Jürg Fröhlich on the deeper meaning of Quantum Mechanics

[akvadrako]

In my opinion, people going on about what is and is not science are basically doing philosophy of science. And badly. At the same time that they are saying how worthless philosophy is.

How can people limit themselves to discussing science when they don't agree what science is? Maybe there should be some reasonable guidelines about where the boundary is. And of course you're right — talking about what constitutes science is clearly in the realm of philosophy; science can't be self-defining.

 

[vanhees71]

No, there is no need for General or Special Relativity. They are important conceptually. You can have the same predictive power in an ad hoc theory that just uses a power series with empirically determined coefficients.

Well, that can only say someone who is inclined to philosophy rather than science, at the same time not knowing about the historical development of physics.

 

[DarMM]

Basically you can still replicate Wigner's friend even under a frequentist view.

No, because both Wigner and his friend only entertain subjective approximations of the objective situation. Subjectively everything is allowed. Even logical faults are subjectively permissible (and happen in real subjects quite frequently).

I've been thinking about this more and I still don't really see what is really changed by a Frequentist view.

So in a Frequentist/Ensemble view, somebody has loads of copies of a quantum system $S$. For a given property $A$ then $P(A_i)$ is the fraction of ensemble members with the $i$ value for $A$. $P(A_i|B_j)$ is the fraction of the subensemble with value $j$ for $B$ who have value $i$ for $A$ etc in the Classical case.

In the quantum case this might be measuring one property $A$ after another property $B$ and you could consider the combination of the measurement of $B$ + the original preparation to be a preparation itself. The difference between $P(A_i)$ and $P(A_i|B_j)$ is then just the differences in the proportions of the ensemble with property $A_i$ given the two preparations.

In the ensemble view the new thing about QM is that $P(A_i)$ and $P(A_i|B_j)$ mesh differently depending on whether you perform the $B$ measurement or not. If you do they are related as:
$$P(A_j) = \sum_{i}P(A_j|B_i)P(B_i)$$
if you do not it is (assuming the $B$ are SIC-POVMs say just to have a simple formula):
$$P(A_j) = (d + 1)\sum_{i}P(A_j|B_i)P(B_i) - 1$$

So including the $B$ measurement is not simply the filtering to a subensemble of the original ensemble but in fact must be considered the preparation of a new ensemble. That's a major difference in the ensembles of Classical and Quantum Physics.

So with all this out of the way I don't think much different is going on in Wigner's friend. The friend performs a measurement that Wigner cannot see. Since Wigner is utterly sealed off from the friend, part of the preparation is that he cannot see the friend's measurement. Hence over an ensemble of "friend labs" and measuring all the observables he has access to, which include some superobservables related to the lab's atomic structure, the statistics he finds are best described by the superposed state:
$$\frac{1}{\sqrt{2}}\left(|\uparrow, D_{\uparrow}, L_1\rangle + |\downarrow, D_{\downarrow}, L_2\rangle\right)$$

 

[stevendaryl]

Well, that can only say someone who is inclined to philosophy rather than science, at the same time not knowing about the historical development of physics.

Einstein was concerned with the concepts of physics, not just deriving the equations. You disagree with that?

I would say that what you're calling philosophy is actually physics. It's what people like Einstein did.

 

[A. Neumaier]

No, because both Wigner and his friend only entertain subjective approximations of the objective situation. Subjectively everything is allowed. Even logical faults are subjectively permissible (and happen in real subjects quite frequently).

the statistics he finds are best described

But subjective assignments of states do not need to be the best ones. Most real subjects assign suboptimal states to complex situations. To require best according to some criterion makes the observers fictitious.

Frequentists only have actual frequencies of actual (though - especially when infinite - not necessarily fully known) ensembles of actual systems, and the probabilities apply to these. Frequentists have a platonic truth about what is real, independent of the approximation they have to use when making numerical calculations. But what they know (and hence assign) are approximations of these probabilities only, based on somewhat subjective assumptions about the estimation procedure, and limited access to the data.

These only give approximate and sometimes quite erroneous states. Just like drawing a concrete circle gives only an approximation to the ideal platonic circle. Blackboard drawings of circles for illustrative purposes may even be quite poor approximations, often not even topolocially equivalent.

 

[DarMM]

How does that affect Wigner's friend though? Just replace the language with "he conjectures the lab is described by the superposed state" which he checks by looking at the statistics.

I mean I understand they are proposing true objective frequencies and in actual applications they only have estimates of that. However what has this got to do with or alter about Wigner's friend?

 

[A. Neumaier]

How does that affect Wigner's friend though? Just replace the language with "he conjectures the lab is described by the superposed state" which he checks by looking at the statistics.

