1|^2 |v1|^2. This is the probability to observe a and b, so P(a and b) = |u1|^2 |v1|^2

Applying our definition of conditional probability, we see that P(a|b) = |v1|^2, and we didn't have to re-normalize the state.

 

[straycat]

Bzzzzt ! The conditional probability P(a|b) is completely known if we know P(a and b) and P(b), because it is equal (by definition) to P(a and b)/P(b).

Now, imagine that the initial state is |psi0>, and that this first evolves into u1|b> + u2|c>, where of course |u1|^2 + |u2|^2 = 1 if psi0 was normalized. This means that we had probability |u1|^2 to observe |b> (so P(b) will be equal to |u1|^2).
Now, imagine that this further evolves into:
u1 (v1|a> + v2 |d>) |b> + u2 |c'>

Clearly, |v1|^2 + |v2|^2 = 1 too if the entire state is to stay normalized, which it will, through unitary evolution. If, after having observed |b>, we now observe |a>, we are in fact in the branch |a>|b>, which has hilbert norm u1 v1 and thus probability |u1|^2 |v1|^2. This is the probability to observe a and b, so P(a and b) = |u1|^2 |v1|^2

Applying our definition of conditional probability, we see that P(a|b) = |v1|^2, and we didn't have to re-normalize the state.

Aack! I think you are correct; my "conditional probability" critique was wrong. You caught me napping :zzz: .

There is still something I do not understand. Suppose we are measuring the spin state of a particle. If it is a spin 1/2 particle, then there are 2 states; spin 1, 3 states; etc. So when we apply the Schrodinger equation to the intial state |psi0>, it evolves into

u1|b> + u2|c> if the particle is spin 1/2, or

u1|b> + u2|c> + u3|e> if it is spin 1, etc.

So my question: how does the Schrodinger equation "know" how many states are possible? Is it part and parcel of our original definition of |psi0> ? Or does it somehow emerge from the Schrodinger equation itself, without our having to inject it externally?

 

[vanesch]

So my question: how does the Schrodinger equation "know" how many states are possible? Is it part and parcel of our original definition of |psi0> ?

?? I'd say it is part of the saying that it is a spin-1 particle in the first place. If it goes into 3 parts, we call it a spin-1 particle !

Sounds like: "how does a green car know it has to reflect green light ?" or something... unless I miss what you want to say.

 

[mbweissman]

agreed

BUT THAT IS NOTHING ELSE BUT NON-CONTEXTUALITY. It is always the same trick (equation 9a).cheers,
Patrick.

Yes, I agree completely. Deutsch, Wallace, Zurek etc do fine through the point of showing that equal-measure outcomes have equal probabilities. The next step involves assuming that probability is fixed after an experiment, and independent of observer. These are exactly the features which one does not find in APP. It's nice to show that Born can be derived from slightly weaker assumptions, but that doesn't mean that those assumptions follow from unitary QM. Worse, it doesn't mean that those assumptions are even consistent with unitary QM and our operational definition of probability.

 

[mbweissman]

Zurek does something else, which is assuming additivity of the probabilities of the "fine-grained" state components and then uses the specific case where there are exactly a sufficient number of fine-grained state components to arrive at the requested Born rule. The request that, for this specific fine-grained situation, the probability of the component in the relevant (coarse-grained) eigenspace is given ONLY by the sum of "component probabilities" within this eigenspace, is yet another form of requiring non-contextuality: namely that the probability is entirely determined by the component in the relevant eigenspace (by taking the sum), and NOT by the context, which is how the rest is sliced up, and how the components are distributed over the rest. So in an involved way, he also requires non-contextuality. And then, by Gleason, you find the Born rule.

Yeah- I thought you might be interested in these excerpts from a I comment wrote on the zurek argument. (I tried a bit to get it published there.) The point is that our arguments are almost identical, which is reassuring.


