A Statistical ensemble interpretation done right

[vanhees71]

I don't understand, how you can build an ensemble to be in a given state, if you are not able to relate it to the preparation procedure on a single system. Otherwise I think @martinbn and I agree on the interpretation of the state, which as any probabilistic description refers to the properties of an ensemble, because you can experimentally test the probabilistic predictions only on an ensemble of equally prepared systems.

On the one hand the state thus is the description of a preparation procedure on a single system, but on the other the physical meaning of the state refers only to ensembles.

 

[martinbn]

I don't understand, how you can build an ensemble to be in a given state, if you are not able to relate it to the preparation procedure on a single system. Otherwise I think @martinbn and I agree on the interpretation of the state, which as any probabilistic description refers to the properties of an ensemble, because you can experimentally test the probabilistic predictions only on an ensemble of equally prepared systems.

On the one hand the state thus is the description of a preparation procedure on a single system, but on the other the physical meaning of the state refers only to ensembles.

You misunderstood me. I am not saying how things are, but how things are according to one interpretation (which is not even my proffered one). You on the other hand are saying how things are according to a different interpretation. But in addition you claim that this is the only possibility and you think that your statements are interpretation independent. That is where we disagree.

 

[Morbert]

I don't understand, how you can build an ensemble to be in a given state, if you are not able to relate it to the preparation procedure on a single system. Otherwise I think @martinbn and I agree on the interpretation of the state, which as any probabilistic description refers to the properties of an ensemble, because you can experimentally test the probabilistic predictions only on an ensemble of equally prepared systems.

On the one hand the state thus is the description of a preparation procedure on a single system, but on the other the physical meaning of the state refers only to ensembles.

A minimal statistical ensemble (MSE) proponent can certainly associate a state with a preparation procedure, but what motivates the MSE interpretation of a state as an infinite ensemble is the avoidance of pitfalls in thought processes. E.g. Both MSE and Copenhagen proponents can associate a state with a preparation procedure, but a Copenhagen proponent would happily proceed to think about measurement propensities in a single experimental run while an MSE proponent would not. They would only concern themselves with relative frequencies in several experimental runs and how they compare with rates in the infinite ensemble, as they are obliged to interpret the state as representing an infinite ensemble. They would hope to avoid ambiguities a Copenhagen proponent might be more vulnerable to.

 

[PeterDonis]

I don't understand, how you can build an ensemble to be in a given state, if you are not able to relate it to the preparation procedure on a single system.

The ensemble is not something that gets "built". It's an abstraction. Please read the passages from Ballentine that I referenced.

 

[vanhees71]

It's not an abstraction. At colliders you prepare particles at pretty well determined momenta and let them collide (in fact it's bunches of many particles, but that are details not too relevant for the general argument). This you repeat zillion of times to "collect enough statistics". Particularly for "rare probes" (as my beloved "dileptons and photons" in heavy-ion collisions) you need a lot of statitics. One point of the recent upgrade of the LHC was to get higher luminosities to get "the statistics" in a shorter time. Another great challenge was to also adapt the detectors to cope with these higher rates!

 

[martinbn]

It's not an abstraction.

You do understand that it can be an abstraction in one interpretation and not in another, don't you?

 

[PeterDonis]

It's not an abstraction.

Again, please read the passages from Ballentine that I referenced. As far as I know Ballentine's definition of "ensemble" is the usual one in ensemble interpretations of QM. And it is inconsistent with what you are saying.

 

[vanhees71]

Of course, from a mathemtical point of view you need infinite abstract ensembles to define probabilities. QT is, however a physical theory and it is successfully applied to real-world experiments, where you deal with real-world finite ensembles, including the corresponding systematical and statistical errors, which you have to carefully estimate as an experimentalist. Of course, Ballentine is right with his formal cautions about the meaning of probabilities, but there's no contradiction to apply probability theory to statistics of real-world experiments/observations. If this were the case, QT (or any other probabilistic theory, as classical statistical physics!) wouldn't have any connection to real-world experiments, and nobody would consider it as a physical theory.

 

[PeterDonis]

Of course, from a mathemtical point of view you need infinite abstract ensembles to define probabilities.

So now you agree that an ensemble is an abstract infinite set?

