Statistical ensemble interpretation done right

[Demystifier]

Concerning statistical ensemble interpretation (SEI), a lot of confusion has been created and lot of nonsense has been said, especially in a recent thread on "missed opportunities in Bohmian mechanics". Even Ballentine himself said many things about it that do not always seem perfectly clear and self-consistent. I want here to formulate the SEI in a clear, simple and self-consistent way.

The main source of confusion is whether SEI talks about individual systems, or only about large ensembles of systems. In particular, within SEI, does it make sense to say that we have one particle in the state $|\psi\rangle$?

Let us start from a purely instrumental point of view. Let $P$ be a preparation procedure, defined in terms of actions by an experimentalist in the laboratory. This procedure $P$ usually prepares a small number of particles, typically one particle, or one pair of entangled particles. For simplicity, let as assume that $P$ prepares one particle. If the procedure $P$ is known to a theorist, then for any such $P$ the theorist can write down an abstract density operator $\rho$. Depending on the $P$, the $\rho$ can be either pure or mixed. In the rest, for simplicity, I shall assume that it is pure. Thus, to any preparation $P$ of a single particle we can associate a unique (up to an irrelevant overall phase) state $|\psi\rangle$ in the Hilbert space. In this sense $|\psi\rangle$ is associated with a single particle, not with an ensemble of particles.

Above, the crucial assumption is that $P$ is known to the theorist. But what if it is not known, what if the theorist does not know how the particle is prepared? In that case he cannot write down $|\psi\rangle$. And what if even the experimentalist does not know $P$ (e.g. because $P$ was performed by another experimentalist)? Can the experimentalist perform a measurement, tell the result to the theorist, so that then the theorist can write down $|\psi\rangle$? The answer is - no! If $P$ is unknown, there is no measurement by which one can determine $|\psi\rangle$.

Nevertheless, there is a way out. Suppose that the same unknown preparation $P$ can be repeated many times. By repeating it many times, we create a large ensemble of particles. In this case there is a measurement procedure performed on the ensemble by which $|\psi\rangle$ can be determined approximately. Larger the ensemble, better the precision of $|\psi\rangle$ determination. In the limit of infinite ensemble, the determination of $|\psi\rangle$ can be exact. In this sense, $|\psi\rangle$ is associated with a large ensemble of particles, not with a single particle.

To summarize, $|\psi\rangle$ (or more generally $\rho$) can be associated with a single system when the preparation procedure $P$ is known, but only with a large ensemble of systems when $P$ is unknown. This is the essence of the statistical ensemble interpretation (SEI).

Now someone will note that the above is just the fact accepted by all interpretations. The above is not a particular interpretation. I would say that the above is the minimal SEI. It is minimal, in the sense that it does not add any additional interpretation claims that may create controversy.

However, it does not mean that one cannot add additional interpretation claims to SEI. One can, and one does. One can do that in more than one way, so there is more than one version of SEI. Some versions may tacitly assume existence of additional unknown variables associated with a single system, other versions may deny their existence. Some versions may accept collapse as an information update, other versions may deny collapse in any form. Some versions may view Bell's theorem as a proof of "nonlocality", other versions may view it as a proof of "nonrealism".

What is common to all versions of SEI, except the facts above shared by all interpretations? All versions of SEI have in common the following:
- The collapse does not exist in the physical sense. (The information update is not considered physical.)
- Additional (hidden) variables (e.g. Bohm) are not considered.

 

[Haborix]

To summarize, |ψ⟩ (or more generally ρ) can be associated with a single system when the preparation procedure P is known, but only with a large ensemble of systems when P is unknown. This is the essence of the statistical ensemble interpretation (SEI).

I don't see the need to go here. Why restrict to this binary categorization of unknown/known preparation procedure and single/(implicitly)infinite particles. I might have a rough idea of the preparation procedure and a semi-large ensemble and thus be able to say more about the state than if I had only the preparation procedure or the semi-large ensemble. And my bookkeeping tool is the state. Does SEI even require us to say something about associating, or not associating, it with a single system?

 

[Demystifier]

I might have a rough idea of the preparation procedure

Can you give an example of such "rough idea"?