I mean I understand they are proposing true objective frequencies and in actual applications they only have estimates of that. However what has this got to do with or alter about Wigner's friend?

From Wikipedia:

An observer W observes another observer F who performs a quantum measurement on a physical system. The two observers then formulate a statement about the physical system's state after the measurement according to the laws of quantum theory.

For a frequentist, there is no updating at all, since what Wigner and/or his friend conjecture about the state is completely irrelevant.

All subjective updating happens outside probability theory when some subject wants to estimate the true probabilities about which the theory is.

What counts is the true state of the full system, and what one gets depends on what is taken to be the full system. The true measurement results (independent of whether and/or how reliably they are observed by anyone) can be used to approximate the true state, and their statistics can be predicted by the true state. Thus we always have one state of the maximal system considered, the reduced states of the subsystems considered, and the actual measurement statistics, which conforms up to errors of $O(N^{-1/2})$ with the prediction from these by the quantum formalism.

On the other hand, the state is altered by the measurement, in a way depending on the details of the measurement. This new state is obtained by the dynamical law of the full system (including the measurement device); by unitary evolution if the full system is isolated. One can approximate this in various ways; choosing an approximation is already subjective.

To find out this state one cannot ask observers but must perform a theoretical calculation based on assumptions made about a model for the measurement, or do quantum tomography. The latter is possible only for the tiny subsystem measured and hence only gives the corresponding reduced state. moreover, all results obtained are approximate only. The former gives under appropriate assumptions in certain approximation schemes a von Neumann collapse - but only of the tiny subsystem. To extend this to an approximate state of the full system requires additional assumptions (max. entropy and the like). In any case, the assumptions and the approximations schemes employed determine which approximation to the state of the full system is obtained, and consistency requires that the subsystem's states are the corresponding reduced states. Thus everything is determined by these assumptions and approximations schemes, and is sub/objective to the extent these assumptions and approximations schemes are considered sub/objective.

What should be added by any updating argument? It affects neither these states nor the statistics.
It affects only how people with different subjective views of the matter approximate these states by their preferred estimation procedure (which may or may not be close to some axioms about rational behavior) form the part of the statistics available to them (plus anything they get to ''know'' from hearsay or pretend to ''know'').

Thus I think that the Wigner's friend puzzle makes no sense from a frequentist perspective. Like most of the foundational puzzles, the paradoxical features come from an overidealization of the true situation.

 

[vanhees71]

Einstein was concerned with the concepts of physics, not just deriving the equations. You disagree with that?

I would say that what you're calling philosophy is actually physics. It's what people like Einstein did.

Of course Einstein was concerned with the concepts of physics, as any theortician is. What I find is ridiculus is the claim that observable facts have played no role in creating is special and general theory of relativity or that these theory are irrelevant for phenomenology.

Indeed Einstein in his younger years did physics and created profound new insight in statistical mechanics, relativity, and quantum theory. Alone each of the 3+1 famous papers of 1905 is worth a Nobel prize. The irony is that his Nobel cerificate is the only one I know, which contains explicitly a statement, for what Einstein has not gotten the Nobel prize, namely for his theories of relativity. Nowadays we know that this was due to philosophical reasons. Rightfully Bergson, who is the culprit in this affair, is forgotten today, but Einstein is not. In later years Einstein got caught in his philosophical prejudices, ignoring the observed facts, and has not contributed much to physics from then on. He's an pardigmatic example for the danger of philosophy in the natural sciences ;-)).

 

[DarMM]

What should be added by any updating argument? It affects neither these states nor the statistics.
It affects only how people with different subjective views of the matter approximate these states by their preferred estimation procedure (which may or may not be close to some axioms about rational behavior) form the part of the statistics available to them (plus anything they get to ''know'' from hearsay or pretend to ''know'').

Thus I think that the Wigner's friend puzzle makes no sense from a frequentist perspective. Like most of the foundational puzzles, the paradoxical features come from an overidealization of the true situation

Sorry but I still don't really understand.

Let me try something more basic.

In the typical presentation the friend models the system as being in the state $\frac{1}{2}\left(|\uparrow\rangle + |\downarrow\rangle\right)$ upon measurement and obtaining the $\uparrow$ outcome he models later experiments with the state $|\uparrow\rangle$. In an ensemble view he could consider the original preparation and his measurement as a single new preparation.

However Wigner uses the superposed state I mentioned above.

Both of these assignments are from using the textbook treatment of QM.

You're saying if you are a frequentist something is wrong with this. What is it? Wigner's state assignment or the friends or both?

If this actually bears out and you are right I think you should consider writing something on this as I've never heard that frequentism alters details of Wigner's friend.

