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.

 

[A. Neumaier]

Are you familiar with de Finetti's 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.

I was familiar with it 20 years ago. But then I lost interest in subjective interpretations, which more and more seemed to me contrived. (A true subjectivist is free to update in any way he likes, but then theory says no longer anything about the temporal fate of the density matrix. Thus we need sort of an objective, optimal, subjectivist. But this means that there is no freedom left - at least not asymptotically. Thus the objective, optimal, subjectivist is sooner or later a frequentist...)

So I no longer recall its contents. (But I'll read the paper you pointed to; you don't need to explain.) For the present discussion I just want an answer to point 1 - an explicit update rule for the density matrix, given the current density matrix, a polarizer setting, and an observation (1 or 0), depending on whether the photon was or wasn't detected.

 

[DarMM]

For the present discussion I just want an answer to point 1 - an explicit update rule for the density matrix, given the current density matrix, a polarizer setting, and an observation (1 or 0), depending on whether the photon was or wasn't detected

1.3 in that paper is the basic rule. There's more details later on in the paper.

 

[A. Neumaier]

1.3 in that paper is the basic rule. There's more details later on in the paper.

Oh, so the subjective Bayesian describes the quantum system not by a density operator but by a probability distribution on the space of density operators? Thus his beliefs have a much bigger state space than that of quantum mechanics, which is described by single density operators.

 

[DarMM]

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?

Well it's not as if Subjective Bayesianism is a statement that knowledge doesn't matter and you can bet when you want.

Rather take a horse race with the horses given various probabilities of winning by the bookies (I'm not talking about the odds, but the probabilities the bookie will use prior to offering odds). To Bayesians these probabilties are coherent judgements about the race rather than properties of ensembles of races with those horses. However there is such a thing as knowing more about those horses, there is a world out there! Thus there are better probability assignments. That's why a Bayesian has Bayes rule, it reflects learning more. Not you must bet whenever you want because all probabilities are the same, even uninformed ones.

All three views will agree on the primacy of frequency data as a major way of testing ones assignments.

 

[A. Neumaier]

Well it's not as if Subjective Bayesianism is a statement that knowledge doesn't matter and you can bet when you want.

Rather take a horse race with the horses given various probabilities of winning by the bookies (I'm not talking about the odds, but the probabilities the bookie will use prior to offering odds). To Bayesians these probabilities are coherent judgments about the race rather than properties of ensembles of races with those horses. However there is such a thing as knowing more about those horses, there is a world out there! Thus there are better probability assignments.

The same state can be a spurious state on which to bet is foolish, or an informative state on which to bet can earn you a living. Thus the complete knowledge about a real situation would consist of (at least) a state and the assessment how informative the state is, as you need both to be successful at betting. But then not all knowledge can be in the state.

However, in quantum mechanics, the state is claimed to encode all knowledge about the system.
Thus there is an inconsistency...

 

[DarMM]

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?

Perhaps a better response would be that a Bayesian has probabilities as states of knowledge. Since there is such a thing as "knowing more" there are better states. However that's not in contradiction to the subjective nature of that knowledge.

 

[A. Neumaier]

a Bayesian has probabilities as states of knowledge. Since there is such a thing as "knowing more" there are better states.

In ''a state of a classical particle'' or ''a state of a beam of light'', the state says everything about the entity of which it is the state, while in your sentence the word "state" just means ''attribute'', it seems.

Without specifying a clear, unambiguous meaning for the concept of ''knowledge", anything based on it has very unsafe foundations.

 

[DarMM]

In ''a state of a classical particle'' or ''a state of a beam of light'', the state says everything about the entity of which it is the state, while in your sentence the word "state" just means ''attribute'', it seems.

Without specifying a clear, unambiguous meaning for the concept of ''knowledge", anything based on it has very unsafe foundations.

I think we're now just back to probability in the foundations.

