Insights Quantum Physics via Quantum Tomography: A New Approach to Quantum Mechanics

[A. Neumaier]

I still do not understand why you say that the content of the review papers by Dehmelt and Brown contain anything denying the validity of Born's rule. For me it's used all the time!

Because Born's rule assumes identical preparations which is not the case when a nonstationary system is measured repeatedly. I am not denying the validity but the applicability of the rule!

I need to read the paper before I can go into details.

 

[vanhees71]

I don't understand this argument. You just measure repeatedly some observable. The measurements (or rather the reaction of the measured system to the coupling to the measurement device) themselves of course have to taken into account as part of the "preparation" too.

 

[A. Neumaier]

I don't understand this argument. You just measure repeatedly some observable. The measurements (or rather the reaction of the measured system to the coupling to the measurement device) themselves of course have to be taken into account as part of the "preparation" too.

It is a preparation, but not one to which Born's rule applies. Born's rule is valid only if the ensemble consists of independent and identically prepared states. You need independence because e.g., immediately repeated position measurements of a particle do not respect Born's rule, and you need identical prepartion because there is only one state in Born's formula.

In the case under discussion, one may interpret the situation as reeated preparation, as you say. But unless the system is stationary (and hence uninteresting in the context of the experiment under discussion), the state prepared before the $k$th measurement is different for each $k$. Moreover, due to the preceding measurement this state is only inaccurately known and correlated with the preceding one. Thus the ensemble prepared consists of nonindependent and nonidentically prepared states, for which Born's rule is silent.

 

[vanhees71]

This would imply that you cannot describe the results about a particle in a Penning trap with standard quantum theory, but obviously that's successfully done for decades!

 

[A. Neumaier]

This would imply that you cannot describe the results about a particle in a Penning trap with standard quantum theory,

This statement is indeed true if you restrict standard quantum theory to mean the formal apparatus plus Born's rule in von Neumann's form. Already the Stern-Gerlach experiment discussed above is a counterexample.

but obviously that's successfully done for decades!

This is because standard quantum theory was never restricted to a particular interpretation of the formalism. Physicists advancing the scope of applicability of quantum theory were always pragmatic and used whatever they found suitable to match the mathematical quantum formalism to particular experimental situations. This - and not what the introductory textbooks tell - was and is the only relevant criterion for the interpretation of quantum mechanics. The textbook version is only a simplified a posteriori rationalization.

This pragmatic approach worked long ago for the Stern-Gerlach experiment. The same pragmatic stance also works since decades for the quantum jump and quantum diffusion approaches to nonstationary individual quantum systems, to the extent of leading to a Nobel prize. They simply need more flexibility in the interpretation than Born's rule offers. What is needed is discussed in Section 4.5 of my paper.

 

[vanhees71]

I don't understand, what the content of Sect. 4.5 has to do with our discussion. I don't see, how you can come to the conclusion that the "pragmatic use" of the formalism contradicts the Born rule as the foundation. To the contrary all these pragmatic uses are based on the probabilistic interpretation of the state a la Born. Also, as I said before, I don't understand how you can say that with a non-stationary source no accuracy is reachable, while the quoted Penning-trap experiments lead to results which are among the most accurate measurements of quantities like the gyro-factor of electrons or, just recently reported even in the popular press, the accurate measurement of the charge-mass ratio of the antiproton.

Nowhere in your paper I can see, that there is anything NOT based on Born's rule, although you use the generalization to POVMS, but I don't see that this extension is in contradiction to Born's rule. Rather, it's based on it.

 

[Fra]

Science has no single betting strategy. Each scientist makes choices of his or her own preference, but published is only what passed the rules of scientific discourse, which rules out most poor judgment on the individual's side. What schience knows is an approximation to what it thinks it knows, and this approximation is quite good, otherwise resulting technology based on it would not work and not sell.