...Here I argue Zurek makes an implicit assumption which runs counter to the explicit assumptions.
...
The difficulty arises in extending the argument to decoherent outcomes whose measures are not equal, i.e. to density matrices whose non-zero diagonals are not equal. Here Zurek introduces a second environment, called C, and proposes that it is possible for E to become entangled with C in such a way that the density matrix for SC traced over E can (almost) be expressed by a collection of equal diagonal terms, with each diagonal term in the density matrix of S expanding into a finite set of equal diagonal terms in the density matrix of SC.. Now applying the swapping-symmetry argument to SC, Zurek gets that these SC outcomes must have equal probabilities.
Since the particular C-E entanglement required will occur on a set of measure zero of starting states, and cannot even approximately arise for the general case by any physical process represented by linear time evolution, the argument is not that such processes will occur but rather that they might sometimes occur, and the probabilities obtained in those special cases must be the same as those obtained in all cases because the density matrix for S is unaffected by the C-E* entanglement .
Treating the probabilities of S outcomes as sums over (more detailed) SC outcomes then gives the Born rule. This step, however, does not amount to simply using additivity of probabilities within a single probability space but rather implicitly assumes that the probabilities defined on S are simply related to the probabilities defined on SC. No matter how much that step accords with our experience-based common sense, it does not follow from the stated assumptions, which are deeply based on the idea that probabilities cannot be defined in general but only on a given system. Thus the question of why quantum probabilities take on Born values, or more generally of why they seem independent of where a line is drawn between system and environment, is not answered by Zurek's argument.
A counterexample may reinforce this point. A simple probability, defined from numbers* of diagonal terms in the density matrix of the system without weighting by measure, is entirely "envariant" and obeys the swapping symmetry. It does not obey any simple general relation between the probabilities defined on a system and on some supersystem. This probability is, of course, none other than 'world counting', which has frequently been argued to be the obvious probability to arise in collapse-free pictures without dynamical or metaphysical addenda. 2-8
Thus the problem of showing why the probabilities we experience emerge from quantum mechanics remains. ...

 

[straycat]

?? I'd say it is part of the saying that it is a spin-1 particle in the first place. If it goes into 3 parts, we call it a spin-1 particle !

Sounds like: "how does a green car know it has to reflect green light ?" or something... unless I miss what you want to say.

Sorry for the delayed reply ... been busy at work.

Let me see if I can explain my question. The operator for the (non-relativistic) Schrodinger equation in general form is

(1) H = T + V

If we are dealing with a spin 1/2 particle, then we know that the coefficients for the spin states take the form:

(2) |a_up|^2 = cos^2(theta), |a_down|^2 = sin^2(theta)

Obviously, these are normalized, meaning that sin^2 + cos^2 = 1 for any theta. But how exactly did we get from the general relation (1) to the specific relation (2)? Somewhere in this process, we had to inject the fact that there are *two* states. My point is that the Schrodinger equation does not tell us that there are two states; rather, this is an additional piece of information that is put in "externally" when we derive (2) from (1) so that we can normalize correctly. Therefore, the length of the component in the relevant eigenspace *does* depend on the total number of eigenspaces.

Unless I am missing something, which is entirely possible. (I have never understood the significance/meaning of "noncontextuality" -- hopefully I can fix that in this thread ...)

Just for reference, this is the statement that prompted my question:

As you know, the APP NEEDS this information: it needs to know HOW MANY OTHER EIGENSPACES have a non-zero component of the state in them. The Born rule doesn't: the length of the component in the relevant eigenspace is sufficient.

David

 

[straycat]

The difficulty arises in extending the argument to decoherent outcomes whose measures are not equal, i.e. to density matrices whose non-zero diagonals are not equal. ... This step, however ... rather implicitly assumes that the probabilities defined on S are simply related to the probabilities defined on SC. ...

I agree with this general line of reasoning. It seems so obvious that, when I first read Zurek's paper, it made me wonder whether I had missed some subtle point. So far I haven't found it though ...

David

 

[vanesch]

Treating the probabilities of S outcomes as sums over (more detailed) SC outcomes then gives the Born rule. This step, however, does not amount to simply using additivity of probabilities within a single probability space but rather implicitly assumes that the probabilities defined on S are simply related to the probabilities defined on SC.

EXACTLY !

No matter how much that step accords with our experience-based common sense, it does not follow from the stated assumptions, which are deeply based on the idea that probabilities cannot be defined in general but only on a given system. Thus the question of why quantum probabilities take on Born values, or more generally of why they seem independent of where a line is drawn between system and environment, is not answered by Zurek's argument.

Yes, that was also the reasoning I had. And IF you make that extra assumption (which, I think, corresponds to non-contextuality) then we *already know* that we will find the Born rule through Gleason's theorem.

A counterexample may reinforce this point. A simple probability, defined from numbers* of diagonal terms in the density matrix of the system without weighting by measure, is entirely "envariant" and obeys the swapping symmetry. It does not obey any simple general relation between the probabilities defined on a system and on some supersystem. This probability is, of course, none other than 'world counting', which has frequently been argued to be the obvious probability to arise in collapse-free pictures without dynamical or metaphysical addenda. 2-8
Thus the problem of showing why the probabilities we experience emerge from quantum mechanics remains. ...

Yes, your counter example is of course the "APP", which we should maybe give the name "RHA" (Revelator of Hidden Assumptions)

 

[vanesch]

Sorry for the delayed reply ... been busy at work.