Although de Finetti does have a decent enough definition I think in terms of coherent numerical beliefs, i.e. ones that can't be Dutch booked. Numerical belief assignments that can't be Dutch booked obey the Kolmogorov axioms and thus one recovers the normal probability axioms.

Coherency even forces the law of large numbers, avoiding Dutch Booking means that if you think event $E$ has probability $P(E)$, then on repeated trails with $E$ as an outcome you should assign a probability approaching $1$ that in $N$ trials as $N \rightarrow \infty$ the ratio of $E$ events to total events will be roughly $P(E)$.

I don't see it as completely arbitrary, he does give an axiomatic statement of what he means. It's just that it permits you to update those belief assignments in light of observations. Indeed the Dutch booking gives you Bayes's rule.

 

[A. Neumaier]

I think we're now just back to probability in the foundations.

No; in the last few mails we were discussing subjective probability only. Subjective probability replaces the basic notion of probability by the even more problematic basic notion of knowledge, which is a step backwards. Frequentist probability has no such problems; its only problem is that what we can know (in the informal sense) about the true state (the subject of quantum mechanics) is limited in accuracy by the law of large numbers.

Although de Finetti does have a decent enough definition I think in terms of coherent numerical beliefs

Thus we need sort of an objective, optimal, subjectivist.

I just found the following here:

In the Brukner–Zeilinger interpretation, a quantum state represents the information that a hypothetical observer in possession of all possible data would have. Put another way, a quantum state belongs in their interpretation to an optimally-informed agent, whereas in QBism, any agent can formulate a state to encode her own expectations.

I don't think that solves much, but at least it is more sensible.

Note that I do not dispute Bayesian probability as a mathematical subject and Bayesian procedures as rules justified for problems of decision making. But they are highly questionable in the foundations of physics.

Indeed the Dutch booking gives you Bayes's rule.

but only in the form (1.3) in post #182. According to this, knowledge is represented not by a density operator but by a probability distribution on density operators. In terms of degrees of freedom (for a qubit, an infinite-dimensional manifold of states $P(\rho)$ of knowledge), this is heavy overkill compared to the parsimony of quantum mechanics (for a qubit, a 3-dimensional manifold of states $\rho$ of the qubit). Thus most of the subjective Bayesian information to be updated is relevant only for modeling mind processes manipulating knowledge, but irrelevant for encoding physics.

Frequentist probability is unaffected by these problems; its place in the foundation is much more acceptable.

 

[DarMM]

but only in the form (1.3) in post #182

That's different. Dutch booking in de Finetti's treatment of probability (see his own monograph or Kaldane's) gives you Bayes's rule for Classical probability in its typical form.

The representation theorem shows that all probability assignments (density matrices in quantum case) have an alternate form (the "representation" to which the theorem's title refers) as a distribution over assignments. The space of states is still the same, e.g. the 3D manifold you mentioned. The alternate form simply shows that one can always think of one's current state as such a distribution and further more show that separate agents with different initial priors can conceive of all their sequence of Bayesian updates as a narrowing distribution over the state of probability assignments. Hence explaining why in a Subjectivist setting they converge to the same results.

The actual state space is not different. It is simply that the alternate representation simply allows a tidy demonstration why subjectivist updating can act like "slowly finding the true state" and why different priors can converge given the same data.

 

[vanhees71]

I don't know, how @A. Neumaier can misunderstand what I wrote in my notes on statistical physics. As he rightfully says, it's in accordance with the standard interpretation, and that's my intention: I don't see any problems with the standard interpretation (which for me is the minimal statistical interpretation).

A system's state is as completely determined as possible according to QT if it is prepared in a pure state. If there is incomplete knowledge about the system one has to describe it with a mixed state, and the problem is, how to choose this mixed state, according to the knowledge about the system at hand, and one objective way is to argue with information theory and the maximum-entropy principle.