Yes. By a similar reasoning, I think observer/agents that fail to adapt to their environment, will not be ubiquitous. But the fitness is relative to the environment only, just as a learning agent will be "trained" to what it's exposed to. What is true in an absolute sense seems be be about as irrelevant as the absolute space is to relative motion.

/Fredrik

 

[Fra]

Nowhere in your paper I can see, that there is anything NOT based on Born's rule, although you use the generalization to POVMS, but I don't see that this extension is in contradiction to Born's rule. Rather, it's based on it.

As I read this again, I think I also may have confused the "issue" with borns rule. Some objections I have in mind(having todo with the choice of optimal compression), seems to be off topic here, but now it seems that the main point here is the generalized "born rule", it the one relevant for mixed states? But as Vanhees says, the core essence of the "born rule" is still there, right?

/Fredrik

 

[A. Neumaier]

I don't understand, what the content of Sect. 4.5 has to do with our discussion. I don't see, how you can come to the conclusion that the "pragmatic use" of the formalism contradicts the Born rule as the foundation.

I didn't claim a contradiction with, I claimed the nonapplicability of Born's rule. These are two very different claims.

all these pragmatic uses are based on the probabilistic interpretation of the state a la Born.

You seem to follow the magic interpretation of quantum mechanics. Whenever you see statistics on measurements done on a quantum system you cast the magic spell "Born's probability interpretation", and whenever you see a calculation involving quantum expectations you wave a magic wand and say "ah, an application of Born's rule". In this way you pave your way through every paper on quantum physics and say with satisfaction at the end, "This paper proves again what I knew for a long time, that the interpretation of quantum mechanics is solely based on the probabilistic interpretation of the state a la Born".

You simply cannot see the difference between the two statements

1. If an ensemble of independent and identically prepared quantum systems is measured then $p_k=\langle P_k\rangle$ is the probability occurrence of the $k$th event.
2. If a quantum system is measured then $p_k=\langle P_k\rangle$ is the probability occurrence of the $k$th event.

The first statement is Born's rule, in the generalized form discussed in my paper.
The second statement (which you repeatedly employed in your argumentation) is an invalid generalization, since the essential hypothesis is missing under which the statement holds. Whenever one invokes Born's rule without having checked that the ensemble involved is actually independent and identically prepared, one commits a serious scientific error.

It is an error of the same kind as to conclude from x=2x through division by x that 1=2, because the assumption necessary for the argument was ignored.

Also, as I said before, I don't understand how you can say that with a non-stationary source no accuracy is reachable, while the quoted Penning-trap experiments lead to results which are among the most accurate measurements of quantities like the gyro-factor of electrons or, just recently reported even in the popular press, the accurate measurement of the charge-mass ratio of the antiproton.

This is not a contradiction since both the gyro-factor of electrons and the charge-mass ratio of the antiproton are not observables in the traditional quantum mechanical sense but constants of Nature.

A constant is stationary and can in principle be arbitrarily well measured, while the arbitrarily accurate measurement of the state of a nonstationary system is in principle impossible. This holds already in classical mechanics, and there is no reason why less predictable quantum mechanical systems should behave otherwise.

Nowhere in your paper I can see, that there is anything NOT based on Born's rule, although you use the generalization to POVMS, but I don't see that this extension is in contradiction to Born's rule. Rather, it's based on it.

This is because of your magic practices in conjunction with mixing up "contradition to" and "not applicable". Both prevent you from seeing what everyone else can see.

 

[vanhees71]

I think the problem is that I understand something completely different when I read this paper than it's the intention of the authors. Particularly I have no clue, why behind the entire formalism of the description of the outcomes of measurements there should not be Born's rule. For me POVMs are just a description of measurement devices and the corresponding experiments, where one does not perform an ideal von Neumann filter measurement, and it's of course right that only a very few real-world experiment are such ideal von Neumann filter measurements, and a more general description of the experiments that have become possible nowadays (starting roughly with the first Bell tests by Aspect et al).