 

[Auto-Didact]

What I find is ridiculus is the claim that observable facts have played no role in creating is special and general theory of relativity or that these theory are irrelevant for phenomenology.

But that isn't the claim at all; the claim is that abstruse mathematics of a novel conceptualization of an observation, with the conceptualization generalizing far beyond the standard notion of how the original observation is perceived, all in order to reach a deeper understanding of things, was the driving force for discovering SR and GR. The key point to take away is that the mathematization only has to come/start once the conceptualization is correct; premature mathematization should be avoided at all costs!

As should be clear we don't need to go as far as Einstein, since as I said before many, many people before Newton had Kepler's data yet they were not actually using the methodology of theoretical physics as we know it today, simply because it wasn't explicitly invented yet by anyone. Newton however made an enormous conceptual leap, and then - being the best mathematician of his time - mathematicized his purely conceptual thoughts to the very extreme.

Conceptualizing and subsequently mathematicizing is de facto the original methodology of mathematical theoretical physics as invented by Newton and explained in detail in the beginning of his Principia. He did this purely to satisfy his own philosophical curiosity i.e. expand his own understanding of the world; he literally didn't even care to share his findings with anyone for years until goaded on by Halley et al.

This kind of extreme conceptualization is characteristic of some particular kinds of mathematicians and non-experimentally thinking physicists - who often become theoreticians - such as Feynman and Einstein as well. Looking back at the mathematicians we see it in Hamilton, Euler, Gauss, Riemann and Poincaré. It is arguable that this manner of thinking is rarely seen in a mathematician after Poincaré. Hadamard did a study on this phenomena and summarized it in a short book.

 

[A. Neumaier]

In the typical presentation the friend models the system as being in the state $\frac{1}{2}\left(|\uparrow\rangle + |\downarrow\rangle\right)$ upon measurement and obtaining the $\uparrow$ outcome he models later experiments with the state $|\uparrow\rangle$. In an ensemble view he could consider the original preparation and his measurement as a single new preparation.

However Wigner uses the superposed state I mentioned above.

Both of these assignments are from using the textbook treatment of QM.

You're saying if you are a frequentist something is wrong with this. What is it? Wigner's state assignment or the friends or both?

Frequentist arguments are about ensembles modeled in the probability space for the maximal domain of discourse fixed once and for all. Conditional probabilities are derived statements about well-specified subensembles.

There are no assignments in the frequentist's description, except arbitrary subjective approximations to the objective but unattainable truth.

But there is much more wrong with the Wigner's friend setting, and even with von Neumann's original simpler discussion of measurement:

1. Quantum mechanics as defined in the textbooks is a theory about a single time-dependent state, (for a quantum system, an ensemble of similarly prepared quantum systems, or the knowledge about a quantum system, depending on the interpretation). But unlike in frequentist probability theory, the traditional foundations make no claims at all about how the state of a subsystem is related to the state of the full system. This introduces a crucial element of ambiguity into the discussion of everything where a system together with a subsystem is considered in terms of their states. In this sense, the standard foundations (no matter in which description) of quantum mechanics (not the practice of quantum mechanics itself) is obviously incomplete.

2. Projective measurements are realistic only for states of very tiny systems, not for systems containing a detector. As long as the state remains in the microscopic domain where projective measurements may be realistic, Wigner friend arguments apply but prove nothing about the measurement situation. Therefore, Wigner's friend in the 2/3-state setting mentioned here is an irrelevant caricature.

But I better refrain from further discussing in detail interpretations which I don't think to be valid. I only get into a state where my mind is spinning - as in the time about 20 years ago when I seriously tried to make sense of other interpretation. At that time I failed because there were too many simplifications of things I deemed essential for understanding, and because the interpretations were at crucial points too vague to say clearly what they imply in a given context, so that each author used the interpretation in a different way. This experience of a few years fruitless, intense effort taught me to stay away from poorly defined interpretations.

A good interpretation must be able to spell out exactly what its terms mean (in the context of a sufficiently rich mathematical model) and how the terms may and may not be applied. That none of the traditional interpretations meets this criterion is the reason for the continued multitude of competing interpretations and modifications thereof. I hope that the thermal interpretation that I developed in response to the above insights will fare better in this respect. Everything is defined in sufficient precision to allow a precisely mathematical analysis, though the latter may be complex. At least there is no ambiguity about what the interpretation claims (apart from the undefined notion of uncertainty which however is familiar from all our knowledge).

 

[DarMM]

Frequentist arguments are about ensembles modeled in the probability space for the maximal domain of discourse fixed once and for all. Conditional probabilities are derived statements about well-specified subensembles

make no claims at all about how the state of a subsystem is related to the state of the full system

Well first of all Quantum Mechanics in its standard formulation doesn't have a single sample space, so there simply isn't "the probability space for the maximum domain of discourse". Also you are quite right that in general it doesn't have a clear relation between the states of subsystems and the states of full systems.