I don't see, where there is a contradiction to what I wrote in one of my today's earlier postings. There I explained the well-known standard procedure, if you want to describe a part of a larger system. The answer in all textbooks I know is that you take the partial trace.

Nothing at all contradicts the statements in my manuscript: If you have a big quantum system, this big quantum system can well be completely prepared, i.e., prepared in a pure state and then the part of the system you describe by tracing out the other part(s) of the system according to this rule, is usually in a mixed state. Of course, tracing out the non-wanted part of the big system and describing only one part means to ignore the rest of the system. This means of course that you lose information, and thus the partial system is not in a pure state. Why should it be? The reduced density matrix is the correct choice based on the knowledge we have in this case, which is that the big system is prepared in some pure state but that I choose to ignore parts of the system and only look at one part, of which we have only partial information and thus describe it by a mixed state.

Take the Bohm's spin-1/2 example, the preparation of a spin-1/2 pair in the singlet state (total spin $S=0$). Then the pair is in the pure state
$$\hat{\rho}=|\Psi \rangle \langle \Psi| \quad \text{with} \quad |\Psi \rangle=\frac{1}{\sqrt{2}} (|1/2,-1/2 \rangle - |-1/2,1/2 \rangle.$$
Tracing out particle 2, i.e., only looking at particle 1, leads to the state for particle 1,
$$\hat{\rho}_1= \mathrm{Tr}_2 \hat{\rho} = \frac{1}{2} (|1/2 \rangle \langle 1/2| + |-1/2 \rangle \langle -1/2|)=\frac{1}{2} \hat{1},$$
i.e., to the state of maximum entropy.

The same of course holds for the reduced stat. op. of particle 2, which is described by
$$\hat{\rho}_2=\mathrm{Tr}_1 \hat{\rho}=\frac{1}{2} \hat{1}.$$

This is in full accordance with my statistics script and (as you claim) with Landau and Lifshitz (I guess, you refer to vol. III, which I consider as one of the better QT books, with somewhat too much overemphasis of wave mechanics, but that's a matter of taste; physicswise it's amazingly up to date given its date of publication; one has just to ignore the usual collapse-hypothesis argument of older QM textbooks ;-))):

You have to distinguish precisely who describes which system and how to associate the statistical operators with the various systems. For the above example you have the following:

(1) An observer Alice, who only measures the spin of particle 1 (you disginguish particle 1 and particle 2 simply by where they are measured; I don't want to make the example to complicated and ignore the spatial part, which however is important when it comes to identical particles in this example). What she shall measure are simply completely unpolarized particles and thus her stat. op. for the spin state is that of maximal entropy, which is $\hat{\rho}_1$ with the maximal possible entropy for a spin 1/2-spin component, $S_1=\ln 2$.

(2) An observer Bob, who only measures the spin of particle 2. What he shall measure
are simply completely unpolarized particles and thus her stat. op. for the spin state is that of maximal entropy, which is $\hat{\rho}_2$ with the maximal possible entropy for a spin 1/2-spin component, $S_2=\ln 2$.

(3) Observer Cecil, who knows that the particle pair was produced through the decay of a scalar particle at rest and thus its total spin is $s=0$. He describes the state of the complete system (consisting of two spins here) by the pure state $\hat{\rho}$, and thus his knowledge is complete and accoringly the entropy is $S=0$.

He is the one who knows, without even knowing the measurement results of A and B, that there's a 100% correlation of the two measured spins, namely if A finds $+1/2$, B must necessarily find $-1/2$ and vice versa. That's independent of the temporal order A and B measure there respective spin and thus there's no causal "action at a distance" of eithers spin measurement on the others particle.

All three description of the situation are thus (a) consistent, (b) there's no non-local action at a distance caused by the local measurement processes of A's and B's spin, (c) there's no contradiction to the statement that A's and B's knowledge prior to their measurement is less complete compared to C's. In this case it's even taken to the extreme that C's knowledge is even complete, i.e., he associates the entropy 0 ("no missing information") to his knowledge, while A and B have the least possible information, and that's what they also will figure out when doing their spin measurements.