QT is, however a physical theory and it is successfully applied to real-world experiments, where you deal with real-world finite ensembles

No, you deal with real-world finite sets of actual experiments, which are not, as Ballentine says, correctly described as "ensembles".

 

[Fra]

Both MSE and Copenhagen proponents can associate a state with a preparation procedure, but a Copenhagen proponent would happily proceed to think about measurement propensities in a single experimental run while an MSE proponent would not.

Except for the wordings, to me the "propensitites" in a single experiment, follwing a preparation still only has a the same operational statisitical meaning.

Or would you suggest that the MSE propoent would refuse to even use the term prospensities, they call it instead probability? How does that make a difference? or are we talking I think about interpretation of "probability" and not just interpretation of QM?

/Fredrik

 

[Fra]

No, you deal with real-world finite sets of actual experiments, which are not, as Ballentine says, correctly described as "ensembles".

I think all would agree with this. The limit is a mathematical object only. But I didnt think this was the topic, it seemed too obvious.

I had another "fiction" in mind, namely wether there is a physical basis to the ensemble, set aside wether infinite or not. Here I see it physically encoded in the environment, as the tuned preparation setup and in that sense not fiction. I thought this was what vanhees tried to say?? to which i agree. The finite ensenble and the imperfect preparation procedure are not perfect, but not fiction?

/Fredrik

 

[gentzen]

Or would you suggest that the MSE propoent would refuse to even use the term prospensities, they call it instead probability? How does that make a difference? or are we talking I think about interpretation of "probability" and not just interpretation of QM?

I think all would agree with this. The limit is a mathematical object only. But I didnt think this was the topic, it seemed too obvious.

This and the previous discussion did not get off-track over physical disagreements, but over disagreements how the mathematical abstractions related to probabilities and ensembles are to be interpreted in concrete physical contexts.

 

[GarberMoisha]

QT is, however a physical theory and it is successfully applied to real-world experiments...

That would be a stretch. Remove the Born's rule and it becomes a non-physical theory altogether.

 

[PeterDonis]

wether there is a physical basis to the ensemble

The physical basis would be the preparation procedure, which involves actual physical equipment and processes. At least, that is how Ballentine explains it.

 

[PeterDonis]

The finite ensenble

There is no "finite ensemble". The term "ensemble" specifically means the abstract infinite set that has been described. It does not mean the actual finite set of actual systems we run the actual preparation procedure on.

 

[vanhees71]

The physical basis would be the preparation procedure, which involves actual physical equipment and processes. At least, that is how Ballentine explains it.

Exactly that's what I also say all the time. I don't see any contradiction between my view that real-world experiments use ensemles to test probabilistic predictions, including those of QT and what you quoted from Ballentine's intro chapter. It's well known that there's a difference between mathematical, idealized infinetely large ensembles and finite real-world empirical ensembles. Hypothesis testing thus is an entire subbranch of applied statistics.

 

[vanhees71]

There is no "finite ensemble". The term "ensemble" specifically means the abstract infinite set that has been described. It does not mean the actual finite set of actual systems we run the actual preparation procedure on.

Then just tell us which term you prefer for real-world collections of data, if we are not allowed to say "ensemble". It's really hard to discuss if one is not allowed to use standard terminology!

 

[vanhees71]

That would be a stretch. Remove the Born's rule and it becomes a non-physical theory altogether.

For me Born's rule is one of the fundamental postulates, and it's indeed crucial to make the theory to a physical theory.

 

[gentzen]

It's really hard to discuss if one is not allowed to use standard terminology!

I sort of agree, even so I don't want to voice an opinion on what is "standard". When I hint that "this and the previous discussion got off-track" in my opinion, it is because we started to disagree over so basic things (or talk past one another) that I don't even see which language we could use to reach a common understanding again.

 

[vanhees71]

Ballentine's book is very clear, as is Peres's. When I talk to experimentalists they all understand under "ensembles" their millions of repetitions of scattering experiments taken with real-world detectors (measurement) using particles from an accelerator (preparation).

