Key problems in classical and quantum measurement

[A. Neumaier]

Quantum mechanical foundations are usually phrased in terms of measurement. I believe this is the main cause why these foundations remain shaky after almost 100 years of a good mathematical basis. Classical mechanics never had any reference to measurement in its foundations, and hence it was always clear what the terms meant on a theoretical level. It should be like that in any foundations that deserve this name.

For measurement is an exceedingly complex process that cannot be taken as unexplained primitive, vague to the point of meaninglessness when used in foundational arguments. In the present foundations of quantum mechanics it looks as if there were a miraculous process that creates measurement results when an experimenter sets up a corresponding setting. No wonder that the traditional interpretations of quantum mechanics are similarly miraculous in one or more respect!

In reality (i.e., in actual practice) measurement results are nothing miraculous at all but appear as the result of complex physical activities. Thus whatever can be said about measurement must be based on a description and analysis of these activities. This makes measurement an area of statistical mechanics that cannot be discussed without having already the whole theoretical set-up of a background theory that defines what the objects measured are and what the items mean that go into the description of a protocol for how valid measurements are created by those (people or automata) who take measurements.

The proper theoretical description and analysis of these measurement activities is the real measurement problem, and it is an unsolved problem both in classical mechanics and in quantum mechanics, and for very similar reasons. Without progress in solving these problems there is little hope of clarifying even the meaning of the present foundational discussions.

 

[A. Neumaier]

From a pragmatic point of view, wikipedia describes the measurement setting quite well. One sees already that it is far more complex than what the traditional axioms of quantum mechanics suggest; indeed the description seems to be almost disjoint from that one finds in the discussion of the latter. This shows that the traditional foundational discussions on quantum measurement are extremely superficial.

 

[ftr]

But isn't it that already in QM we assume that the object is in a superposition of the observable (some say AND , and some say OR). So before the measurement the object had an undefined value, that is already way weird and not straightforward. Then on top of that you have HUP and its interpretation, now you have a wild party:)

 

[A. Neumaier]

in QM we assume that the object is in a superposition of the observable

No. Observables cannot be in superposition.

before the measurement the object had an undefined value, that is already way weird and not straightforward. Then on top of that you have HUP and its interpretation, now you have a wild party:)

Yes, if you put it like that, the foundations are completely crazy. But this is only the view taken in some of the many written foundations.

The unwritten foundations - the actual practice of interpreting experimental arrangements and results - shows a quite different picture:
Here the unmeasured particle is in an ion trap, or in a beam, and people perfectly know (or at least assume) that it is there! Otherwise they couldn't make experiments with it. Thus the particles have a well-defined position to within some uncertainty, and this is in full accord with Heisenberg's uncertainty relation.

The particles have all the properties their state has, just as in classical physics! No less and no more!

 

[Mentz114]

No. Observables cannot be in superposition.

Yes, if you put it like that, the foundations are completely crazy. But this is only the view taken in some of the many written foundations.

The unwritten foundations - the actual practice of interpreting experimental arrangements and results - shows a quite different picture:
Her the unmeasured particle is in an ion trap, or in a beam, and people perfectly know (or at least assume) that it is there! Otherwise they couldn't make experiments with it. Thus the particles have a well-defined position to within some uncertainty, and this is in full accord with Heisenberg's uncertainty relation.

The particles have all the properties their state has, just as in classical physics! No less and no more!

I could have written the bold sentences myself, but (naturally) I wouldn't dare.

It makes nonsense of 'unmeasured' position has no value - which I never believed in any case. This is linked to the superstitious belief that only an act of measurement can allow us to make factual statements. Another casualty of the hallowed ( and incorrect) measurement theory.

 

[dextercioby]

Classical mechanics in any of its formulations (Newton, Lagrange, Hamilton, Hamilton-Jacobi) makes no reference to measurements of observables. Why should the quantum mechanics do?