This example shows that there are no contradictions within minimally interpreted QT nor between Einstein causality and QT.

The fact that a part of a bigger system prepared completely is not prepared completely, by the way, was Einstein's true quibble with QT, not what's written in this (in)famous EPR paper, which Einstein himself didn't like much, being quite unhappy with Podolsky's formulations when writing it up. He called this feature of quantum theory "inseparability", and that's what's the real profound physical value of this debate: It triggered Bell to develop his famous inequality valid for all local deterministic hidden-variable models and to the empirical conclusion that all these are wrong but QT is right, and Einstein's quibble, the inseparability, is an empirically validated fact.

 

[vanhees71]

Now there's also this strange idea about "subjective probabilities" in this thread. Whatever this might be, it's not modern quantum theory, which to the contrary (together with information theory) is a method to provide objective probabilities, reflecting precisely what the observers know about the system and not something "subjective" by choosing an inappropriate probability description introducing some bias, that is not justified according to what's known about the system.

 

[DarMM]

@A. Neumaier see this quote from the paper on p.13:

The upshot of the theorem, as already advertised, is that it makes it possible to think of an exchangeable quantum-state assignment as if it were a probabilistic mixture characterized by a probability density $P(\rho)$ for the product states $\rho^{\otimes N}$

 

[DarMM]

Now there's also this strange idea about "subjective probabilities" in this thread. Whatever this might be, it's not modern quantum theory, which to the contrary (together with information theory) is a method to provide objective probabilities, reflecting precisely what the observers know about the system and not something "subjective" by choosing an inappropriate probability description introducing some bias, that is not justified according to what's known about the system.

Well I don't know if it's a "strange idea" simply because it mightn't be useful in modern quantum theory. However it is, since it's just an alternate motivation for statistical tools that you can use regardless of what you think of probability theory. Such an application is here:
https://journals.aps.org/pra/abstract/10.1103/PhysRevA.93.012103

 

[A. Neumaier]

That's different. Dutch booking in de Finetti's treatment of probability (see his own monograph or Kaldane's) gives you Bayes's rule for Classical probability in its typical form.

Once one has the rules of probability theory (which any foundation of probability should produce), Bayes rule is a triviality. So why do you claim its derivation through Dutch booking as an asset?

all their sequence of Bayesian updates as a narrowing distribution over the state of probability assignments.

But this update is still an update analogous to (1.3), and my critique applies, though now to a classical bit: Uncertain knowledge is represented not by a classical density but by a probability distribution on classical densities. In terms of degrees of freedom (for a bit, an infinite-dimensional manifold of states P(p) of knowledge), this is even more heavy overkill compared to the parsimony of uncertain classical mechanics (for a bit, the interval [0,1] of probabilities p of the bit being 1). Thus most of the subjective Bayesian information to be updated is relevant only for modeling mind processes manipulating knowledge, but irrelevant for encoding physics.

The actual state space is not different.

Then please answer again my question in 1. of post #179, in terms of the actual state space. if the knowledge is $\rho$, how is it updated when a new measurement result comes in? What is the updated $\rho$?

Now there's also this strange idea about "subjective probabilities" in this thread. Whatever this might be, it's not modern quantum theory, which to the contrary (together with information theory) is a method to provide objective probabilities, reflecting precisely what the observers know about the system and not something "subjective" by choosing an inappropriate probability description introducing some bias, that is not justified according to what's known about the system.

Well, we are discussing here (in the whole thread) various interpretations of quantum mechanics, and some of them are based on subjective probability. I find it strange, too, but one cannot usually discuss other interpretations by casting them in ones own differing interpretation without losing important features - one must use the language in which they describe themselves.

 

[microsansfil]

Or what else should distinguish the frequentist from the Bayesian?