 

[lodbrok]

The quantum state is the formal description of a preparation procedure on a single system. In this sense the quantum state refers to the single system. The meaning of the quantum state is, of course, entirely statistical

Focusing on the second part first -- statistics. You can generate statistics by (a) measuring a million different single systems once each or by (b) measuring the same single system a million different times (not always possible in practice, but in principle). Your measured statistics at CERN are of type (a), but by saying the system you are measuring is "an electron," you imply that it is of type (b).

The distinction you make in post #62 is not the appropriate one. It isn't between one million CERNS measuring one electron each vs one CERN measuring a million electrons. Rather, it's between one CERN measuring a single electron one million times vs one CERN measuring one million different similarly prepared electrons. Therefore, it is a fact that your measurements refer to an ensemble of similarly prepared elections or to the preparation procedure. To say the system being measured is "an electron" requires additional justification.

You are obviously free to interpret the QM state as referring to "an electron" rather than "an ensemble of similarly prepared electrons" (assuming you agree there's a difference between the two). That is not the issue. But then, how do you link the theory to the experiments? Only one of those "interpretations" is directly consistent with what you measure; you always measure an ensemble of similarly prepared electrons. Though each individual detector click refers to a particular electron, what justifies you putting all the results together, doing statistics on them, and then ascribing the resulting statistics to "an electron"? You will never ascribe the average height of people in a population to "an individual." Why do it here?

I guess I'm suggesting that you may have multiple ways of interpreting the math but only one way of interpreting the experiment.

 

[PeterDonis]

It's well known that there's a difference between mathematical, idealized infinetely large ensembles and finite real-world empirical ensembles.

My point is that Ballentine (who, I think, is a reasonable representative example of the literature) does not use the word "ensemble" at all to describe the latter (the "finite real-world" sets of systems). There might be some other standard term that is used for that, but it is not "ensemble". And we should be using terms the way they are used in the literature.

 

[PeterDonis]

Then just tell us which term you prefer for real-world collections of data, if we are not allowed to say "ensemble". It's really hard to discuss if one is not allowed to use standard terminology!

I am disputing whether "ensemble" is "standard terminology" for real-world collections of data, at least not in QM. That is why I gave Ballentine as a counter-example. I do not know another standard term for real-world collections of data. I just know that, at least according to Ballentine, "ensemble" is not that term.

 

[PeterDonis]

Ballentine's book is very clear

Not if you are claiming he uses "ensemble" to mean finite real-world collections of data. I have already given specific quotes to show that he does not.

When I talk to experimentalists they all understand under "ensembles" their millions of repetitions of scattering experiments taken with real-world detectors (measurement) using particles from an accelerator (preparation).

Then perhaps there is a difference of terminology between different parts of the physics community in this regard. Is there a standard reference that experimentalists use for such terminology?

 

[vanhees71]

I have read what you quoted. Ballentine's book is not the bible, you have to follow literarly. QT is a successful physical theory and not an empty mathematical scheme. It is also clear that any measurement has its statistical and systematical errors, and you never literary measure probabilities but statistical approximations to them. That's well understood by any serious experimentalist.

 

[Fra]

In fields which do real signal processing (not just symbolic math) of data where the patterns are masked by high variability, ensembles and ensemble averaging is indeed a standard term, in the signal processing options in various softwares this is also how it's called. Wether you acquire the data from multiple "objects/subjects" or from repeated sampling on the same object the signal processing terminology is the same, and these are of course never abstractions or infinite. You get the average and statistical variations.

(To decompose a mixed ensemble into say standard distributiosn are handled by other algorithms as a separate issue.)

https://www.sciencedirect.com/topics/engineering/ensemble-average

matlab uses the term, so does software for neuroscience i used.

/Fredrik

 

[gentzen]

When I talk to experimentalists they all understand under "ensembles" their millions of repetitions of scattering experiments taken with real-world detectors (measurement) using particles from an accelerator (preparation).

Then perhaps there is a difference of terminology between different parts of the physics community in this regard. Is there a standard reference that experimentalists use for such terminology?

I know that you are asking vanhees71 for a "reference of experimentalists terminology". I cannot help with that. But maybe I can help with terminology common among (applied) statisticians:

I'm reading The art of statistics by Spiegelhalter now. Fun book which stresses conceptual aspects of statistics and data analysis.

