A Statistical ensemble interpretation done right

[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?

 

[gentzen]

No. The system defines the observables I can measure on it.

That is just playing with words. What defines what you consider to be the system? Good, you may prefer the word "the system" over my "The degrees of freedom and the Hilbert space you choose", but in the end you have chosen what you consider to be your system.

 

[Haborix]

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.

The "my" should have been the royal "our." I didn't mean anything technical by bookkeeping. It was my first try at getting at the different states you might think about as you go about the preparation procedure and eventual experiments. How each of the preparation procedure and experimental measurements would change the knowledge of the state. I think I made too much of the use of the word "associate" when I initially commented, or at least I latched onto it as a bad proxy for what I was trying to get at.

 

[vanhees71]

The system is what I'm investigating, e.g., an electron.

In non-relativistic physics the observable algebra of an electron is generated by the position, momentum and the spin observables, $\vec{x}$, $\vec{p}$, and $\vec{s}$. Each of these observables is represented by an essentially self-adjoint operator $\hat{\vec{x}}$, $\hat{\vec{p}}$, and $\hat{\vec{s}}$. The Hilbert space can be represented by Pauli-spinor valued wave functions, $\psi_{\sigma}(\vec{x})$, choosing the set $\vec{x}$ and $s_z$ as a complete minimal set of compatible observables, defining a (generalized) basis $|\vec{x},\sigma \rangle$. For $|\psi \rangle$ then you have the wave function $\psi_{\sigma}(\vec{x}) = \langle \vec{x},\sigma|\psi \rangle$. The scalar product is defined as
$$\langle \psi_1|\psi_2 \rangle=\int_{\mathbb{R}^3} \sum_{\sigma=\pm 1/2} \psi_{1 \sigma}^*(\vec{x}) \psi_{2 \sigma}(x).$$

A state of the electron, i.e., a preparation procedure, is described by a statistical operator $\hat{\rho}$, which is a positive semidefinite operator obeying $\mathrm{Tr} \hat{\rho}=1$. Having prepared the system in this state, you can measure any observable $A$, represented by a self-adjoint operator $\hat{A}=\hat{A}(\hat{\vec{x}},\hat{\vec{p}},\hat{\vec{s}})$, which defines a (generalized) complete orthonormal set of eigen vectors $|a,\alpha \rangle$. The possible values the observable can take are the eigenvalues $a$, and $\alpha$ labels the orthornomal vectors spanning the subspace of eigenvectors of this eigenvalue.

In the minimal statistical interpretation the meaning of the system being prepared in the state $\hat{\rho}$ is (exclusively!) that the probability (density) for finding the value $a$, when measuring $A$ is given by
$$P(a|\hat{\rho})=\sum_{\alpha} \langle a,\alpha|\hat{\rho}|a,\alpha \rangle,$$
where the sum over $\alpha$ can also contain integrals if there are continuous parts in the parameters written as $\alpha$. You can think of $\alpha$ given by the common eigenvalues making $\hat{A}$ and some other observables $\hat{A}_k$ ($k \in \{1,2,3\}$) with eigenvalues $\alpha=(\alpha_1,\ldots,\alpha_{3})$ a complete set of independent compatible observables.

Since these probabilities can only be experimentally verified by preparing a large ensemble of electrons, each being prepared in the state $\hat{\rho}$ through a corresponding appropriate preparation procedure, concerning the outcome of measurements on an arbitrary observable, the state only describes such an ensemble. For the single electron prepared in this state the observable $A$ only takes a determined value if there is one eigenvalue $a_0$ of $\hat{A}$ for which $P(a_0|\hat{\rho})=1$, which implies that $P(a|\hat{\rho})=0$ for all eigenvalues $a \neq a_0$ of $\hat{A}$.

Last but not least it should be clear that a generalized eigenvector for an "eigenvalue" in the continuous part of the spectrum, is not normalizable and thus there's no state (preparation), for which the observable can take this determined value.

 

[vanhees71]

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.

Maybe

A. Peres, Quantum Theory: Concepts and Methods, Kluwer
Academic Publishers, New York, Boston, Dordrecht, London,
Moscow (2002).

 

[martinbn]

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?