 

[Haborix]

Hmm, I'm just making things up, but say the particle source in a Stern-Gerlach experiment was wonky and is producing particles with unusual momentum distributions. Assume also the experimenter was unsure about how well characterized the source was but that he nevertheless has good, well-characterized magnets. The final distribution on the plate after the magnet is going to tell the experimenter something about his preparation procedure related to the particle source. Anyway, I'm willing to grant for sake of argument you can only know or not know the preparation procedure. I'm still uncertain about the need for the binary association rule for the state with the single particle.

 

[Demystifier]

the particle source in a Stern-Gerlach experiment was wonky and is producing particles with unusual momentum distributions.

If we know how much winky, and if this is the only source of uncertainty, we can associate a well defined mixed state with the source.

Assume also the experimenter was unsure about how well characterized the source was

This is still too vague, so I cannot comment it.

Anyway, I'm willing to grant for sake of argument you can only know or not know the preparation procedure. I'm still uncertain about the need for the binary association rule for the state with the single particle.

I didn't intend to be binary. I explained two cases, but I didn't say that intermediate cases don't exist. Quite generally, If you are uncertain about the preparation of the pure state, you can always describe your incomplete knowledge by a mixed state.

 

[gentzen]

And my bookkeeping tool is the state.

It is unclear to me what you mean. (Are you a Bayesian or a Dutch bookie from those Dutch book arguments?)

Does SEI even require us to say something about associating, or not associating, it with a single system?

If you look at SEI from an operational verification perspective, then yes, you must associate a single system with a state before you know the results of the non-preparation measurements. This allows it to take part in some verification. Of course, no statistical verification can ever fully reject your state assignments, at most it can tell you that winning the jackpot of a lottery would have been more probable than your obtained measurement results given your previous state assignments.

 

[Morbert]

The main source of confusion is whether SEI talks about individual systems, or only about large ensembles of systems. In particular, within SEI, does it make sense to say that we have one particle in the state $|\psi\rangle$?

Let us start from a purely instrumental point of view. Let $P$ be a preparation procedure

To summarize, $|\psi\rangle$ (or more generally $\rho$) can be associated with a single system when the preparation procedure $P$ is known, but only with a large ensemble of systems when $P$ is unknown. This is the essence of the statistical ensemble interpretation (SEI).

The subtlety is in the distinction between "we consider an ensemble of particles, each prepared in the state $|\psi\rangle$" and "we consider an ensemble $|\psi\rangle$ of similarly prepared particles" and "we consider a class of preparation procedures $|\psi\rangle$". In all three cases, we can associate a single microscopic system with $|\psi\rangle$, but this doesn't collapse the distinction between them. The distinction is why I would hesitate to say the first statement is strictly compatible with SEI.

 

[martinbn]

I don't see any inyerpretation of quantum mechanics here. In fact i don't see anything quantum specific. What you are describing is how statistics work when there is lack of knowlage.

 

[vanhees71]

Concerning statistical ensemble interpretation (SEI), a lot of confusion has been created and lot of nonsense has been said, especially in a recent thread on "missed opportunities in Bohmian mechanics". Even Ballentine himself said many things about it that do not always seem perfectly clear and self-consistent. I want here to formulate the SEI in a clear, simple and self-consistent way.

The main source of confusion is whether SEI talks about individual systems, or only about large ensembles of systems. In particular, within SEI, does it make sense to say that we have one particle in the state $|\psi\rangle$?

This never makes sense, because a quantum state is described by a statistical operator not a state vector (or if you restrict yourself to pure state only by a unit ray in Hilbert space, not the state vector itself, but that's in fact not so important for the basics we are discussing here).

Let us start from a purely instrumental point of view. Let $P$ be a preparation procedure, defined in terms of actions by an experimentalist in the laboratory. This procedure $P$ usually prepares a small number of particles, typically one particle, or one pair of entangled particles. For simplicity, let as assume that $P$ prepares one particle. If the procedure $P$ is known to a theorist, then for any such $P$ the theorist can write down an abstract density operator $\rho$. Depending on the $P$, the $\rho$ can be either pure or mixed. In the rest, for simplicity, I shall assume that it is pure. Thus, to any preparation $P$ of a single particle we can associate a unique (up to an irrelevant overall phase) state $|\psi\rangle$ in the Hilbert space. In this sense $|\psi\rangle$ is associated with a single particle, not with an ensemble of particles.