However I think a few things here.

I think it would be more accurate to characterize your position as stating that frequentism is not possible in the standard reading of QM, as opposed to Wigner's friend doesn't appear in a frequentist version of the standard approach. In other words you are rejecting a whole line of thinking related to the standard approach meaning there are so many elements of the standard way of thinking about the subject jettisoned that one never even gets near being able to formulate Wigner's friend.

Thus this is in a sense parallel to my post #80 that you responded to and in fact your parallel line of thinking is admitted in the last line.

However it is possible to give Wigner's friend a frequentist reading, but it's not one you would enjoy. Essentially Wigner and the friend are dealing with two separate ensembles. After the friend obtains a result, essentially preparing a different ensemble by obtaining the $\uparrow$ outcome, Wigner does not have a separate ensemble. He still retains the original one because he can still obtain outcomes compatible with the superposition when he looks at super-observables relating to the lab's subatomic structure. He is also capable of performing measurements that can completely rewire his friends material state as if it had followed from the $\downarrow$ outcome (this is easier to see in Spekkens Toy Model than in QM itself). Thus the ensemble of labs is still the same, even if a magical being filtered to only those friends who obtained $\uparrow$ and thus had the $|\uparrow\rangle$ ensemble of systems.

I agree that this can sound daft, but it is essentially the frequentist reading of the standard formalism.

As I said in #80 this is all more a problem with attempting to give a statistical reading in any sense to the standard way of doing QM. I don't think what you're doing here is an attempt to show that there is a valid frequentist reading of the standard formalism, but rather a rejection utterly of probability in the foundations. As such it is compatible with what I wrote.

 

[A. Neumaier]

I think it would be more accurate to characterize your position as stating that frequentism is not possible in the standard reading of QM

No. Once one selects a particular set of commuting observables of the complete system which has a state, one has a consistent setting in which one can do frequentist reasoning. Such a setting is fully consistent with the standard presentations of the foundations of quantum mechanics.

Of course quantum mechanics is applied in more complicated settings, but for these I believe the standard presentations of the foundations are already incomplete, so this is not special to the frequentist assumption.

 

[A. Neumaier]

Once one selects a particular set of commuting observables of the complete system which has a state, one has a consistent setting in which one can do frequentist reasoning. Such a setting is fully consistent with the standard presentations of the foundations of quantum mechanics.

Indeed, this is the way Born's original rule (applying to $p$, $H$, $J^2$, and $J_z$) and its various generalizations were conceived by the founders in 1926/1927, together with a transformation theory between interpretations in different sets of commuting variables. But the interpretation of the transformation theory at the time was murky and - at least until (including) Hilbert, von Neumann and Nordheim 1928 -, it was not recognized that the probability spaces belonging to these are incompatible.

Incompatible means that to get a sensible probabilistic interpretation one has to pick one of them, and mixing arguments about probabilities from different sets of commuting variables may easily lead to nonsense. At least for the (among the founding fathers of quantum mechanics unversally assumed) frequentist interpretation of probability. [Subjective probabilities may lead to nonsense anyway, since subjective assignments need not be consistent. The rational subjectivist pictured by de Finetti etc. is a theoretical fiction.]

 

[DarMM]

No. Once one selects a particular set of commuting observables of the complete system which has a state, one has a consistent setting in which one can do frequentist reasoning. Such a setting is fully consistent with the standard presentations of the foundations of quantum mechanics.

Of course quantum mechanics is applied in more complicated settings, but for these I believe the standard presentations of the foundations are already incomplete, so this is not special to the frequentist assumption.

Sorry of course one can give a context a frequentist reading, I meant frequentist probabilistic reading of the standard presentation "in general" for complicated settings as you mention in your final paragraph.

 

[A. Neumaier]

I don't think what you're doing here is an attempt to show that there is a valid frequentist reading of the standard formalism, but rather a rejection utterly of probability in the foundations.

In my own interpretation I indeed reject this, but when I put myself into the shoes of other interpretations I argue from their (vague or incomplete) premises and point out what their problems are.

 

[DarMM]

I should say I'm not arguing against Frequentist readings of the standard formalism. In fact I think how they present Wigner's friend in #152 is very interesting.

 

[A. Neumaier]

Sorry of course one can give a context a frequentist reading, I meant frequentist probabilistic reading of the standard presentation "in general" for complicated settings as you mention in your final paragraph.

There is no standard presentation "in general" for complicated settings.