You said that if the preparation procedure is known, then the state is associated with the individual system. If the preparation procedure is not known, then you need many systems to be able to figure out what the sate is. So for you the ensemble interpretation associates a state to an ensemble only because of lack of knowledge. You do not list the possibility that the preparation is fully known and yet the ensemble is in the corresponding state not each member. You also seem to think of the ensemble in a very practical way and not the abstract equivalence class of systems.

 

[Morbert]

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 $S$.

Only 2. frames the state as an ensemble of similarly prepared systems. I do not see this framing in 1. or 3.

 

[PeterDonis]

So for you the ensemble interpretation associates a state to an ensemble only because of lack of knowledge. You do not list the possibility that the preparation is fully known and yet the ensemble is in the corresponding state not each member.

Also not included is that in order to test the predictions of the theory, you need to make the same measurement on a large number of similarly prepared systems, because the theory predicts probabilities. That is how Ballentine justifies thinking of the state as referring to an ensemble (but note that this is an addition to what he calls the primary meaning of the state, which is that it describes the probabilities of measurement results associated with a given preparation procedure).

 

[gentzen]

The system is what I'm investigating, e.g., an electron.

You are just switching the focus to the next word. Now you prefer your "what I'm investigating" over my "what you intent to measure":

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

Or maybe you are unhappy because I also used the word "system" in that sentence.

But in the end, this is a fight over words, or perhaps about "how to talk about that stuff". It seems mostly unrelated to the physics.

In non-relativistic physics the observable algebra of an electron is generated by the position, momentum and the spin observables, ...

Not sure whether talking about physics is helpful at that point.

 

[Demystifier]

Only 2. frames the state as an ensemble of similarly prepared systems. I do not see this framing in 1. or 3.

But the difference is only in rhetoric, I don't see any substantial physical difference, at least in the classical case. Do you see a substantial physical difference? Or maybe, as an adherent of consistent histories, you see framing, i.e. rhetoric, as physical?

 

[PeterDonis]

I don't see any substantial physical difference, at least in the classical case.

Since we're talking about QM here, the classical case is irrelevant.

Do you see a substantial physical difference?

We are talking about interpretations of QM, which is a model, not reality. If there were a physical difference involved at all here, it would mean we would have different "interpretations" making different predictions for experimental results, which would make them not different interpretations of QM, but different theories.

In other words, the whole concept of an interpretation "done right", which is the phrase you chose in the title of this thread, cannot involve any physical difference; it is purely a matter of personal preference and opinion. Or "rhetoric", if you want to call it that. (The guidelines for this subforum explicitly recognize this.)

 

[Demystifier]

You said that if the preparation procedure is known, then the state is associated with the individual system. If the preparation procedure is not known, then you need many systems to be able to figure out what the sate is. So for you the ensemble interpretation associates a state to an ensemble only because of lack of knowledge. You do not list the possibility that the preparation is fully known and yet the ensemble is in the corresponding state not each member. You also seem to think of the ensemble in a very practical way and not the abstract equivalence class of systems.

Yes, you summarized it very well. I see SEI as a rather practical approach, it always seemed to me that SEI is an attempt to formulate QM with a minimal amount of philosophy. Now I am becoming aware that not everybody sees SEI that way.

 

[Demystifier]

Since we're talking about QM here, the classical case is irrelevant.

It's relevant, if we want to understand how exactly statistical aspects of QM differ from statistical aspects of classical physics. In my opinion, the former cannot be understood correctly before first understanding the latter.

We are talking about interpretations of QM, which is a model, not reality. If there were a physical difference involved at all here, it would mean we would have different "interpretations" making different predictions for experimental results, which would make them not different interpretations of QM, but different theories.

In the context of quantum interpretations, the word "physical" has a wider meaning than that. By "physical", realists often mean ontological, qbists often mean informational, and consistent-historians perhaps often mean something related to framing.

In other words, the whole concept of an interpretation "done right", which is the phrase you chose in the title of this thread, cannot involve any physical difference; it is purely a matter of personal preference and opinion. (The guidelines for this subforum explicitly recognize this.)

In the first post of this thread, my intention was to clearly separate the non-controversial parts of SEI, from its philosophical/interpretational controversial parts. Both parts need to be discussed, but in such a discussion one needs to be aware which is which, that's what I meant by "done right". It was not my intention to say which philosophical/interpretational version of SEI is "right".

 

[PeterDonis]

I see SEI as a rather practical approach, it always seemed to me that SEI is an attempt to formulate QM with a minimal amount of philosophy. Now I am becoming aware that not everybody sees SEI that way.