Let me see if I can explain my question. The operator for the (non-relativistic) Schrodinger equation in general form is

(1) H = T + V

If we are dealing with a spin 1/2 particle, then we know that the coefficients for the spin states take the form:

(2) |a_up|^2 = cos^2(theta), |a_down|^2 = sin^2(theta)

Obviously, these are normalized, meaning that sin^2 + cos^2 = 1 for any theta. But how exactly did we get from the general relation (1) to the specific relation (2)? Somewhere in this process, we had to inject the fact that there are *two* states.

We don't derive (2) from (1). (2) is part of the interpretation of the Hilbert space of states. In order to set up a quantum theory, we need to start NAMING the independent degrees of freedom, and *assign* them operational meanings. This is a part that is often skipped in textbooks, because they are usually interested in quantizing classical systems, and in this case, the independent degrees of freedom are given by the points in configuration space of the classical system, so it is "automatic". And then we STILL need to put in, by hand, a few extra degrees of freedom, such as spin.
Things like (2) are usually called the "kinematics" of the theory, while (1) is the "dynamics".

This compares, in classical physics, to: (1) specifying the number of particles and degrees of freedom of the mechanical system and (2), writing down Newton's equation or its equivalent.
Clearly, you cannot derive the number of planets from Newton's equation, and you cannot derive the number of degrees of freedom for spin from the Schroedinger equation.

 

[straycat]

...
This compares, in classical physics, to: (1) specifying the number of particles and degrees of freedom of the mechanical system and (2), writing down Newton's equation or its equivalent.

Clearly, you cannot derive the number of planets from Newton's equation, and you cannot derive the number of degrees of freedom for spin from the Schroedinger equation.

This is a good analogy. To calculate the orbital of a planet (say Mars), we take as input the masses, intial positions, and initial velocities of the sun, planets, and other objects in the solar system, and plug these into Newton's equation, which spits out the solution x_mars(t). (We could then talk about the "Isaac rule" which states that "x(t) is interpreted as the trajectory.") Likewise, to calculate the probability associated with a given observation, we take the initial state of the system, which includes stuff like the particle's spin, which includes the total number of possible spin states, and plug this into the Schrodinger equation, which spits out the amplitudes a_n. Finally, we apply the Born rule, which tells us that |a|^2 gives us the probability.

Now you said earlier that

As you know, the APP NEEDS ... to know HOW MANY OTHER EIGENSPACES have a non-zero component of the state in them. The Born rule doesn't: the length of the component in the relevant eigenspace is sufficient.

So you are saying that the probability of, say, spin up is independent of the total number of eigenstates (two in this case), because it depends only upon a_up. But isn't this like saying that the trajectory of Mars is independent of the trajectories of the other planets, because it depends only on x_mars(t)? Obviously this statement is false, because you cannot change (say) x_jupiter(t) without inducing a change in x_mars(t) as well.

To sum up: if we assume non-contextuality, this means (correct me if I am wrong) that we are assuming that the probability of the n^th outcome depends only on a_n and not on anything else, such as the total number N of eigenspaces. My difficulty is that I do not see how this assumption is tenable, given that you cannot calculate a_n without knowing (among other things) the total number of eigenspaces. It would be analogous to the assumption that the trajectory of Mars is independent of the trajectory of Jupiter.

So what am I missing?

David

 

[vanesch]

This is a good analogy. To calculate the orbital of a planet (say Mars), we take as input the masses, intial positions, and initial velocities of the sun, planets, and other objects in the solar system, and plug these into Newton's equation, which spits out the solution x_mars(t). (We could then talk about the "Isaac rule" which states that "x(t) is interpreted as the trajectory.")

Worse! The number of planets will even CHANGE THE FORM of Newton's equation: the number of 1/r^2 terms in the force law for each planet will be different to whether there are 2, 3, 5, 12, 2356 planets. In other words, the DIMENSION OF THE PHASE SPACE defines (partly) the form of Newton's equation.

Likewise, to calculate the probability associated with a given observation, we take the initial state of the system, which includes stuff like the particle's spin, which includes the total number of possible spin states, and plug this into the Schrodinger equation, which spits out the amplitudes a_n.

In the same way, the FORM of the Schroedinger equation (or better, of the Hamiltonian) will depend upon the spins of the particles ; and this time not only the number of terms, but even the STRUCTURE of the Hamiltonian: a Hamiltonian for a spin-1/2 particle CANNOT ACT upon the Hilbert space of a spin-1 particle: it is not the right operator on the right space.