The setting of the standard interpretation is (if it wants to be consistent) always a single experiment with a single time-dependent state, interpreted in terms of a single set of commuting variables, the ''whole experiment'' of Bohr [Science, New Ser. 111 (1950), 51--54].

Phrases often found in the physical literature as 'disturbance of phenomena by observation' or 'creation of physical attributes of objects by measurements' represent a use of words like 'phenomena' and 'observation' as well as 'attribute' and 'measurement' which is hardly compatible with common usage and practical definition and, therefore, is apt to cause confusion. As a more appropriate way of expression, one may strongly advocate limitation of the use of the word phenomenon to refer exclusively to observations obtained under specified circumstances, including an account of the whole experiment.

This is the simple setting I referred to. The complicated setting is quantum mechanics as it is actually used in practice. This does not follow the textbook foundations but is quite a different thing, mixing at the liberty of the interpreter (i.e., applied paper writer, not foundational theorist) incompatible pieces as is deemed necessary to get a sensible match of experiment and theory. Some of this is well described in the paper ''What is orthodox quantum mechanics?'' by Wallace.

orthodox QM, I am suggesting, consists of shifting between two different ways of understanding the quantum state according to context: interpreting quantum mechanics realistically in contexts where interference matters, and probabilistically in contexts where it does not. Obviously this is conceptually unsatisfactory (at least on any remotely realist construal of QM) — it is more a description of a practice than it is a stable interpretation.

 

[DarMM]

There is no standard presentation "in general" for complicated settings

I'd need to think about that, but regardless your second paragraph is compatible with what I said in #80.

 

[A. Neumaier]

I'd need to think about that, but regardless your second paragraph is compatible with what I said in #80.

Well, in your post #80 you claim that agents are essential in probability.

Any probability model contains the notion of an "agent" who "measures/learns" the value of something.

But this is not the case. In the book Probability via expectation by Peter Whittle, my favorite exposition of the frequentist approach, the only mention of 'agent' has quite a different meaning. (He contrasts the frequentist and the subjective point of view in Section 3.5.)

Your statement is valid for the Bayesian but not for the frequentist. For him there is a true model, and various (even incompatible) ways of estimating the parameters of this model. The agent who trusts one or the other of these estimators is always outside the theory. Statistical theory is only about the consistency of estimators, for a fixed but unknown model from a given class.

 

[vanhees71]

Frequentist arguments are about ensembles modeled in the probability space for the maximal domain of discourse fixed once and for all. Conditional probabilities are derived statements about well-specified subensembles.

There are no assignments in the frequentist's description, except arbitrary subjective approximations to the objective but unattainable truth.

But there is much more wrong with the Wigner's friend setting, and even with von Neumann's original simpler discussion of measurement:

1. Quantum mechanics as defined in the textbooks is a theory about a single time-dependent state, (for a quantum system, an ensemble of similarly prepared quantum systems, or the knowledge about a quantum system, depending on the interpretation). But unlike in frequentist probability theory, the traditional foundations make no claims at all about how the state of a subsystem is related to the state of the full system. This introduces a crucial element of ambiguity into the discussion of everything where a system together with a subsystem is considered in terms of their states. In this sense, the standard foundations (no matter in which description) of quantum mechanics (not the practice of quantum mechanics itself) is obviously incomplete.

This is not true. There's a standard rule, and it's derived from probability theory applied to probabilities given by Born's rule.

Let's assume you have a big system for which you want to consider two subsystems (you can generalize everything to more subsystems if necessary of course). It is completely at your choice which two subsystems you study. It's given by the physical question you want to address.

The system is described by some Hibert space $\mathcal{H}$, and the subsystems are defined by writing $\mathcal{H}=\mathcal{H}_1 \otimes \mathcal{H}_2$. A general state ket is thus a superposition of product state kets $|\psi_1 \rangle \otimes \psi_2 \rangle=:|\psi_1,\psi_2 \rangle$.

Then let $|u_j,v_k \rangle$ be a complete orthonormal set of observables. Then, if the big system is prepared in the state described by the Stat. Op. $\hat{\rho}$, then the Stat. Op. of the subsystem 1 is
$$\hat{\rho}_1=\mathrm{Tr}_2 \hat{\rho} = \sum_{j,k,l} |u_j \rangle \langle u_k,v_k|\hat{\rho}|u_l,v_k \rangle \langle u_l|.$$
This is the "reduced state", describing the state the subsystem 1 is prepared in, provided the big system is prepared in the state described by $\hat{\rho}$. Analogously you define
$$\hat{\rho}_2=\mathrm{Tr}_1 \hat{\rho}.$$
Note that $\hat{\rho}_1$ is a statistical operator operating in $\mathcal{H}_1$ and $\hat{\rho}_2$ in $\mathcal{H}_2$ as it should be.