My understanding of the paper is that it is very close to the view as provided, e.g., by Asher Peres in his book

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

What's new is the order of presentation, i.e., it is starting from the most general case of "weak measurements" (described by POVMs) and then brings the standard-textbook notion of idealized von Neumann filter measurements as a special case, and this makes a lot of sense, if you are aiming at a deductive (or even axiomatic) formulation of QT. The only problem seems to be that this view is not what the author wants to express, and I have no idea what the intended understanding is.

Maybe it would help, when a concrete measurement is discussed, e.g., the nowadays standard experiment with single ("heralded") photons (e.g., produced with parametric down conversion using a laser and a BBO crystal, using the idler photon as the "herald" and then doing experiments with the signal photon). In my understanding such a "preparation procedure" determines the state, i.e., the statistical operator in the formalism. Then one can do an experiment, e.g., a Mach-Zender interferometer with polarizers, phase shifters etc. in the two arms and then you have photon detectors to do single-photon measurements. It should be possible to describe such a scenario completely with the formalism proposed in the paper and then pointing out where, in the view of the author, this contradicts the standard statistical interpretation a la Born.

 

[Fra]

the arbitrarily accurate measurement of the state of a nonstationary system is in principle impossible.

This fact is one reason for my own views. What is "stationary or not", is I think also relative. Ie. relative to the speed of information processing of the observer. This is why IMO what is "stationary enough" is observer dependent.

Any realistic scenario is necessarily about decision making and placing best under incomplete information, of the sort that we can not even measure the incompleteness. This is why seek an instrinsic starting point.

Just as the case of the impure state, real total limitations of predictability have two reasons, one is the classical one which we can think of as just ignorance of agent (or it beeing misinformed etc), and the other thing which has to do with dependence between pieces of information, which is the essence of quantum mechanics. Certainly both issues are important in a real inference, and perheps also it's interplay.

/Fredrik

 

[vanhees71]

How then can it be that, e.g., the measurement of the gyrofactor of the electron using a Penning trap is as precise as it is?

 

[Fra]

IF the precessing electron is "stationary enough", if they are able to keep a single electron precessing for a month?

/Fredrik

 

[A. Neumaier]

IF the precessing electron is "stationary enough", if they are able to keep a single electron precessing for a month?

Electrons in accelerators come in large bunches, not a single electrons...

What is "stationary or not", is I think also relative. Ie. relative to the speed of information processing of the observer.

It is only relative to the speed and accuracy with which reliable measurements can be taken. This is independent of any information processing on the side of the agent.

 

[Fra]

Electrons in accelerators come in large bunches, not a single electrons...

I didn't analyze this in depth, or perhaps I misseed soemthing as it's not my focus, but what I was thinking of was for example this:

New Measurement of the Electron Magnetic Moment and the Fine Structure Constant
"A measurement using a one-electron quantum cyclotron gives the electron magnetic moment in Bohr magnetons, g/2 = 1.001 159 652 180 73 (28) [0.28 ppt], with an uncertainty 2.7 and 15 times smaller than for previous measurements in 2006 and 1987."
-- https://arxiv.org/abs/0801.1134

It is only relative to the speed and accuracy with which reliable measurements can be taken. This is independent of any information processing on the side of the agent.

I think this answer makes pretty good sense, if we view the "information processing" as a classical conventional processing of "detector data". Ie. for example they way a physicists processes experimental data. I agree to this extent.