So you are saying that the probability of, say, spin up is independent of the total number of eigenstates (two in this case), because it depends only upon a_up. But isn't this like saying that the trajectory of Mars is independent of the trajectories of the other planets, because it depends only on x_mars(t)? Obviously this statement is false, because you cannot change (say) x_jupiter(t) without inducing a change in x_mars(t) as well.

I was referring to the number of eigenspaces of the hermitean measurement operator, NOT about the unitary dynamics (which is of course sensitive to the number of dimensions of HILBERTSPACE, which is nothing else but the number of physical degrees of freedom). The eigenspaces of the hermitean measurement operator, however, depend entirely on the measurement to be performed (the different, distinguishable, results). When the measurement is complete, then all eigenspaces are one-dimensional. But clearly that's not possible, because that means an almost infinite amount of information.
The hermitean measurement operator is in fact just a mathematically convenient TRICK to say which different eigenspaces of states correspond to distinguishable measurement results (and to include a name = real number for these results). What actually counts is the slicing up of the hilbert space in slices of subspaces which "give the same results".

The extraction of probabilities in quantum theory, comes from the confrontation of two quantities:
1) the wavefunction |psi> and 2) the measurement operator (or better, the entire set of eigenspaces) {E_1,E_2...E_n},
and the result of this confrontation has to result in assigning a probability to each distinct outcome.

I tried to argue in my paper that every real measurement always has only a finite number of eigenspaces {E_1...E_n} ; that is to say, the result of a measurement can always be stored in a von Neumann computer with large, but finite, memory. You can try to fight this, and I'm sure you'll soon run into thermodynamical problems (and you'll even turn into a black hole ).

As such, when I do a specific measurement, so when I have a doublet: {|psi>,{E_1...E_n}}, then I need to calculate n numbers, p_1,... p_n, which are the predicted probabilities of outcomes associated, respectively, to E_1 ... E_n.

This means that p_i ({|psi>,{E_1...E_n}}) in general.
This means that p_i can change completely if we change ANY of the E_j. On the other hand, FOR A GIVEN SET OF {E_1...E_n}, they have to span a Kolmogorov probability measure. But FOR A DIFFERENT SET, we can have ANOTHER probability measure.

The Born rule says that p_i is given by <psi|P_i|psi> (if psi is normalized), where P_i is the projector on E_i. The APP says that p_1 = 1/k, with k the number of projectors which do not annihilate |psi>.

Non-contextuality is the claim that p_i can only depend upon |psi> and E_i. Gleason's theorem says then, that IF WE REQUIRE that p_i is ONLY a function of |psi> and E_i (no matter what the other E_k are, and how many there are), then the ONLY solution is the Born rule. If the probability for a measurement to say that the state is in E_i can only depend upon E_i itself, and the quantum state |psi>, then the only solution is the Born rule.

This rules out the APP, for the APP needs to know the number k of eigenspaces which contain a component of |psi>. The APP is hence a non-non-contextual rule. It needs to know 'the context' of the measurement in WHICH we are trying to calculate the probability of |psi> to be in E_i.

To sum up: if we assume non-contextuality, this means (correct me if I am wrong) that we are assuming that the probability of the n^th outcome depends only on a_n and not on anything else, such as the total number N of eigenspaces. My difficulty is that I do not see how this assumption is tenable, given that you cannot calculate a_n without knowing (among other things) the total number of eigenspaces. It would be analogous to the assumption that the trajectory of Mars is independent of the trajectory of Jupiter.

No, that's because you're confusing two different sets of dimensions. The physical situation determines the number of dimensions in Hilbert space, and the dynamics (the unitary evolution) is dependent upon that only. But there's no discussion about the number of dimensions of Hilbert space (the number of physical degrees of freedom).
The number of EIGENSPACES is related to the resolution and kind of the measurement, in that many physical degrees of freedom will give identical measurement results, which are then lumped into ONE eigenspace. It is more about HOW WE DISTRIBUTE the physical degrees of freedom over the DIFFERENT measurement outcomes, and how this relates to the probability of outcome.
Non-contextuality says that if we perform TWO DIFFERENT measurements, but which happen to have a potential outcome in common (thus, have one of their E_i in common), that we should find the same probability for that outcome, for the two different cases. It is a very reasonable requirement at first sight.

 

[straycat]

Sorry again for the much delayed response.

I've been pondering your last post. It seems you are drawing a distinction between (1) the number of dimensions of Hilbert space and (2) the number of eigenspaces of the measurement operator. I realize these are different, but there is still something I am not quite grokking. I'll have to re-read this thread and ponder more when I get more time. In the meantime, a few comments/questions.

Non-contextuality is the claim that p_i can only depend upon |psi> and E_i. ...