That is indeed a very good and readable book, and it contains a 24-page glossary and a 7-page index. Here are some examples from the glossary:
aleatory uncertainty: unavoidable unpredictability about the future, also known as chance, randomness, luck and so on.
epistemic uncertainty: lack of knowledge about facts, numbers or scientific hypotheses.
probability: the formal mathematical expression of uncertainty. Let P(A) be ...
probability distribution: a generic term for a mathematical expression of the chance of a random variable taking on particular values. A random variable X has ...
sampling distribution: the probability distribution of a statistic.
statistic: a meaningful number derived from a set of data.
Neither the glossary nor the index contain the word "ensemble," and the word "empirical" only appears in the index as:
empirical distribution 197, 404
sampling distribution 197, 404

My feeling would have been that the actual finite experimental ensemble would be called "empirical ensemble," but maybe this usage of "empirical" was only common 25 years ago in Germany, when I had statistics and probability courses at university. But "sampling ensemble" would not sound nice in my ears. But even back then, we were told that of course the clarifying word "empirical" would often be omitted, but that we should still try to distinguish between the formal mathematical expression and the actual empirical data.

 

[WernerQH]

For me Born's rule is one of the fundamental postulates, and it's indeed crucial to make the theory to a physical theory.

I agree. Unfortunately Born's rule seems inextricably linked to the term "measurement" (and, in consequence, decoherence, macroscopic apparatus, etc.).

Measurements are nothing else than interactions between the system and a measurement device, which obey the same "quantum rules" as any other interaction.

In the paper we describe of course an open system, because we couple the particle to a heat bath, but the underlying equations are quantum time evolution

Excellent. Your paper shows that (using the Schwinger-Keldysh formalism) one can write down expressions for observable quantities without the need to talk about measurements and (god forbid!) collapsing wavefunctions.

 

[vanhees71]

All of physics is about observation and thus measurement. A physical theory has to predict what's measured from some fundamental principles, and that does all of theoretical physics since Newton, and so does Q(F)T in the 21st century.

 

[WernerQH]

All of physics is about observation and thus measurement

Sounds tautological. And beside the point. Obviously you disagree with John Bell ("Against Measurement").

 

[vanhees71]

I agree with John Bell's physics but I don't agree with his philosophy ;-).

 

[A. Neumaier]

Then just tell us which term you prefer for real-world collections of data, if we are not allowed to say "ensemble". It's really hard to discuss if one is not allowed to use standard terminology!

I do not know another standard term for real-world collections of data.

That's called a statistical sample.

 

[PeterDonis]

Ballentine's book is not the bible

I understand that, but it is still a published reference. If there is indeed a difference of terminology in this area in different parts of the relevant scientific community, you should give published references for your preferred usage of the term "ensemble", as I have asked you.

 

[vanhees71]

I don't have specific references. Just tell me, how I'm allowed to call collected data of real-world experiments, that are taken from "equally prepared systems" in this forum, so that we can discuss the real issues rather than pure semantics. I only said, it's very hard to get really to the interesting core of things, when you are not allowed to use the standard terminology everybody uses when discussing physics among physicists. You can really get into a state of "rigor mortis" being too rigorous! At least in my community we often call collections of experimental data as data taken from "ensembles" or one simply says things like: "We have collected data about 1 Mio. $J/\psi$ decays in pp collisions at the LHC" etc. That of course refers to many pp collisions and in this sense to an "ensemble" described in QFT by asymptotic in states (two protons colliding head on at a given center-momentum energy).

 

[PeterDonis]

I don't have specific references.

Then I'm very confused about how you can be so confident about your usage, with no reference to back it up. I have given a reference to back up mine.

Just tell me, how I'm allowed to call collected data of real-world experiments, that are taken from "equally prepared systems" in this forum

@A. Neumaier gave a term for it in post #132.

 

[PeterDonis]

so that we can discuss the real issues rather than pure semantics

We can only discuss real issues if we agree on the terminology to describe them. If a particular term has multiple, incompatible usages, we don't have such agreement. That appears to be the case for the term "ensemble".