Above, the crucial assumption is that $P$ is known to the theorist. But what if it is not known, what if the theorist does not know how the particle is prepared? In that case he cannot write down $|\psi\rangle$. And what if even the experimentalist does not know $P$ (e.g. because $P$ was performed by another experimentalist)? Can the experimentalist perform a measurement, tell the result to the theorist, so that then the theorist can write down $|\psi\rangle$? The answer is - no! If $P$ is unknown, there is no measurement by which one can determine $|\psi\rangle$.

Nevertheless, there is a way out. Suppose that the same unknown preparation $P$ can be repeated many times. By repeating it many times, we create a large ensemble of particles. In this case there is a measurement procedure performed on the ensemble by which $|\psi\rangle$ can be determined approximately. Larger the ensemble, better the precision of $|\psi\rangle$ determination. In the limit of infinite ensemble, the determination of $|\psi\rangle$ can be exact. In this sense, $|\psi\rangle$ is associated with a large ensemble of particles, not with a single particle.

To summarize, $|\psi\rangle$ (or more generally $\rho$) can be associated with a single system when the preparation procedure $P$ is known, but only with a large ensemble of systems when $P$ is unknown. This is the essence of the statistical ensemble interpretation (SEI).

Now someone will note that the above is just the fact accepted by all interpretations. The above is not a particular interpretation. I would say that the above is the minimal SEI. It is minimal, in the sense that it does not add any additional interpretation claims that may create controversy.

That's a perfect description of the minimal interpretation (you should make an Insight article out of it, with the little correction in the beginning concerning what uniquely represents a state, which is a crucial part of the mathematical foundations).

That also underlines that this is the "minimal interpretation" and thus the interpretation which restricts the quantum formalism to the objective scientific meaning of this formalism. Everything going beyond this is philosophy or religion and leads to confusion.

However, it does not mean that one cannot add additional interpretation claims to SEI. One can, and one does. One can do that in more than one way, so there is more than one version of SEI. Some versions may tacitly assume existence of additional unknown variables associated with a single system, other versions may deny their existence. Some versions may accept collapse as an information update, other versions may deny collapse in any form. Some versions may view Bell's theorem as a proof of "nonlocality", other versions may view it as a proof of "nonrealism".

No, this goes beyond the scientific part of the formalism. E.g., as long as nobody observes "hidden variables", you don't need them. There's no need of a collapse assumption. What state is "prepared" after some measurements depends on the physical situation due to this measurement. The von Neumann "filter measurement" (which is rather a preparation procedure than just a measurement) is very rarely really possible.

What is common to all versions of SEI, except the facts above shared by all interpretations? All versions of SEI have in common the following:
- The collapse does not exist in the physical sense. (The information update is not considered physical.)
- Additional (hidden) variables (e.g. Bohm) are not considered.

Indeed! If you don't need to introduce non-observable elements, you shouldn't do it. It's just Occam's razor, avoiding unnecessary confusion.

 

[vanhees71]

Can you give an example of such "rough idea"?

That's the most common thing in practice. You have some "rough" preparation procedure. Then you have to make an educated guess, which statistical operator you should associate with this preparation procedure. Usually you have to figure this out experimentally. E.g., when you build some detector in a particle-collision experiment you have to test its properties carefully. It's part of the "callibration procedure" of the measurement device.

 

[vanhees71]

Hmm, I'm just making things up, but say the particle source in a Stern-Gerlach experiment was wonky and is producing particles with unusual momentum distributions. Assume also the experimenter was unsure about how well characterized the source was but that he nevertheless has good, well-characterized magnets. The final distribution on the plate after the magnet is going to tell the experimenter something about his preparation procedure related to the particle source. Anyway, I'm willing to grant for sake of argument you can only know or not know the preparation procedure. I'm still uncertain about the need for the binary association rule for the state with the single particle.

That's an interesting example. Indeed, Stern got the description of the preparation procedure wrong in the beginning of his "molecular beam experiments". This was an experiment done before the famous SG experiment but was not less fundamental, i.e., it was a direct confirmation of the Maxwell-Boltzmann distribution for the momenta of particles in a thermal source. Now this was done by using vapor (I think also silver) released from a oven through a hole. To describe this correctly you must take into account an additional velocity factor, ironically precisely because you have separated out a beam from the thermal ensemble. That was figured out by Einstein, and after Stern corrected for this glitch in his evaluation of his meausurement results he indeed got a much better agreement with the expectation than with the errorneous assumption.

 

[vanhees71]

It is unclear to me what you mean. (Are you a Bayesian or a Dutch bookie from those Dutch book arguments?)