... the APP needs to know the number k of eigenspaces which contain a component of |psi>. The APP is hence a non-non-contextual rule.

Suppose we assume the APP. Given a particular measurement to be performed, suppose we have K total fine-grained outcomes, with k_i the number of fine-grained outcomes corresponding to the i^th coarse-grained result. eg, we have N position detector elements, i an integer in [1,N], and the sum of k_i over all i equals K. So the probability of detection at the i^th detector element is k_i / K, and we define:
E_i = k_i / K

So if I claim that p_i can depend only upon E_i (ie p_i = E_i), then it seems to me that I could argue, using the same reasoning that you use above for the Born rule, that the APP is non-contextual. What is wrong with my reasoning? I suppose you might say that you cannot calculate E_i (in the framework of the APP) without knowledge of K, ie without knowledge of the context. But it still seems to me that you likewise cannot calculate E_i in the framework of the Born rule, without knowledge of the measurement operator. iow, I'm trying to argue that the APP and the Born rule are either both contextual, or both non-contextual, depending on how exactly you define contextual, and you can't distinguish them based on "contextuality."

Perhaps I should study Gleason's theorem in greater detail than I have done so far. I actually think it is somewhat remarkable that it leads to the Born rule. However, it still seems to me that the assumption of a normalized Hilbert space for state representation is where the Born rule sneaks in. That is, Hilbert space assumes the state is represented by f, and the sum of |f|^2 over all states equals 1 (by normalization). So really it's not that surprising that |f|^2 is the only way to get a probability.

It is a very reasonable requirement at first sight.

Do you say "at first sight" because a careful analysis indicates that it's not all that reasonable?

I tried to argue in my paper that every real measurement always has only a finite number of eigenspaces {E_1...E_n} ; that is to say, the result of a measurement can always be stored in a von Neumann computer with large, but finite, memory. You can try to fight this, and I'm sure you'll soon run into thermodynamical problems (and you'll even turn into a black hole ).

I actually agree with you here. The argument in my mind goes like this: consider a position measurement. If you want to come up with a continuous measurement variable, this would be it. But from a practical perspective, a position measurement is performed via an array or series of discrete measurement detectors. The continuous position measurement is then conceived as the theoretical limit as the number of detector elements becomes infinite. But from a practical, and perhaps from a theoretical, perspective, this limit cannot ever be achieved: the smallest detector element I can think of would be (say) an individual atom, for example the atoms that make up x-ray film.

David

 

[vanesch]

I've been pondering your last post. It seems you are drawing a distinction between (1) the number of dimensions of Hilbert space and (2) the number of eigenspaces of the measurement operator. I realize these are different,

Yes, this is essential. The number of dimensions in Hilbert space is given by the physics, and by physics alone, of the system, and might be very well infinite-dimensional. I think making assumptions on the finiteness of this dimensionality is dangerous. After all, you do not know what degrees of freedom are hidden deep down there. So we should be somehow independent of the number of dimensions of the Hilbert space.

However, the number of eigenspaces of the measurement operator is purely determined by the measurement apparatus. It is given by the resolution by which we could, in principle, determine the quantity we're trying to measure, using the apparatus in question. You and I agree that this must be a finite number, and a rather well-determined one. This is probably where we are differing in opinion, and where you seem to claim "micromeasurements" of eventually unknown physics of which we are not aware versus "macromeasurements" which are just our own coarse-graining of these micromeasurements- while I claim that with every specific measurement goes a certain, well-defined number of outcomes (which could eventually be more fine-grained than the observed result but that this should not be dependent on "unknown physics", but that a detailled analysis of the measurement setup should reveil that to us). I would even claim that a good measurement apparatus makes the observed number of outcomes about equal to the real number of eigenspaces.

Suppose we assume the APP. Given a particular measurement to be performed, suppose we have K total fine-grained outcomes, with k_i the number of fine-grained outcomes corresponding to the i^th coarse-grained result. eg, we have N position detector elements, i an integer in [1,N], and the sum of k_i over all i equals K. So the probability of detection at the i^th detector element is k_i / K, and we define:
E_i = k_i / K

Yes, but I'm claiming now that for a good measurement system, k_i = 1 for all i, and even if it isn't (for instance, you measure with a precision of 1 mm, and your numerical display only displays up to 1 cm resolution), you're not free to fiddle with k_i as you like.
Also, you now have a strange outcome! You ALWAYS find probability E_i for outcome i, no matter what was the quantum state ! Even if the quantum state is entirely within the E_i eigenspace, you'd still have a fractional probability ? That would violate the rule that two measurements applied one after the other will give the same result.