But my point was to add a possible other perspective. From the perspective of my interpretation, "measurement" and "decision making" and "informaiton processing" is all part of the agents general inference (of which we of course don't have a theory for, at least not yet). If one takes the agent to be a real part of the interaction, then the agents "decisions and preparations" for the next interaction (measurement) should be constrained. So for the agent to be able to entertain an theory, at least approximately isomorphic to QM, the agents capacaity to detect, recode and production and action (a measurement) the amount of information and the confidence of the information implied in the measurement must be comparatively insignificant. Otherwise the theory will need to be evolve or deform before a reliable "statistics" is acquired. This makes this insanely complicated and self-referencing indeed. But I think it's how nature is - I want to undertand the more robust QM/QFT as the limiting caser in such a bigger picture. I think one can appreciate the structure of QM, but still entertain other possibilities for the purpose of seeking more explanatory value. There is no contradiction between the success of QM, and thinking that there is a better paradigm. It may be problematic only if one does not see QM as an effective theory but as logical strucutre that is proven perfect and that can never change, but only the be added upon. Like as if science is about unravelling pieces of absolute truth, one bit at a time without ever needing to revise what old pieces of the "truth".

/Fredrik

 

[vanhees71]

I don't know, what you mean by "agent". Is it the physicist sitting at a computer evaluating the "raw data on tape" given some scheme to extract the measurements of observables he wants to measure? Then I'd say it's completely irrelevant how this is described by quantum theory. Here we are really in the realm, where classical physics is the only necessary description. The physicist just uses stored irreversible facts (data on some storage device like a hard disk) and evaluates them with some (classical) algorithm to extract the data in a form he wants for his analysis of the (quantum) physical experiment.

 

[CoolMint]

At the turn of the 19th century, black body radiation was the only thing classical physics could not explain and most scientists agreed that classical physics was the best explanation and nothing further was was needed.

 

[vanhees71]

There was also no theoretical understanding of the discrete line spectra. Since 1911 also the stability of matter was also no longer describable within classical physics, leading to Bohrs suggestion of "old quantum mechanics", worked then further out by Sommerfeld.

 

[Fra]

I don't know, what you mean by "agent". Is it the physicist sitting at a computer evaluating the "raw data on tape" given some scheme to extract the measurements of observables he wants to measure? Then I'd say it's completely irrelevant how this is described by quantum theory. Here we are really in the realm, where classical physics is the only necessary description. The physicist just uses stored irreversible facts (data on some storage device like a hard disk) and evaluates them with some (classical) algorithm to extract the data in a form he wants for his analysis of the (quantum) physical experiment.

What I mean by agent is in the context of weird variant of interacting generalized qbism agents with algorithmtic angles. My main point is that I still see a a physicists in the scientific community, is a "special case" of such an agent. Such an agent is what "observer" sort of means in standard QM (as the operator of the detector) as we know it. In this case, I agree that information processing power of the physicists (ie what kind of tools or super computer it has) has nothing todo with QM interacting as such. In this view, agent-agent interactions are all classical communication, and they share the same illusion of reality (modulo classical relativity of course).

To understand the analog you need to make the "physicist" part of the physical interaction, and then the speed of the inferences will necessarily depend on the physicists "technology", which in turn will influence the whole interaction between physicist-environment.

So in the way you an Neumaier refers to it, I agree. But I still suggest that is a possible "simplicifaction" that for ME, I can not let go of.

/Fredrik

 

[A. Neumaier]

You use yourself Born's rule all the time since everything is based on taking averages of all kinds defined by $\langle A \rangle=\mathrm{Tr} \hat{\rho} \hat{A}$ (if you use normalized $\hat{\rho}$'s).

Born's rule is not just taking averages of anything!

I use quantum expectations all the time, but Born's rule only when I interpret a quantum expectation in terms of measuring independent and identical prepared systems - which is a necessary requirement for Born's rule to hold.

How do you define the experimental meaning of $\langle A\rangle$ when $A$ is not normal, which is often the case in QFT?

 

[A. Neumaier]

don't understand, what the content of Sect. 4.5 has to do with our discussion.

There I discuss the case of nonstationary quantum systems.

then pointing out where, in the view of the author, this contradicts the standard statistical interpretation a la Born.

Please do not confuse contradictions and non-applicability! These are two very different things!