If you look at SEI from an operational verification perspective, then yes, you must associate a single system with a state before you know the results of the non-preparation measurements. This allows it to take part in some verification. Of course, no statistical verification can ever fully reject your state assignments, at most it can tell you that winning the jackpot of a lottery would have been more probable than your obtained measurement results given your previous state assignments.

This is another as interesting question as the state definition as representing preparation procedures. It's the other direction, i.e., you have given some preparation procedure, how to figure out the state. Since according to QT in the minimal statistical interpretation all you have are the probabilistic outcome of measurements. To determine the state of the system according to this preparation procedure experimentally you necessarily need an ensemble (that's why the statistical interpretation is often somewhat imprecisely quoted as only referring to ensembles rather than single systems, which without the here discussed more precise statements, would make it impossible to define ensembles, because you'd not have a relation to the single systems making up the ensembles to begin with), and you need a sufficient set of measurements of different, incompatible observables, each measured on large enought ensembles to get good enough statistics to be able to reconstruct the quantum state from all these measurements, with a given statistical significance. This socalled "state tomography" is discussed very carefully in Ballentines book too.

 

[vanhees71]

The subtlety is in the distinction between "we consider an ensemble of particles, each prepared in the state $|\psi\rangle$" and "we consider an ensemble $|\psi\rangle$ of similarly prepared particles" and "we consider a class of preparation procedures $|\psi\rangle$". In all three cases, we can associate a single microscopic system with $|\psi\rangle$, but this doesn't collapse the distinction between them. The distinction is why I would hesitate to say the first statement is strictly compatible with SEI.

Once more, a pures state is described by $\hat{\rho}=|\psi \rangle \langle \psi|$, not $|\psi \rangle$ itself, but that's a minor point in this discussion.

You need both, the association of $\hat{\rho}$ with a preparation procedure, which refers to a single system to be able to define which ensemble is described by this statistical operator. It describes an ensemble in the sense that this state describes nothing else than the probabilities for the outcome of measurements on each of the so prepared single systems. To verify these probabilities you need an ensemble and statistical methods.

 

[gentzen]

I don't see any inyerpretation of quantum mechanics here. In fact i don't see anything quantum specific. What you are describing is how statistics work when there is lack of knowlage.

Are you replying to Morbert, or to Demystifier?

Assuming you are replying to Demystifier's initial post:

• The serious discussion about the statistical interpretation was triggered by Demystifier querying "us" (directly only me, but indirectly also vanhees71 and you) about our opinion on the collapse in the statistical interpretation, and on some dubious claims Ballentine seems to have made in that context.
• I think it is a misunderstanding to regard the collapse as unrelated to how statistics works, and try to give it some quantum specific meaning.
• Even more general, all those misunderstanding and disagreements regarding the "minimal statistical interpretation" seem to come from problems of "how to talk about statistics" and "how to interpret statistical predictions", and rarely from anything quantum specific. The only quantum specific part seems to be the confusion about which parts are specific to quantum mechanics.

 

[gentzen]

Even more general, all those misunderstanding and disagreements regarding the "minimal statistical interpretation" seem to come from problems of "how to talk about statistics" and "how to interpret statistical predictions", and rarely from anything quantum specific. The only quantum specific part seems to be the confusion about which parts are specific to quantum mechanics.

There is actually one quantum specific part (of the state assignment, not necessarily of the confusion) that I only realized after reading some of vanhees71's explanations in the other thread. It is related to:

If you look at SEI from an operational verification perspective, then yes, you must associate a single system with a state before you know the results of the non-preparation measurements. This allows it to take part in some verification.

You not only have to associate a single system with a state before you know the results of the non-preparation measurements, but also before you know the precise settings of those measurements.
Or maybe even this part is not quantum specific, and could also occur in other statistical contexts. But the specific nature of a quantum state seems to make this part always necessary in the quantum case, while it is often not necessary in other statistical contexts.

 

[Demystifier]

I don't see any inyerpretation of quantum mechanics here. In fact i don't see anything quantum specific. What you are describing is how statistics work when there is lack of knowlage.

Maybe, but I see it as a virtue, not a drawback.

 

[Morbert]

You need both, the association of $\hat{\rho}$ with a preparation procedure, which refers to a single system to be able to define which ensemble is described by this statistical operator. It describes an ensemble in the sense that this state describes nothing else than the probabilities for the outcome of measurements on each of the so prepared single systems. To verify these probabilities you need an ensemble and statistical methods.