So if I claim that p_i can depend only upon E_i (ie p_i = E_i), then it seems to me that I could argue, using the same reasoning that you use above for the Born rule, that the APP is non-contextual.

No, non-contextuality has nothing to do with the number E_i you're positioning here, it is a property of being only a function of the eigenspace (spanned by the k_i subspaces) and the quantum state, no matter how the other eigenspaces are sliced up. Of course, in a way, you're right: if the outcome is INDEPENDENT on the quantum state (as it is in your example), you are indeed performing a non-contextual measurement. In fact, the outcome has nothing to do with the system: outcome i ALWAYS appears with probability E_i. But I imagine that you only want to consider THOSE OUTCOMES i THAT HAVE A PART OF the quantum state in them, right ? And THEN you become dependent on what happens in the other eigenspaces.

That is, Hilbert space assumes the state is represented by f, and the sum of |f|^2 over all states equals 1 (by normalization). So really it's not that surprising that |f|^2 is the only way to get a probability.

Well, there's a difference in the following sense: if you start out with a normalized state, you will always keep a normalized state under unitary evolution, and if you change basis (change measurement), you can keep the same normalized vector. That cannot be said for the E_i and k_i construction, which needs to be redone after each evolution, and after each different measurement basis.

Do you say "at first sight" because a careful analysis indicates that it's not all that reasonable?

Well, it is reasonable, but it is an EXTRA assumption (and, according to Gleason, logically equivalent to postulating the Born rule). It is hence "just as" reasonable as postulating the Born rule.
What I meant with "at first sight" is that one doesn't realize the magnitude of the step taken! In unitary QM, there IS no notion of probability. There is just a state vector, evolving deterministically by a given differential equation of first order, in a hilbert space. From the moment that you require, no matter how little, a certain quality of a probability issued from that vector, you are in fact implicitly postulating an entire construction: namely that probabilities ARE going to be generated from this state vector (probabilities for what, for whom?), that only part of the state vector is going to be observed (by whom?) etc... So the mere statement of a simple property of the probabilities postulates in fact an entire machinery - which is not obvious at first sight. Now if your aim is to DEDUCE the appearance of probabilities from the unitary machinery, then implicitly postulating this machinery is NOT reasonable, because it implies that you are postulating what you were trying to deduce in one way or another.

I actually agree with you here. The argument in my mind goes like this: consider a position measurement. If you want to come up with a continuous measurement variable, this would be it. But from a practical perspective, a position measurement is performed via an array or series of discrete measurement detectors. The continuous position measurement is then conceived as the theoretical limit as the number of detector elements becomes infinite. But from a practical, and perhaps from a theoretical, perspective, this limit cannot ever be achieved: the smallest detector element I can think of would be (say) an individual atom, for example the atoms that make up x-ray film.

That's what I meant, too. There's a natural "resolution" to each measurement device, which is given by the physics of the apparatus. An x-ray film will NOT be in different quantum states for positions which differ much less than the size of an atom (or even a bromide xtal). This is not "unknown physics" with extra degrees of freedom. I wonder whether a CCD type camera will be sensing on a better resolution than one pixel (meaning that the quantum states would be different for hits at different positions on the same pixel). Of course, there may be - and probably there will be - some data reduction up to the display, but one cannot invent, at will, more fine-grained measurements than the apparatus is actually naturally performing. And this is what determines the slicing-up of the Hilbert space in a finite number of eigenspaces, which will each result in macroscopically potentially distinguishable "pointer states". And I think it is difficult (if not hopeless) to posit that these "micromeasurements" will arrange themselves each time in such a way that they work according to the APP, but give rise to the Born rule on the coarse-grained level. Mainly because the relationship between finegrained and coarse grained is given by the measurement apparatus itself, and not by the quantum system under study (your E_i = k_i/K is fixed by the physics of the apparatus, independent of the state you care to send onto it ; the number of atoms on the x-ray film per identified "pixel" on the scanner is fixed, and not depending on how it was irradiated).

cheers,
Patrick.

 

[Hurkyl]

You can try to fight this, and I'm sure you'll soon run into thermodynamical problems (and you'll even turn into a black hole ).

Proof by threat of black hole!

 

[vanesch]

Proof by threat of black hole!

I'm proud to have found a new rethorical technique

 

[Hurkyl]

It takes a singular mind to come up with such things! (Okay, I'll stop now)

 

[Tez]

Treating the probabilities of S outcomes as sums over (more detailed) SC outcomes then gives the Born rule. This step, however, does not amount to simply using additivity of probabilities within a single probability space but rather implicitly assumes that the probabilities defined on S are simply related to the probabilities defined on SC. No matter how much that step accords with our experience-based common sense, it does not follow from the stated assumptions, which are deeply based on the idea that probabilities cannot be defined in general but only on a given system. Thus the question of why quantum probabilities take on Born values, or more generally of why they seem independent of where a line is drawn between system and environment, is not answered by Zurek's argument.