 

[A. Neumaier]

Classical mechanics in any of its formulations (Newton, Lagrange, Hamilton, Hamilton-Jacobi) makes no reference to measurements of observables. Why should the quantum mechanics do?

They shouldn't. But the unfortunate fact is that the traditional foundations do make reference to it, and thus make the foundations fuzzy.

 

[zonde]

Classical mechanics in any of its formulations (Newton, Lagrange, Hamilton, Hamilton-Jacobi) makes no reference to measurements of observables. Why should the quantum mechanics do?

Classical mechanics makes predictions about physical facts. These predictions can be compared to measurements to test the theories.
I suppose that quantum mechanics do not have to refer to measurements as long as it talks about electron states in atoms. But to go further you have to have predictions about particle positions as physical facts that can be compared with measurements.

 

[A. Neumaier]

But to go further you have to have predictions about particle positions as physical facts that can be compared with measurements.

One only needs to have predictions about things like black spots in a Stern-Gerlach experiment and photocurrents in a Bell-inequality test experiment. The latter are well-described by the expectations calculated in statistical mechanics.

Classical mechanics makes predictions about physical facts. These predictions can be compared to measurements to test the theories.

The measurement problem in classical mechanics consists of explaining how a multiparticle system called the observer can collect perfect information about the physical facts defining the few-particle system measured. In fact, perfect information cannot be obtained. Whatever is obtained experimentally needs a justification why it deserves being called a particle position or momentum and how uncertain it is.

Giving this justification is also a problem in statistical mechanics. But it has attracted almost no attention by the philosophically minded (only pragmatically from a few people such as Suppes) since it doesn't get the huge weirdness publicity that the traditional foundations give to quantum mechanics.

Thus the measurement problem is a difficult statistical mechanics problem in both the classical and the quantum situation.

The so-called foundations of quantum mechanics (and the much slimmer Hamiltonian or Lagrangian foundations of classical mechanics) sidestep this problem by assuming that the measurement results appear miraculously upon performing an experiment. Those working in quantum foundations refer to the classical view by drastically simplifying the picture, assuming that, miraculously, the measurement result is identical to the exact particle position. But they treat the quantum view differently - they explain the miracles inherited by ignoring the true nature of measurement by equipping the quantum world with all sorts of miraculous things, such as wave function collapse, splitting of worlds, or the invention of a submicroscopic level of particles with highly nonlocal, in principle unobservable properties.

Thus everyone invents miracles to explain miracles. Hardly anybody tries to explain the miracles by looking at their origin - the idealization that ignores the fact that measurement means that one subsystem of a big system ''has'' some information about another subsystem in the form of correlations, due to couplings for which we already know the laws.

The only rational approach to the measurement problem is to study how these laws give rise to these correlations.

 

[A. Neumaier]

Thus the measurement problem is a difficult statistical mechanics problem in both the classical and the quantum situation.

For a discussion in the quantum case see this thread.

 

[PGaccount]

The reason we don't have to talk about measurement in classical physics is the fact that we can always control and account for the influence of the measuring bodies on the objects under investigation. For example we can make the effect of the measuring bodies as small as we want, or if it is finite, we can control and take that finite effect into account in our description. This means that we can talk about the state of a system as something that exists independently of observation.

This is not possible in quantum physics because the effect of the measuring bodies is uncontrollable. If a body is to serve as a clock, then there will be an uncontrollable exchange of energy with the clock, which cannot be separately taken into account in order to specify the state of the objects. Any experiment where we attempt to prove that "an amount of energy E went into the clock" will destroy the original phenomenon.

 

[A. Neumaier]

we can always control and account for the influence of the measuring bodies on the objects under investigation.

Only under the idealized assumption that the bodies are macroscopic. If you try to measure a microscopic point particle with a classical $N$-particle system you get very similar (and unsolved) statistical mechanics problems as in the quantum case.