 

[DarMM]

I suppose I'll have to read Whittle's book as I still don't understand a real sense in which these objections are about Bayesianism alone rather than probability in general. Especially for Wigner's friend where there simply isn't one true model.

 

[DarMM]

I guess a cleaner way to say what I intended in #80 is that when the quantum state is understood probabilistically (regardless of how one does this, Bayesian or Frequentist) I don't think there is a inconsistency or paradox related to the measurement as expressed in Wigner's friend. At least I have never had somebody clearly express the inconsistency within the context of a probabilistic view.

This is separate from such a view being problematic for other reasons.

 

[vanhees71]

There is no standard presentation "in general" for complicated settings.

The setting of the standard interpretation is (if it wants to be consistent) always a single experiment with a single time-dependent state, interpreted in terms of a single set of commuting variables, the ''whole experiment'' of Bohr [Science, New Ser. 111 (1950), 51--54].

Sure, what else should "experiment mean".

This is the simple setting I referred to. The complicated setting is quantum mechanics as it is actually used in practice. This does not follow the textbook foundations but is quite a different thing, mixing at the liberty of the interpreter (i.e., applied paper writer, not foundational theorist) incompatible pieces as is deemed necessary to get a sensible match of experiment and theory. Some of this is well described in the paper ''What is orthodox quantum mechanics?'' by Wallace.

It is true that in introductory textbooks first the ideal case of complete measurements are discussed, i.e., you prepare a system (in the introductory part of textbooks even restricted to pure states) and then measure one or more observables precisely. This is to start with the simplest case to set up the theory. You also do not start with symplectic manifolds, Lie derivatives and all that to teach classical Newtonian mechanics ;-)).

Later you extent the discussion to mixed states and all that. There's nothing incompatible in the standard interpretation (and I consider the collapse hypothesis as NOT part of the standard interpretation). You know Bohr's papers better than I, but as far as I know, Bohr never emphasized the collapse so much. The only inconsistent thing in some flavors of Copenhagen is the collapse hypothesis. It's inconsistent with the very construction of relativistic local (microcausal) QFT's, according to which no instantaneous collapse is possible since as any other interaction also the interaction of the measurement device with the measured object is local and thus it cannot lead to some causal effect with faster-than light signal propagation.

Particularly that holds true for (local!) measurements at far distances of parts of an entangled quantum systems (e.g., a typical Bell measurement of single-photon polarization on two polarization entangled photons). This is ensured by the formalism via the proof that local microcausal QFT fulfills the linked-cluster property of the S-matrix. So there cannot be any instantaneous collapse by construction.

 

[A. Neumaier]

I'd need to think about that, but regardless your second paragraph is compatible with what I said in #80.

well, there you claim:

Any probability model contains the notion of an "agent" who "measures/learns" the value of something.

But this is not the case for the frequentist. For him there is a true model, and various (even incompatible) ways of estimating the parameters of this model. The agent who trusts one or the other of these estimators is always outside the theory. Statistical theory is only about the consistency of estimators, for a fixed model.

This is not true. There's a standard rule, and it's derived from probability theory applied to probabilities given by Born's rule.

Let's assume you have a big system for which you want to consider two subsystems (you can generalize everything to more subsystems if necessary of course). [...]

$$\hat{\rho}_1=\mathrm{Tr}_2 \hat{\rho} = \sum_{j,k,l} |u_j \rangle \langle u_k,v_k|\hat{\rho}|u_l,v_k \rangle \langle u_l|.$$
This is the "reduced state", describing the state the subsystem 1 is prepared in

But this is a mixed state, not a state in the sense of the standard foundations, which say (in almost all textbooks) that the state of a system is given by a state vector. It is not even a classical mixture of such states but an improper mixture only - and is usually compatible with all possible state vectors for the subsystem.

Of course I know that one can patch the standard foundations to make it work with a wider scope, and I indicated this in the comments to the 7 basic rules Insight article (given in the above link). Then your construction is valid. But one needs to patch quite a lot, and must undo along the way some of the damage introduced by the standard foundations.

 

[DarMM]

But this is a mixed state, not a state in the sense of the standard foundations, which say (in almost all textbooks) that the state of a system is given by a state vector. It is not even a classical mixture of such states but an improper mixture only - and is usually compatible with all possible state vectors for the subsystem

I think maybe I'm not sure of what is meant by "standard" here as I would have never seen mixed states as outside standard QM. So perhaps what you and Wallace say applied to this other "standard" view.

Maybe it's like computer languages with several competing standards!