Hi Michael,

5 or so years ago when I was visiting Paul Kwiat you gave me a preprint of how you thought the Born rule could/should be derived. I remember there was a cute idea in there somewhere, though I can't remember what it was! How did it pan out?

Tez

 

[straycat]

And I think it is difficult (if not hopeless) to posit that these "micromeasurements" will arrange themselves each time in such a way that they work according to the APP, but give rise to the Born rule on the coarse-grained level. Mainly because the relationship between finegrained and coarse grained is given by the measurement apparatus itself, and not by the quantum system under study (your E_i = k_i/K is fixed by the physics of the apparatus, independent of the state you care to send onto it ; the number of atoms on the x-ray film per identified "pixel" on the scanner is fixed, and not depending on how it was irradiated).

Well, maybe it's not as difficult / hopeless as you might think! Let's play around for a moment with the idea that all measurements boil down to one particle interacting with another. That is, the fundamental limit of resolution of a particle detector is governed by the fact that the detector is made of individual particles. So if we look at the micro-organization at the fine-grained level, we see micro-structure that is determined by the properties of the particles in question; let's say, some property that is characteristic of fermions / bosons for fermi /bose statistics, respectively. When a particle hits an atom in a CCD detector, then there is a corresponding micro-structure that always follows some particular pattern, and it gives rise to the Born rule when you look at it from a coarse-grained perspective. So if particles were "constructed" differently, then we might not have the Born rule, we might have some other rule. This, in fact, is exactly how my toy scheme works!

This view is consistent with the notion that it does not matter whether there is data reduction up to the display. That is, it does not matter whether the CCD has resolution of 1 mm or 1 cm; if two different CCD's have different pixel resolution, but are made of the same types of atoms, then they will have the same fundamental fine-grained "resolution" when we look at the micro-structure.

I'm starting to contemplate a thought experiment, not sure where it will take me. Suppose we have a CCD camera (length, say, 10 cm) and we remove a 2 cm chunk of it which we replace with a lens that focuses all particles that would have hit the plate on that 2 cm stretch onto (for the sake of argument) a single atom. What effect do we expect this will have on our measurement probabilities? Contrast that to a different scenario: we have a CCD camera, length 10 cm, with resolution 1 mm. Remove a 2 cm chunk and replace it with a single pixel, ie 2 cm resolution. But both CCD setups are made of the same types of atoms. I would expect that the probability of detection over the 2 cm single pixel equals the sum of the probability of detection of all 20 of the individual 1 mm pixels; my reasoning is that in both setups, we have the same density and type of atoms in the CCD's. But I would imagine that using the lens setup, we would get something completely different, since we are effectively replacing detection over a 2 cm stretch using lots of atoms with detection using only one atom.

 

[straycat]

I'm starting to contemplate a thought experiment, not sure where it will take me. Suppose we have a CCD camera (length, say, 10 cm) and we remove a 2 cm chunk of it which we replace with a lens that focuses all particles that would have hit the plate on that 2 cm stretch onto (for the sake of argument) a single atom. What effect do we expect this will have on our measurement probabilities? Contrast that to a different scenario: we have a CCD camera, length 10 cm, with resolution 1 mm. Remove a 2 cm chunk and replace it with a single pixel, ie 2 cm resolution. But both CCD setups are made of the same types of atoms. I would expect that the probability of detection over the 2 cm single pixel equals the sum of the probability of detection of all 20 of the individual 1 mm pixels; my reasoning is that in both setups, we have the same density and type of atoms in the CCD's. But I would imagine that using the lens setup, we would get something completely different, since we are effectively replacing detection over a 2 cm stretch using lots of atoms with detection using only one atom.

Actually this reminds me of the quantum zeno effect ( http://en.wikipedia.org/wiki/Quantum_Zeno_effect ), which I mentioned in post #37 of this thread. From the wiki description, the experiment they do is sort of similar to the thought experiment I outlined above, except that I am playing around with resolution of the position measurement, whereas they were playing around with the resolution of a time measurement in the experiment described in wiki. The point of the zeno effect is that if you change the resolution of the time measurement at the fine grained level, then you change the probability distribution as a function of time. Similarly, I expect that if you change the resolution of the position measurement in a fundamental sense, ie using the lens setup, then you should change the probability distribution as a function of position. But if you simply swap the 1 mm pixel with the 2 cm pixel, then (I expect) you will not change the probability as a function of position, because you have done nothing to change the fundamental micro-structure, since the 1 mm and 2 cm CCD detectors have the same density of atoms.