 

[A. Neumaier]

New Measurement of the Electron Magnetic Moment and the Fine Structure Constant
"A measurement using a one-electron quantum cyclotron gives the electron magnetic moment in Bohr magnetons, g/2 = 1.001 159 652 180 73 (28) [0.28 ppt], with an uncertainty 2.7 and 15 times smaller than for previous measurements in 2006 and 1987."
-- https://arxiv.org/abs/0801.1134

How then can it be that, e.g., the measurement of the gyrofactor of the electron using a Penning trap is as precise as it is?

The measurement of the gyrofactor of the electron using a Penning trap is as precise as it is
because certain experimental situations happen to have very accurate descriptions in terms of a few-parameter quantum stochastic process, and the gyrofactor is one of these parameters.

Though not interpretable in terms of Born's rule or POVMs, such processes are able to describe single time-dependent quantum systems, just as classical stochastic process are able to describe single time-dependent classical systems.

The facts that there are only very few parameters and that one can measure arbitrarily long time series imply that one can use statistical parameter estimation techniques to find the parameters to arbitrary accuracy. The fact that the models are accurate imply that the parameters found for the gyrofactor accurately represent the gyrofactor.

I am now reading the papers you and Fra cited and will give details once I have digested them.

 

[CoolMint]

What I mean by agent is in the context of weird variant of interacting generalized qbism agents with algorithmtic angles. My main point is that I still see a a physicists in the scientific community, is a "special case" of such an agent. Such an agent is what "observer" sort of means in standard QM (as the operator of the detector) as we know it. In this case, I agree that information processing power of the physicists (ie what kind of tools or super computer it has) has nothing todo with QM interacting as such. In this view, agent-agent interactions are all classical communication, and they share the same illusion of reality (modulo classical relativity of course).

To understand the analog you need to make the "physicist" part of the physical interaction, and then the speed of the inferences will necessarily depend on the physicists "technology", which in turn will influence the whole interaction between physicist-environment.

So in the way you an Neumaier refers to it, I agree. But I still suggest that is a possible "simplicifaction" that for ME, I can not let go of.

/Fredrik

I thought you meant it as in the participatory universe argument since a case can be made that knowledge plays a role at the lowest scales.

 

[Fra]

I thought you meant it as in the participatory universe argument since a case can be made that knowledge plays a role at the lowest scales.

I haven't analyzed Wheelers historical writing as such, but surely it's related to this. But quite often, this is mistunderstood as that the physicists or "human observer" creates reality. If you put it like that, it soon gets silly. This misunderstanding is completely analogous to those that misunderstand the Heisenberg cut as something involving human consciousness. Analogies can help but also and create misinterpretations. When I read about OTHER people describing wheelers meaning, it comes out like a potential mischaracterisation - or Wheeler actually meant it like that. I don't konw.

I mean it in a deep sense. That any piece of matter is such a participatory observer. And the common reality is formed and evolved as they interact and selective pressure of the agents.

In my view, I take this very seriously, and seek the formalism that implements and are able to provide prodictions or explanatory value. Until then, it's admittedly a soup of words. And the current mathematics of QM, can not make sense of these ideas - except intuitively in a limiting sense. If we could infer that limit, from the general case, chances are it will come with additional explanatory power, such as reduction fo free parameters etc. But trying to understand, an build intuition and analogies is a good start.

/Fredrik

 

[vanhees71]

Born's rule is not just taking averages of anything!

I use quantum expectations all the time, but Born's rule only when I interpret a quantum expectation in terms of measuring independent and identical prepared systems - which is a necessary requirement for Born's rule to hold.

How do you define the experimental meaning of $\langle A\rangle$ when $A$ is not normal, which is often the case in QFT?