I agree it's unfortunate that there are such cases, but that doesn't change the fact that there are. When such a case arises, it would be very helpful if you would not continue to use the disputed term in your preferred way when it has already been pointed out that that usage is in dispute, without even acknowledging that there is such a dispute. We could have gotten to this point quite a few posts ago if you had stated earlier what you stated in the first sentence of post #134.

 

[vanhees71]

Just tell me, how I should call, real-lab measurement results, which are evaluated by statistical means to compare them to the predicted probabilities. I try to learn the special "Physics Forum's terminology" defined by @PeterDonis. As I said, I like to discuss physics and not get lost in semantics and linguistics.

 

[weirdoguy]

I try to learn the special "Physics Forum's terminology" defined by @PeterDonis.

This is just ridiculous, @PeterDonis backed up his usage, and you are just quibbling all over the place. Do you even read what others are saying?

 

[GarberMoisha]

Could be a language issue. There is plenty of loose language being used in the physics comminities to compound an already brewing translation difficulty from other languages to English.

 

[vanhees71]

This is just ridiculous, @PeterDonis backed up his usage, and you are just quibbling all over the place. Do you even read what others are saying?

Yes I do! Just tell me, how you want me to phrase things, so that we can avoid this nonsensical debates about semantics!

I'm well aware that empirical finite measurement results is not the same as an abstract mathemtical ensemble!

 

[PeterDonis]

Just tell me, how I should call, real-lab measurement results, which are evaluated by statistical means to compare them to the predicted probabilities.

I have already responded to this.

 

[PeterDonis]

Yes I do!

No, you don't. You have repeated a question that has already been answered, as I posted just now.

Just tell me, how you want me to phrase things, so that we can avoid this nonsensical debates about semantics!

Again, this has already been answered. But apparently you are not reading the answers, although you seem to believe that you are.

 

[vanhees71]

I have overlooked the answer then. Sorry for that. Can you just tell me, which word(s) to use, so that we can come back to a constructive discussion?

 

[PeterDonis]

I have overlooked the answer then. Sorry for that.

Apologies for overlooking something mean nothing if you don't go back and do the thing you overlooked.

Can you just tell me, which word(s) to use, so that we can come back to a constructive discussion?

Can you just read the answer that has already been given in this thread, instead of making people repeat themselves?

 

[vanhees71]

Apologies for overlooking something mean nothing if you don't go back and do the thing you overlooked.Can you just read the answer that has already been given in this thread, instead of making people repeat themselves?

I have done so right now. You haven't answer my question explicitly. So once more: Which term do you want me to use when I refer to a real-lab table of measurement results on a (necessarily finite) set of "equally prepared" systems. In everyday used language among my colleagues that's often called an "ensemble". It's of course only an approximation to a mathematical abstract infinitely large ensemble. I admitted this right from the beginning.

 

[PeterDonis]

You haven't answer my question explicitly.

You're right: I, personally, didn't. I just pointed you to a post by another person, @A. Neumaier, who did. Please go and read what he posted and respond to it, instead of quibbling.

 

[vanhees71]

I've read it. He calls it "statistical sample". Fine! Can we FINALLY get on-topic again now?

 

[A. Neumaier]

I've read it. He calls it "statistical sample". Fine! Can we FINALLY get on-topic again now?

A statistical sample is the standard name for a finite, empirically observed approximation to the theoretical ensemble characterized classically by a probability measure and in quantum mechanics by a mixed (or in special cases pure) state.

From p.26 of Volume 5 of the treatise by Landau and Lifschitz:

we may consider simultaneously, in a formal manner, a very large (in the limit infinite) number of exactly identical subsystems, [footnote:] Such an imaginary set of identical systems is usually called a statistical ensemble.

Thus the statistical ensemble is a merely imagined idealization, a formal construct, whereas a statistical sample consists of the results of actual observations.

Loosely speaking (as often done by physicists), one can use both terms interchangeably, but when interpretation details matter, one should distinguish the two.

 

[vanhees71]

The point is that we got totally off topic although it's fully clear, how I meant the term "ensemble" here. Thanks for giving the simple answer. I try to call it "statisical sample" in this forum from now on, and we can get back to the really interesting discussions.

 

[PeterDonis]

we can get back to the really interesting discussions

What interesting points do you think we haven't covered yet?