 

[selfAdjoint]

Doesn't your spatial-resolution fiddling bear a family resemblance to Asfhar's analysis? I believe that was described here recently in Quantum Zeno terms.

 

[straycat]

Doesn't your spatial-resolution fiddling bear a family resemblance to Asfhar's analysis? I believe that was described here recently in Quantum Zeno terms.

Hmm, I've never thought of comparing the two. It's been a long time since I've thought about the Afshar experiment. I always belonged to the camp that thought there was a flaw somewhere in his analysis, though. That is, I tend to think that the CI and the Everett interpretation each make exactly the same predictions as any of the other formulations of QM (see, eg, the wonderful paper [1]) -- so I am a bit biased against Afshar's (and Cramer's) claims to the contrary.

As for the Zeno effect, I have actually not really pondered it really really deeply. But from my cursory contemplation, the existence of the Zeno effect does not surprise me all that much. To me, the main lesson of the Zeno effect could be stated loosely: how you measure something (the resolution of the time measurements) has an effect on the probability distribution (probability of decay as a function of time). But that is simply Lesson # 1 (in my mind) in quantum mechanics. eg, the 2-slit exp tells us that how we measure something (whether we do or do not look at the slits) has an effect on the resulting probability distribution (where it hits the screen). So perhaps the Zeno effect is just teaching us the same lesson as the 2-slit exp, but dressed up differently.

So my knee jerk reaction to your question would be that Afshar's analysis is based in a (somehow) flawed reading/implementation of the CI (and MWI), but the Zeno effect is founded upon a correct implementation of quantum theory. I'd have to take another look at Afshar though to see the comparison with Zeno ...

David

[1] Styer et al. Nine formulations of quantum mechanics. Am J Phys 70:288-297, 2002

 

[straycat]

Doesn't your spatial-resolution fiddling bear a family resemblance to Asfhar's analysis? I believe that was described here recently in Quantum Zeno terms.

Oh yea! Afshar used a lens in his setup too -- now I remember -- duhhh

D

http://en.wikipedia.org/wiki/Afshar_experiment

 

[alfredblase]

wow I just the wiki article on the ashfar experiment... mmm.. so proc. spie is an optical engineering journal and not a physics journal...

I guess it must be generally believed by the physics powers that be that ashfar's interpretation of the experiment is erroneous.

good enough for me i guess.. hehe

 

[straycat]

I guess it must be generally believed by the physics powers that be that ashfar's interpretation of the experiment is erroneous.

Yea, I just linked from wiki to Lubos Motl's blog article [1] criticising the Afshar analysis, and I see with some amusement that Lubos' critique is essentially the same critique that I made myself [2] over in undernetphysics ... except that I made my critique a month earlier!

That's right, I beat him to the punch ... who's ya' daddy now?

DS <ducking in case Lubos is lurking about somewhere ...>

[1] http://motls.blogspot.com/2004/11/violation-of-complementarity.html

[2] http://groups.yahoo.com/group/undernetphysics/message/1231

 

[vanesch]

Afshar's experiment has been discussed here before also:

https://www.physicsforums.com/showthread.php?t=59795

 

[mbweissman]

non-linear decoherence

Hi Michael,

5 or so years ago when I was visiting Paul Kwiat you gave me a preprint of how you thought the Born rule could/should be derived. I remember there was a cute idea in there somewhere, though I can't remember what it was! How did it pan out?

Tez

Hi Tez- Sorry for the delay- haven't been checking the forum. The idea was that if there were a non-linear decoherence process, the proper ratio of world-counts could arise asymptotically without fine-tuning. Basically it runs like this: if large-measure branches decohere faster than small-measure ones the limiting steady-state distributions would have the same average measure per branch. Hence branch count is simply proportional to measure.

How'd it work out? It was published in Found Phys. Lett., after some extraordinary constructive criticism from a referee. So far there are no obvious holes in it- e.g. no problem with superluminal communication, unlike some types of non-linear dynamics. On the other hand, it proposes extra machinery not in ordinary quantum mechanics, without giving a specific theory. Although the extra gunk is much less Rube-Goldbergish than in explicity collapse theories, it would be nice not to have to propose something like that at all.

I'm about to post a follow-on, in which I point out that once the non-linear processes have been proposed to rescue quantum measurement, they give the second law at no extra cost. A similar point was made by Albert in his book Time and Chance, but he was referring to non-linear collapse (much uglier) rather than simple non-linear decoherence.