I still think the relevant subtlety is being overlooked here. We have a single microscopic system, a preparation procedure and a quantum state. The microscopic system has been subject to the preparation procedure by a competent physicist.

i) "We can associate the single microscopic system with the quantum state."
ii) "The quantum state represents an ensemble of similarly prepared microscopic systems"
iii) "The microscopic system is prepared in the quantum state"

If you are arguing that you need both i) and ii), that's fine, but it is not clear that iii) is compatible with the SEI. Especially when we are discussing state collapse. Loosely, iii) is fine. When @Demystifier originally said "Suppose that you prepare 100 particles, each in the state $|\psi\rangle$", I instantly knew what he meant, but it is still good to stay cognizant of subtle differences. Just as, and I'm sure you agree, it might not always be important to describe a quantum state with $|\psi\rangle\langle\psi |$ and not $|\psi\rangle$, but it is good to remember the difference.

 

[vanhees71]

I don't know what you think the difference between i) and iii) is.

It's utmost important to keep in mind that the one-to-one correspondence between pure quantum states and the formalism is to $|\psi \rangle \langle \psi|$ and not $|\psi \rangle$, but that has not much to do with the interpretational issues we are discussing here.

It's also clear that one has to consider "mixed states". E.g., if you take "spontaneous parametric down conversion of a photon to an entangled two-photon state" as the preparation procedure for one of the photons in the two-photon state ("idler") using the other photon in the pair for "heralding" this photon ("signal photon"). Then this photon is prepared necessarily in a mixed state.

 

[Demystifier]

The subtlety is in the distinction between "we consider an ensemble of particles, each prepared in the state $|\psi\rangle$" and "we consider an ensemble $|\psi\rangle$ of similarly prepared particles" and "we consider a class of preparation procedures $|\psi\rangle$". In all three cases, we can associate a single microscopic system with $|\psi\rangle$, but this doesn't collapse the distinction between them. The distinction is why I would hesitate to say the first statement is strictly compatible with SEI.

I'm not sure that the difference is relevant. Let us consider a classical analogy, in which preparation is a coin flipping. The flipping prepares the state
$$S=[p({\rm heads})=1/2,p({\rm tails})=1/2]$$
where $p$ is the probability. What is the distinction between the following claims?
1. We consider an ensemble of coins, each prepared in the state $S$.
2. We consider an ensemble $S$ of similarly prepared coins.
3. We consider a class of preparation procedures $S$.

 

[gentzen]

iii) "The microscopic system is prepared in the quantum state"

I don't know what you think the difference between i) and iii) is.

It sounds somewhat strange to me to say: "The microscopic system is prepared in the quantum state, but only if I don't know yet the precise settings of some measurements I plan to do later on this quantum state".

 

[Haborix]

Sorry to interrupt the flow of the thread.

I didn't intend to be binary. I explained two cases, but I didn't say that intermediate cases don't exist. Quite generally, If you are uncertain about the preparation of the pure state, you can always describe your incomplete knowledge by a mixed state.

I think this helps make my question more clear. I agree about the experimenter using the mixed state. Let's assume after running the experiment in various ways we figure out that the state of the system was, within experimental uncertainties, a pure state. Is the state we "associate" with the system in SEI, the idealized true state (the pure state), the best approximated state based on the measurements and preparation procedure, or, possibly before running the experiment, the mixed state? From my limited and scattered reading, I get the impression SEI proponents would pick between the first two.

 

[vanhees71]

It sounds somewhat strange to me to say: "The microscopic system is prepared in the quantum state, but only if I don't know yet the precise settings of some measurements I plan to do later on this quantum state".

That's not what I'm saying. I've not even a clue, what you mean by this.

Given the interpretation of the state as formally describing a preparation of a single system, this doesn't say anything about what I intend to measure on this system. In fact I'm completely free to measure any observable on this system, which defines what an observable is, i.e., it is some quantified property of the system (like the position, momentum, and so on of a (massive) particle).

 

[martinbn]

I'm not sure that the difference is relevant. Let us consider a classical analogy, in which preparation is a coin flipping. The flipping prepares the state
$$S=[p({\rm heads})=1/2,p({\rm tails})=1/2]$$
where $p$ is the probability. What is the distinction between the following claims?
1. We consider an ensemble of coins, each prepared in the state $S$.
2. We consider an ensemble $S$ of similarly prepared coins.
3. We consider a class of preparation procedures $S$.