I think Ballentine does see his formulation of SEI that way. He explicitly links the interpretation directly to the practical fact that QM predicts probabilities and the only way to test those predictions is to run the same experiment on a large number of similarly prepared systems so you can do statistics. The reason he defines an ensemble using an abstraction is that the ensemble is a thing in the model, not reality: an ensemble as he defines it is part of the theoretical framework we use to make predictions, not the experimental framework we use to test them.

But apparently you don't agree with Ballentine's formulation or see it as he does. So I don't think there is even agreement on what "a minimal amount of philosophy" means.

 

[Demystifier]

But apparently you don't agree with Ballentine's formulation or see it as he does.

My impression (which could be wrong) is that my view of SEI is very similar to that of Ballentine, but that he sometimes uses wording that creates more confusion than clarity, so I tried to further simplify the explanation of SEI in a way which avoids confusion. Now it seems that I wasn't very successful in my intentions, but at least I tried.

 

[PeterDonis]

My impression (which could be wrong) is that my view of SEI is very similar to that of Ballentine

If that is the case, then I'm not sure why you had to ask this in your OP:

within SEI, does it make sense to say that we have one particle in the state $\ket{\psi}$?

Ballentine is quite clear that the answer to this is "no". I've already quoted several passages showing that.

he sometimes uses wording that creates more confusion than clarity

Some specific examples would help. The ones I gave, as I noted just now, seem to imply the opposite.

 

[Demystifier]

If that is the case, then I'm not sure why you had to ask this in your OP:

Ballentine is quite clear that the answer to this is "no". I've already quoted several passages showing that.

Some specific examples would help. The ones I gave, as I noted just now, seem to imply the opposite.

Here is one example. At page 207 of his book he writes:

"It is possible to prepare the lowest energy state of a system simply by waiting for the system to decay to its ground state."

He says "system", not "ensemble". To me, it looks as confirmation of my claim that in the case of known preparation he associates the state with a single system, not with an ensemble. But he is not perfectly clear and explicit about that, which I think creates confusion, so I wanted in this thread to make such things clear and explicit.

One should also have in mind what he says at page 46 (boldings are mine):

"The empirical content of a probability statement is revealed only in the
relative frequencies in a sequence of events that result from the same (or an
equivalent) state preparation procedure. Thus, although the primary definition of a state is the abstract set of probabilities for the various observables, it is also possible to associate a state with an ensemble of similarly prepared systems. However, it is important to remember that this ensemble is the conceptual infinite set of all such systems that may potentially result from the state preparation procedure, and not a concrete set of systems that coexist in space. In the example of the scattering experiment, the system is a single particle, and the ensemble is the conceptual set of replicas of one particle in its surroundings. The ensemble should not be confused with a beam of particles, which is another kind of (many-particle) system. Strictly speaking, the accelerating and collimating apparatus of the scattering experiment can be regarded as a preparation procedure for a one-particle state only if the density of the particle beam is so low that only one particle at a time is in flight between the accelerator and the detectors, and there are no correlations between successive particles."

Thus he makes clear that the ensemble interpretation of the state is not the only (in fact, not even the primary) interpretation of the state, and also that "system" is not the ensemble.

 

[Morbert]

But the difference is only in rhetoric, I don't see any substantial physical difference, at least in the classical case. Do you see a substantial physical difference? Or maybe, as an adherent of consistent histories, you see framing, i.e. rhetoric, as physical?

The difference would be the conceptualizations and intuitions invoked to reason about the single system. In all three statements, an ensemble is conceptualized, so they are quite similar, but only in 2. is this conceptualization insisted upon, in the sense that 1. and 3. imply we could discuss the state, divorced from the context of an ensemble. This is not necessarily a bad thing of course.

If by physical we mean real, there is no difference. What is real is the single system of interest.

As an aside: You could apply a statistical ensemble interpretation to the CH formalism. Given a set of histories, the probability of a history would be the relative frequency of that history in an ensemble of similarly prepared systems. Gell-Mann and Hartle went one step further and published an "extended probability ensemble decoherent histories" which embeds the real fine-grained history of a system in an ensemble of alternatives.

 

[Demystifier]

Gell-Mann and Hartle went one step further and published an "extended probability ensemble decoherent histories"

They lost me when they said that probability can be negative or larger than one.