 

[PGaccount]

That is of course only because you are working with a situation where you voluntarily choose not to know the mechanical state of the N-particle system, and you voluntarily choose a thermodynamic or statistical mechanical description. Nothing prevents you in classical physics from knowing the exact mechanical state of something you might describe using thermodynamics.

 

[A. Neumaier]

That is of course only because you are working with a situation where you voluntarily choose not to know the mechanical state of the N-particle system, and you voluntarily choose a thermodynamic or statistical mechanical description. Nothing prevents you in classical physics from knowing the exact mechanical state of something you might describe using thermodynamics.

This is not true, independent of any thermodynamics. Knowing is a physical process, and the knowledge must somehow enter the knower. We are all born knowing nothing, and learn only through interaction with the environment.

Moreover, even when there is exact knowledge of the initial condition and exact deterministic dynamics, the measurement device is a complex system that cannot be made to exactly represent the state of the particle solely through the physical interactions. God may know the state of every particle and the precise nature of all the interactions of a classical universe under his control. But if it is governed by classical mechanics without supernatural intervention, his knowledge will invariably tell him that there is no $N$-particle system whose pointers would exactly reproduce the state of any given particle brought into interaction with it. Thus he knows the state information but he cannot measure it using the tools of his classical universe!

 

[PGaccount]

I don't know why you say that, maybe you can give an example or calculation which shows this.

But in quantum theory, for example, if you assume that the particle passes through one of the slits of a double-slit experiment, then there is a logical consequence, which is that there is no interference pattern. Since however you do observe an interference pattern in certain situations, this assumption is wrong. If you measure which slit it passes through, then you will destroy the interference. This is why you can't talk about the state of the particle as something independent of what you are experimentally doing. You simply cannot assume that the particle passes through one slit or another if you are not measuring it, because the conclusions from it will be wrong.

This situation is clearly different from what you describe, where there is some exact state which you are unable to measure.

 

[RockyMarciano]

This is not true, independent of any thermodynamics. Knowing is a physical process, and the knowledge must somehow enter the knower. We are all born knowing nothing, and learn only through interaction with the environment.

Moreover, even when there is exact knowledge of the initial condition and exact deterministic dynamics, the measurement device is a complex system that cannot be made to exactly represent the state of the particle solely through the physical interactions. God may know the state of every particle and the precise nature of all the interactions of a classical universe under his control. But if it is governed by classical mechanics without supernatural intervention, his knowledge will invariably tell him that there is no $N$-particle system whose pointers would exactly reproduce the state of any given particle brought into interaction with it. Thus he knows the state information but he cannot measure it using the tools of his classical universe!

I think your view here clearly exceeds the limits of science to enter the personal beliefs.

 

[A. Neumaier]

maybe you can give an example or calculation which shows this.

In contrast, any claim that it can be done would have to be shown by an argument, as this would be a very miraculous thing!

This is why you can't talk about the state of the particle as something independent of what you are experimentally doing.

? Physicists routinely talk about states independent of experiments. States do not depend on experiments for their existence. The $N$-particle system called the Earth has a well-defined state long before there were physicists who did experiments.

This situation is clearly different from what you describe, where there is some exact state which you are unable to measure.

Well, you had assumed that one can know the exact state!

 

[A. Neumaier]

I think your view here clearly exceeds the limits of science to enter the personal beliefs.

This has nothing to do with a personal belief. God is the conventional label attached to an all-knowing entity, assumed to exist for the sake of an argument. It is used as metaphorical as the demon in Laplace's argument.

 

[RockyMarciano]

This has nothing to do with a personal belief. God is the conventional label attached to an all-knowing entity, assumed to exist for the sake of an argument. It is used as metaphorical as the demon in Laplace's argument.

I'm not referring to the use of that word as a conventional label, I understood it as such. I' referring to the belief that there are certain complexities that are unkowable in principle, or that are a given. Precisely it is the role of physics to question this.

? Physicists routinely talk about states independent of experiments. States do not depend on experiments for their existence. The $N$-particle system called the Earth has a well-defined state long before there were physicists who did experiments.

