A Jürg Fröhlich on the deeper meaning of Quantum Mechanics

[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!

 

[A. Neumaier]

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".

For someone who thinks that the state is associated with the observer (a subject) rather than the experiment (an object) there is no true state, only subjective assignments. But for a frequentist, the state contains true information about an ensemble of experiments. Or what else should distinguish the frequentist from the Bayesian?

In a frequentist approach they'd be ensembles with minimal entropy. Either way they're not ignorance of the true pure state.

Pure and mixed states are different ensembles, representing different statistics (if one could do experiments differentiating the two) and hence different objective realities. Only one of them is real.

For a 2-state system (polarized beams of light) one can easily differentiate between light prepared in an unpolarized state (true density matrix = 1/2 unit matrix) and a completely polarized state (true density matrix of rank 1), and - in the limit of an unrestricted number of experiments - one can find out the true state by quantum tomography.

On the other hand, a consequent Bayesian who doesn't know how the light is prepared and thinks to be entitled by Jaynes or de Finetti to treat his complete lack of knowledge in terms of the natural noninformative prior will assign to both cases the same density matrix (1/2 unit matrix), and will lose millions of dollars in the second case should he bet that much on the resulting statistics.

Thus the correct state of a 2-state system, whether pure or mixed, conveys complete knowledge about the objective information that can possibly be obtained, while any significantly different state will lead to the wrong statistics. This must be part of any orthodoxy that can claim agreement with experiment.

I don't think that anything changes for bigger quantum systems simply because quantum tomography is no longer practically feasible. (The example of interference of quantum systems, which can be shown for larger and larger systems, suggests that there is no ''complexity border'' beyond which the principles change.)

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

You could instead go forward and solve the mathematical challenges involved in the thermal interpretation! There everything is as well-defined as in Constructive Field Theory but as the subject matter is new, it is not as hard to make significant progress!

 

[DarMM]

For a 2-state system (polarized beams of light) one can easily differentiate between light prepared in an unpolarized state (true density matrix = 1/2 unit matrix) and a completely polarized state (true density matrix of rank 1), and - in the limit of an unrestricted number of experiments - one can find out the true state by quantum tomography.

On the other hand, a consequent Bayesian who doesn't know how the light is prepared and thinks to be entitled by Jaynes or de Finetti to treat his complete lack of knowledge in terms of the natural noninformative prior will assign to both cases the same density matrix (1/2 unit matrix), and will lose millions of dollars in the second case should he bet that much on the resulting statistics

An Objective Bayesian isn't too different from a Frequentist here. They think there is a single "best" set of beliefs given the currently observed statistics. A Subjective Bayesian will be permitted any prior initially, but via the representation theorem (a generalization of de Finetti's classical one, there are a few different proofs of this by now) will update toward a different state if the observations do not match their proposed state.

I don't think quantum tomography differs much between the three views as all three are used in the Quantum Information literature. Though the Bayesian views are more common. There's a major paper in Quantum Information on this topic: https://arxiv.org/abs/quant-ph/0104088

 

[A. Neumaier]

(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.

Bohr didn't mention the collapse in his published writings (only in an unpublished draft, just once).
But some form of collapse is needed at least in some situations, to be able to know what state to assume after a quantum system passes a filter (such as a slit or a polarizer). This cannot be derived from Born's rule without collapse.

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.

One could instead start with the simplest case of a 2-state system, a beam of natural light passing through a polarizer and detected by a photocell. It features a density matrix corresponding to a mixed state that collapses to a pure state through the interaction with the filter. Once one has discussed the properties of polarizers one can discuss quantum tomography, and finds an objective notion of a state (if one is a frequentist). Using a little theory as described in my Insight article on the quibit, one can derive the Schrödinger equation, and everything else that matters for a single qubit.

From this single and elementary example one gets mixed states, collapse, Born's rule, and the Schrödinger equation (and if you like, the thermal interpretation) - everything needed for a good and elementary introduction to quantum mechanics, without having to tell a single children's fable.

I shouldn't waste my time anymore to discuss philosophical issues in this forum. It's kind of fighting against religious beliefs rather than having a constructive scientific discussion.

The problem is that in terms of the philosophy of physics you are a religious zealot fighting other religious zealots with a different religion...

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.

 

[A. Neumaier]

There's a major paper in Quantum Information on this topic: https://arxiv.org/abs/quant-ph/0104088

I'll have a look at it...

A Subjective Bayesian will be permitted any prior initially, but via the representation theorem (a generalization of de Finetti's classical one, there are a few different proofs of this by now) will update toward a different state if the observations do not match their proposed state.

1. Please tell me what the standard update rule for the mixed state $\rho$ of the 2-state system is when a measurement of a test for a particular polarization state becomes available? I think there is no canonical (optimal) way of making the update; or please correct me.

2. The update does not help when the bet has to be made before further knowledge can be accumulated. A subjective Bayesian will bet (or why shouldn't he, according to the Bayesian paradigm?). A frequentist will acknowledge that he knows nothing and the law of large numbers (on which he relies for his personal approximations to the true state) is not yet applicable. Thus he will not accept any bet.

3. Suppose that the light is prepared using photons on demand (one per second) by a device that rotates the polarizer every second by an angle of $\alpha=\pi(\sqrt{5}-1)/2$.

The subjective Bayesian, following the recipe for Bayesian state updates to be revealed to me as the answer to 1., will only get random deviations from his initially unpolarized state.
But the frequentist can apply whatever statistical technique he likes to use to form his personal approximation, and can verify the preparation scheme (and then achieve better and better prediction) by an autoregressive analysis combined with a cyclically repeated tomographic scheme that provides the data for the former.

 

[DarMM]

Are you familiar with de Finetti's representation theorem in the case of classical statistics before I begin an exposition?

To some degree there isn't anything shocking about the quantum case once you know the analogous theorem holds.