 

[martinbn]

Yes, you summarized it very well. I see SEI as a rather practical approach, it always seemed to me that SEI is an attempt to formulate QM with a minimal amount of philosophy. Now I am becoming aware that not everybody sees SEI that way.

Then why did you start this thread?! You agreed with me, that you did not describe any interpretation of QM, and that what you wrote is not even specific to QM. So your view is that the ensemble interpretation is not really an interpretation of QM, it is just statistics. Now you have become aware that some people view the ensemble interpretation as an actual interprwtation!

 

[Demystifier]

Then why did you start this thread?!

See my post #47, the last paragraph. And also #49.

 

[martinbn]

Here is one example. At page 207 of his book he writes:

"It is possible to prepare the lowest energy state of a system simply by waiting for the system to decay to its ground state."

He says "system", not "ensemble". To me, it looks as confirmation of my claim that in the case of known preparation he associates the state with a single system, not with an ensemble. But he is not perfectly clear and explicit about that, which I think creates confusion, so I wanted in this thread to make such things clear and explicit.

He could have been more pedantic, but he probable thought it wouldnt cause a confusion.

One should also have in mind what he says at page 46 (boldings are mine):

"The empirical content of a probability statement is revealed only in the
relative frequencies in a sequence of events that result from the same (or an
equivalent) state preparation procedure. Thus, although the primary definition of a state is the abstract set of probabilities for the various observables, it is also possible to associate a state with an ensemble of similarly prepared systems. However, it is important to remember that this ensemble is the conceptual infinite set of all such systems that may potentially result from the state preparation procedure, and not a concrete set of systems that coexist in space. In the example of the scattering experiment, the system is a single particle, and the ensemble is the conceptual set of replicas of one particle in its surroundings. The ensemble should not be confused with a beam of particles, which is another kind of (many-particle) system. Strictly speaking, the accelerating and collimating apparatus of the scattering experiment can be regarded as a preparation procedure for a one-particle state only if the density of the particle beam is so low that only one particle at a time is in flight between the accelerator and the detectors, and there are no correlations between successive particles."

Thus he makes clear that the ensemble interpretation of the state is not the only (in fact, not even the primary) interpretation of the state, and also that "system" is not the ensemble.

Exactly! This show the difference between your view and his.

 

[martinbn]

See my post #47, the last paragraph. And also #49.

Yes, you say that, but it seems that you were cometely unaware of the interpretation itself!

 

[Demystifier]

Exactly! This show the difference between your view and his.

How? I see this as a confirmation that his view agrees with mine.

 

[vanhees71]

You are just switching the focus to the next word. Now you prefer your "what I'm investigating" over my "what you intent to measure":

By "investigating" I meant of course "doing an experiment".

Or maybe you are unhappy because I also used the word "system" in that sentence.

I'm unhappy about your formulation

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

because it seems to state a very common misconception about quantum mechanics. It seems as if you have an interpretation of the Heisenberg uncertainty relations in mind as if it would prevent from meausring the one or the other observable with arbitrary precision. This is, however, in no way what's implied by the uncertainty relation. I can always measure any observable of a system with as high a precision I want, and I'm not in any way restricted in the ability to choose to measure any observable of the system I like due to the state preparation.

The uncertainty relation also does not say anything about the disturbance of the system by measurement. It can't, because it's a fundamental principle derived without any reference to a specific measurement procedure, and how the system is disturbed by the interaction with the measurement apparatus of course depends on the specific device.

But in the end, this is a fight over words, or perhaps about "how to talk about that stuff". It seems mostly unrelated to the physics.

The above is of utmost significance for the correct interpretation of the formalism, and it doesn't in any way depend on the specific interpretation you prefer. It's one of the objective scientific properties of the theory.

 

[PeterDonis]

He says "system", not "ensemble". To me, it looks as confirmation of my claim that in the case of known preparation he associates the state with a single system

No, he associates the state with the preparation procedure that was used, exactly as he said in what you quoted from his p. 46. He uses "system" to refer to the thingie that comes out of the preparation procedure, precisely in order to distinguish that thingie from the preparation procedure and the abstract ensemble that results from it, which are what he says on p. 46 that the state describes.

Thus he makes clear that the ensemble interpretation of the state is not the only (in fact, not even the primary) interpretation of the state

I don't think so. See above.