 

[A. Neumaier]

for Wigner's friend where there simply isn't one true model.

In a frequentist (objective) setting, the true model is always that of the biggest, most encompassing system, of which the others are subsystems. There cannot be two objective truths about this system, and any truth about a subsystem must be a truth about the big system.

 

[vanhees71]

No, a state is not necessarily a pure state. How do you come to this conclusion? As I said, usually in the beginning of QT textbooks one discusses only pure states for simplicity. This is for didactical reasons only.

However, as the example of entangled subsystems show, the state concept is utterly incomplete if you stop at that level: Even if the big system is prepared in a pure state, any subsystem is not in a pure state, according to the above definition (which in my opinion is the only definition that makes sense in view of the probabilistic interpretation, i.e., is in accordance with the usual axioms of probability theory).

Again you insist on something, some strange "standard representation" would claim, which however is not the case!

The final definition of the state is that in the formalism it's described as a statistical operator. The pure states are special cases, where the Stat. Op. becomes a projection operator. That's the solution of this apparent "problem" of the "standard formalism". BTW that's the reason why I insisted so much on this point when discussing your Insight article "7 Rules of Quantum Mechanics". In the final version that's while it's carefully written that PURE states are represented by a state ket. Unfortunately it has not been said that in general you NEED mixed states and that the most complete and correct description is with a Stat. Op. rather than a "state ket" (or better a unit ray).

 

[DarMM]

Genuinely the discussion has become hard for me to follow, especially with different notions of "standard" QM. Like @vanhees71 I would have thought standard "statistical" QM has mixed states. What you seem to call standard I would have just have thought of as QM as presented in a simple form early on in some textbooks.

This "undergraduate" form of QM probably does have the inconsistency discussed, because if you have only pure states as valid physical states and from Wigner's perspective the friends device is necessarily a mixed state, then that is an inconsistency.

However if you have mixed states as physical states and all the stuff from modern quantum theory, I don't think there is an inconsistency in the statistical view.

I'll read Whittle's book.

 

[vanhees71]

Of course, Standard QM has mixed states. Concerning Wigner's friend the problem seems to me that only strange thought expreiments are thought about (like this Frauchinger paper, which in my opinion simply makes assumptions that are not compatible with QT to begin with, particularly the assumption of an almighty super-observer who can observe other observers, their lab, and the measured system without disturbing the state of the whole or the subsystems, assuming that he can have incompatible observables all determined in one state, etc.).

There's no inconsistentcy with the statistical view.

 

[DarMM]

Of course, Standard QM has mixed states. Concerning Wigner's friend the problem seems to me that only strange thought expreiments are thought about (like this Frauchinger paper, which in my opinion simply makes assumptions that are not compatible with QT to begin with, particularly the assumption of an almighty super-observer who can observe other observers, their lab, and the measured system without disturbing the state of the whole or the subsystems, assuming that he can have incompatible observables all determined in one state, etc.).

There's no inconsistentcy with the statistical view.

This is close to what I think. I suspect superobservers might be impossible, with calculations by Omnes suggesting something along these lines. I've often wonder is there a problem with reversability of measurements and relativity as you enter into a strange "relativity of correlations" as discussed here in a criticism of Frauchiger-Renner:https://arxiv.org/abs/1901.10331

Basically if reversal occurs after two spacelike separated events $a$ and $b$ they have a correlation, but if it occurs between them in time they are uncorrelated. Of course time ordering depends on the reference frame so you seem to have no clear correlation between the events.

So I'm not really sure superobservers with all these powers make sense.

 

[vanhees71]

The very point is that time ordereing, as far as it is relevant for the S-matrix, does not depend on the reference frame. That's the reason why theories of interacting tachyons are not working, and only massive and massless fields occur in the Standard Model of elementary particle physics.

It's very carefully and well explained in

Weinberg, QT of Fields, Vol. 1

 

[DarMM]

Oh the paper above isn't about scattering, it's related to Bell type tests so no S-matrix. I've read all of Weinberg, you mean Chapters 2 and 3 I assume.

 

[A. Neumaier]

I would have never seen mixed states as outside standard QM.

It is not outside standard QM.

But the standard view is that mixed states are proper mixtures, i.e., classical mixtures of pure states, needed to model the uncertainty of not knowing exactly which pure state a system is in.

What you seem to call standard I would have just have thought of as QM as presented in a simple form early on in some textbooks.

@vanhees71 introduces mixtures in this way in Chapter 2 of his lecture notes on statistical physics, without caveats, although this is not an introductory quantum mechanics book.

• The state of a quantum system is described completely by a ray in a Hilbert space [p.19]
• In general, for example if we like to describe macroscopic systems with quantum mechanics, we do not know the state of the system exactly. In this case we can describe the system by a statistical operator ρ [...] It is chosen such that it is consistent with the knowledge about the system we have and contains no more information than one really has.