The point is that in QM there are other possible interpretations of the state. For example claim 4) the state describes the full ensamble only, not the individual systems.

 

[gentzen]

That's not what I'm saying. I've not even a clue, what you mean by this.

I explained why I gave a thumbs-up to Morbert's comment.

Given the interpretation of the state as formally describing a preparation of a single system, this doesn't say anything about what I intend to measure on this system.

The degrees of freedom and the Hilbert space you choose to describe them says something about what you intent to measure on the system.

 

[PeterDonis]

I want here to formulate the SEI in a clear, simple and self-consistent way.

Is this not done to your satisfaction anywhere in the literature? Ballentine has been a popular topic of discussion here, but it's not the only source that expounds a statistical or ensemble interpretation.

 

[Demystifier]

Is this not done to your satisfaction anywhere in the literature? Ballentine has been a popular topic of discussion here, but it's not the only source that expounds a statistical or ensemble interpretation.

It's probably written somewhere in a way I would be satisfied, but I don't know where exactly. In any case, I wanted to write it by myself, in a way I would like someone explained it to me.

 

[Demystifier]

The point is that in QM there are other possible interpretations of the state. For example claim 4) the state describes the full ensamble only, not the individual systems.

Do you think that 4) makes sense in the classical case of coin flipping?

 

[martinbn]

It's probably written somewhere in a way I would be satisfied, but I don't know where exactly. In any case, I wanted to write it by myself, in a way I would like someone explained it to me.

if you are not following any source how do you know that you are writing an exposition of the ensemble interpretation? And not just some other interpretation.

 

[PeterDonis]

I wanted to write it by myself, in a way I would like someone explained it to me.

Even in this forum, discussion is supposed to be based on what is published in the literature. So we should have some kind of grounding in what has been published on statistical or ensemble interpretations. For example:

What is common to all versions of SEI, except the facts above shared by all interpretations? All versions of SEI have in common the following:
- The collapse does not exist in the physical sense. (The information update is not considered physical.)
- Additional (hidden) variables (e.g. Bohm) are not considered.

Whether this is the case depends on what has actually been published. Otherwise we're just throwing around each person's personal definition of what "SEI" means, which is pointless (and against the forum rules).

 

[martinbn]

Do you think that 4) makes sense in the classical case of coin flipping?

No, because there are no true probabilities there. There is a state for the coin in each flip that fully describes its future.

 

[Demystifier]

No, because there are no true probabilities there. There is a state for the coin in each flip that fully describes its future.

OK, but Ballentine does not deny that something similar exists also behind the Born rule in QM. On the contrary, in several places he writes (I can make quotes if someone is interested) that it is a reasonable possibility. So the Ballentine's version of SEI at least does not involve a denial of it.

 

[vanhees71]

I explained why I gave a thumbs-up to Morbert's comment.The degrees of freedom and the Hilbert space you choose to describe them says something about what you intent to measure on the system.

No. The system defines the observables I can measure on it. It's formalized by an observable algebra realized by representations of this observable algebra on a separable Hilbert space. There is no restriction on what I can decide to measure by my decision on how I prepare the system I want to perform measurements on. That's often mixed up, leading to further confusion.

E.g., the standard Heisenberg uncertainty relation is a restriction on the preparability concerning a given pair of observables ($\Delta A \Delta B \geq |\langle [\hat{A},\hat{B}] \rangle|/2$) and not any restriction on how accurately I can measure the one or the other observable on the system.

 

[Demystifier]

Is the state we "associate" with the system in SEI, the idealized true state (the pure state), the best approximated state based on the measurements and preparation procedure, or, possibly before running the experiment, the mixed state? From my limited and scattered reading, I get the impression SEI proponents would pick between the first two.

In my view, all three are legitimate applications of SEI.

 

[martinbn]

OK, but Ballentine does not deny that something similar exists also behind the Born rule in QM. On the contrary, in several places he writes (I can make quotes if someone is interested) that it is a reasonable possibility. So the Ballentine's version of SEI at least does not involve a denial of it.

Yes, one can be agnostic about it, but you deny the other possibility.

 

[Demystifier]

Yes, one can be agnostic about it, but you deny the other possibility.

Can you be more explicit, what exactly do I deny? In the first post I wrote about several versions of SEI, is there a possibility that I didn't mention there?