To get expectation values you need the probabilities/probability distributions, which are given by Born's rule in the formalism. That interpretation of the state, $\hat{\rho}$, leads immediately to $\langle A \rangle=\mathrm{Tr}(\hat{\rho} \hat{A})$. For me all that is subsumed under "Born's rule". Instead of saying "Born's rule" I also could say "the probabilistic interpretation of $\hat{\rho}$", but that's very unusual among physicists.

 

[vanhees71]

There I discuss the case of nonstationary quantum systems.

Please do not confuse contradictions and non-applicability! These are two very different things!

If Born's rule were not applicable here, the experimental results couldn't be understood with standard QT, but they obviously are!

 

[vanhees71]

Though not interpretable in terms of Born's rule or POVMs, such processes are able to describe single time-dependent quantum systems, just as classical stochastic process are able to describe single time-dependent classical systems.

How then can it be that these results are very accurately described by Q(F)T, which uses Born's rule to predict this value of (g-2)?

 

[A. Neumaier]

To get expectation values you need the probabilities/probability distributions, which are given by Born's rule in the formalism.

No.

1. To get quantum expectations one just needs a density operators and the trace formula. This is not Born's rule. But it is what is used everywhere in the formalism of quantum mechanics and quantum field theory
2. To get statistical expectations one just needs to average over a sample of measurement values. This is not Born's rule. But it is what is used everywhere in the analysis of statistical data.
3. To relate the two one needs an assumption - the assumption that the measurements come from independent and identical realizations of the quantum system. In this case (and only in this case!) one can equate quantum expectations and statistical expectations. This is Born"s rule.
4. In general, and in particular whenever the measurements are taken on a single quantum system, the relation between quantum expectations and statistical expectations is complicated. One needs sophisticated statistical techniques to extract from measurements useful information about states or model parameters.

Point 3 is a mathematically precise version of your statement that a state is given by an equivalence class of identically prepared systems.

That interpretation of the state, $\hat{\rho}$, leads immediately to $\langle A \rangle=\mathrm{Tr}(\hat{\rho} \hat{A})$. For me all that is subsumed under "Born's rule". Instead of saying "Born's rule" I also could say "the probabilistic interpretation of $\hat{\rho}$", but that's very unusual among physicists.

These are your magic wand and your magic spell, with which everything done in quantum mechanics looks as being based on Born's rule.

But your magic ignores the assumptions in Born's rule, hence is like concluding $1=2$ from $x=2x$ by division through $x$ without checking the assumption $x\ne 0$.

If Born's rule were not applicable here, the experimental results couldn't be understood with standard QT, but they obviously are!

They are understandable with the pragmatic use of the quantum formalism that uses whatever interpretation explains an experiment. They are not understandable in terms of only Born's rule, since In experiments with single quantum systems, the assumption in Born's rule cannot be satisfied.

How then can it be that these results are very accurately described by Q(F)T, which uses Born's rule to predict this value of (g-2)?

These results are very accurately described by Q(F)T, which uses only mathematics (and not Born's rule) to predict this value of g-2. QED predicts the correct value of g-2 from the QED action purely by mathematical calculations, without any reference to measurement. Hence one has nowhere an opportunity to use Born's rule, since the latter only says something about quantum observables measured by means of averaging over measurement results obtained from independent and identically prepared.

Born's rule would however be needed to interpret probabilities measured from scattering experiments, for which Weinberg correctly invokes Born's rule. This is a typical case where the assumption present in Born's rule is satisfied.

 

[vanhees71]

I give up obviously I'm unable to understand your point of view.

 

[A. Neumaier]

I give up obviously I'm unable to understand your point of view.

Is it so difficult to understand that

• I make a difference between two kinds of expectation (statistical - related to measurement only and quantum - related to the formalism only), to get more clarity into the foundations, while
• you conflate the two and hence have Born's rule even in purely mathematical calculations that have nothing at all to do with measurement?

Once you can accept that one can make this difference, you'll be able to understand everything I said. And you'll benefit a lot from this understanding!