Any perceptive reader will interpret this as that the exact state of the system is a ray, but because we don't know it exactly we replace the exact state by an approximate state given by a density operator, explicitly given later in (2.2.5) as a classical mixture of eigenstates (that we did not but could in principle have measured to get complete knowledge) based on a Bayesian argument of classical probability and incomplete knowledge:

[p.27:] it seems to be sensible to try a description of the situation in terms of probability theory on grounds of the known information. [...] We do not know which will be the state the system is in completely and thus we can not know in which state it will go when measuring

[p.29:] we have to determine the statistical operator with the properties (2.2.11-2.2.13) at an initial
time which fulfills Jaynes’ principle of least prejudice from (1.6.17-1.6.18)

[p.16] how to determine the distribution without simulating more knowledge about the system than we have really about it. Thus we need a concept for preventing prejudices hidden in the wrong choice of a probability distribution. The idea is to define a measure for the missing information about the system provided we define a probability distribution about the outcome of experiments on the system. Clearly this has to be defined relative to the complete knowledge about the system.

This is fully consistent with how Landau and Lifschitz introduce the density operator on pp.16-18 of
their Course of Theoretical Physics (Vol. 3: Quantum mechanics, 3rd ed., 1977), confirming this interpretation.

The quantum-mechanical description based on an incomplete set of data concerning the system is effected by means of what is called a density matrix [...] The incompleteness of the description lies in the fact that the results of various kinds of measurement which can be predicted with a certain probability from a knowledge of the density matrix might be predictable with greater or even complete certainty from a complete set of data for the system, from which its wave function could be derived. [...] The change from the complete to the incomplete quantum-mechanical description of the subsystem may be regarded as a kind of averaging over its various $\psi$ states. [...] The averaging by means of the statistical matrix according to (5.4) has
a twofold nature. It comprises both the averaging due to the probabilistic nature of the quantum description (even when as complete as possible) and the statistical averaging necessitated by the incompleteness of our information concerning the object considered. For a pure state only the first averaging remains, but in statistical cases both types of averaging are always present. It must be borne in mind, however, that these constituents cannot be separated; the whole averaging procedure is carried out as a single operation,and cannot be represented as the result of successive averagings, one purely quantum-mechanical and the other purely statistical.

Thus according to Hendrik van Hees, backed up by the authority of Landau and Lifschitz, the only reason one uses a density matrix is because one lacks the complete information about the true, pure state of the system and hence needs to average over different such states.

Given this, it is illegitimate to interpret improper mixtures obtained for a subsystem through a reduction process in this way - it simply has no physical interpretation in the terms in which the density operator was introduced. Thus @vanhees71 (aka Hendrik van Hees) is inconsistent; he first teaches a childhood fable and later says (as in the above posts #161, #164, and #168) that it is not to be taken serious. But, being orthodox in his own eyes, he complains that I distort the story:

Again you insist on something, some strange "standard representation" would claim, which however is not the case!

Maybe it's like computer languages with several competing standards!

No, it is far worse. Each conscientious individual studying both the standard foundations and the practice of (more than textbook) quantum mechanics soon finds out that the foundations are sketchy only, and sees the need to fix it. They all fix it in their individual way, leading to a multitude (and frequently incompatible) of mutations of the standard.

Those like @vanhees71 and Englert (see post #14) , who found an amendment that they personally find consistent and agreeing with their knowledge about the use of quantum mechanics then think they have solved the problem, think of their version as the true orthodoxy and then claim that there is no measurement problem. But these othodoxies are usually mutually incompatible, and are often flawed in points their inventors did not thoroughly inspect for possible problems. This can be seen from how the proponents of some othodoxy speak about the tenets of other orthodoxies that don't conform to their own harmonization. (I can give plenty of examples...)

This is also the reason why there is a multitude of variants of the Copenhagen interpretation and a multitude of variants of the statistical interpretation.

 

[DarMM]

I always thought denity matrices can't be just classical ignorance because you'd expect $\mathcal{L}^{1}\left(\mathcal{H}\right)$ as opposed to $Tr\left(\mathcal{H}\right)$ to be their space. It always seemed to me if you were going to view the quantum state in a probabilistic way then pure states are states of maximal knowledge rather than the "true state". Of course this is a Bayesian way of seeing things. In a frequentist approach they'd be ensembles with minimal entropy. Either way they're not ignorance of the true pure state.

Well now I've learned even standard QM is hard to define. Does the confusion ever end in this subject?

I'm going back to simpler topics like Constructive Field Theory!