In the context of statistics, these are two different approaches to inference. In hypothesis (or theory for Karl Popper) testing, the frequentist statistician computes a p value, which is Pr( data|H0 ) (e.g probabilities of events according to a certain theory), but the Bayesian statistician computes Pr( H0|Data ) (e.g probabilities of the theories in view of certain events).

https://www.austincc.edu/mparker/stat/nov04/talk_nov04.pdf

/Patrick

 

[DarMM]

Once one has the rules of probability theory (which any foundation of probability should produce), Bayes rule is a triviality. So why do you claim its derivation through Dutch booking as an asset?

When did I claim that? It's how de Finetti does it, I'm not sure what I would mean to say it's an asset, but it's necessary. It's how this approach derives it, it's not "better" though if that's what "asset" is meant to mean. I think how he derives it "neat" as in the proof is a nice way to look at it, but that's about it.

But this update is still an update analogous to (1.3), and my critique applies, though now to a classical bit: Uncertain knowledge is represented not by a classical density but by a probability distribution on classical densities

No. Uncertain knowledge is represented by a classical density as it is always. However one's uncertain knowledge for $N$ sequences, which is also a classical density, can be shown to be equivalent to a Probability distribution over classical densities. Via this alternate representation one can demonstrate convergence from different starting priors given the same data for large $N$.

It's an alternate form used to prove that in Subjective Bayesianism people with the same large set of data will tend towards agreement. It's not what a classical probability assignment is in Subjective Bayesianism.

Then please answer again my question in 1. of post #179, in terms of the actual state space. if the knowledge is $\rho$, how is it updated when a new measurement result comes in? What is the updated $\rho$?

I should have answered this better. The form given in (1.3) is the representation that allows one to show the regular form of updating used in quantum tomography is valid.

 

[A. Neumaier]

When did I claim that?

he does give an axiomatic statement of what he means. It's just that it permits you to update those belief assignments in light of observations. Indeed the Dutch booking gives you Bayes's rule.

But never mind, it is not a critical issue.

Then please answer again my question in 1. of post #179, in terms of the actual state space. if the knowledge is $\rho$, how is it updated when a new measurement result comes in? What is the updated $\rho$?

I should have answered this better. The form given in (1.3) is the representation that allows one to show the regular form of updating used in quantum tomography is valid.

This still leaves me completely in the dark. Suppose that I want to program a subjective Bayesian observer and assign him as prior state for a particular stationary qubit source the state $\rho$. Now my robot observer tests the qubit for being up, and gets a positive result. As a subjective Bayesian, what should be the robot's updated state $\rho'$ in the light of the new information gathered?

You had objected to my suggestion that a subjective Bayesian could update arbitrarily. So how should my robot update rationally? I need an explicit formula to be able to program it, not an abstract theory that produces meta results about Bayesian consistency. Please help me.

 

[DarMM]

But never mind, it is not a critical issue

Sorry I don't understand, where am I saying it's an asset? I'm just saying (in Subjective Bayesianism) Dutch booking provides you with Bayes's theorem, i.e. it's the method of its derivation. Am I misunderstanding the English word "asset"?

You had objected to my suggestion that a subjective Bayesian could update arbitrarily. So how should my robot update rationally? I need an explicit formula to be able to program it, not an abstract theory that produces meta results about Bayesian consistency. Please help me.

Lüders rule in the simple case of iterated measurements not using POVMs.

 

[A. Neumaier]

Lüders rule in the simple case of iterated measurements not using POVMs.

Lüder's rule does not apply here; it is not about updating a poor prior state for the source but about finding the state prepared after passing the test given that the state of the source is already fully known,

But the robot uses destructive tests on qubits sequentially emitted by the source, just to learn (as in quantum tomography) about the state prepared by the source. I want to know how the robot should modify his subjective density matrix in the light of the result of a single destructive test, in order to improve it, in such a way that by repeating the procedure sufficiently often it predicts better and better approximations of the observed statistics.