Previously you referred to getting information through interaction with the environment, so you must concede that it is not necessary to invoke arguments about independence from experiments performed by humans here. Interaction with the environment are a much broader notion and it only depends on the existence of the universe.

 

[A. Neumaier]

Interaction with the environment are a much broader notion and it only depends on the existence of the universe.

But experiments (i.e, what you had talked about and what I was referring to) require an educated experimenter.

 

[RockyMarciano]

But experiments (i.e, what you had talked about and what I was referring to) require an educated experimenter.

No, that was future. I answered one of your replies to him.

 

[thephystudent]

I'm not sure if I really get the point you are making, but my current understanding of a quantum measurement is the following:

1)You entangle the system to be measured with the arrow on the scale of the measurement device and wait a bit.
2)The position of the measurement arrow is a 'pointer state', so survives decoherence.
3) Coupling with the rest of the universe->decoherence, arrow position (and corresponding state of the system) becomes a statistical mixture, corresponding to your lack of knowledge.
4) Look at the arrow and get rid of your statistical uncertainty.

Is this correct/naive/false? What precisely makes a state a pointer state is not entirely clear to me so far, though.

 

[Prathyush]

But if it is governed by classical mechanics without supernatural intervention, his knowledge will invariably tell him that there is no N-particle system whose pointers would exactly reproduce the state of any given particle brought into interaction with it.

Nothing prevents you in classical physics from knowing the exact mechanical state of something you might describe using thermodynamics.

What Future is saying is basically correct.

There is nothing deep lurking in the "measurement problem" in Classical mechanics. In practice it is true that there is always some error because one can never construct an apparatus with infinite precision. The point is that given infinite resources(or infinitely light probes) the error can be made as small as possible. This statement is not wrong. It is irrelevant to talk about if we have infinite resources or not when we want to analyze a question such as "does classical mechanics imply an that there exists an underlying objective reality"

However In quantum mechanics, there is no such thing as objective reality. It is impossible to construct an experiment where one can both measure which slit the particle went while observing interference between paths. There is a fundamental difference between the 2 scenarios.

 

[A. Neumaier]

Is this correct/naive/false?

This depends on whom you ask.

What precisely makes a state a pointer state

In my view, a pointer state is a mixed state of a macroscopic object whose expectation is (withing the measurement uncertainty) the measured pointer value. This is the view most consistent with actual experimental practice. Most physicists working on foundations, however, idealize the situation so much that it bears no longer any clear relation to experiment. They take a pointer state to be an eigenstate of the pointer observable (which makes no sense for continuous-valued pointers) and assume that measured values are infinitely accurate.

 

[A. Neumaier]

What Future is saying is basically correct.

There is nothing deep lurking in the "measurement problem" in Classical mechanics. In practice it is true that there is always some error because one can never construct an apparatus with infinite precision. The point is that given infinite resources(or infinitely light probes) the error can be made as small as possible.

Your ''basically correct'' is valid only in a highly idealized sense that sweeps all problems under the carpet.

In a classical universe there are no infinite resources. The universe is described by an $N$-particle system with huge $N$. Some part of it is the measured particle, some other part the observer. The flow of information between the two is defined through the standard interactions.

It is irrelevant to talk about if we have infinite resources or not when we want to analyze a question such as "does classical mechanics imply an that there exists an underlying objective reality"

But according to post #1 which defines the topic of this thread, we do not want to analyze this question; traditionally classical mechanics simply assumes an underlying objective reality. The goal of the thread is to analyze the question ''What is a classical (or quantum) measurement and how can it be that we can infer from something read from an apparatus an exact property of a single particle?''.

The key paradox is: We measure a property of the pointer (or screen, or current, etc.) but we then claim a property of the particle. The problem posed in #1 is to resolve this paradox by using classical (or quantum) dynamics to show why this claim is valid in model problems that preserve the key properties of realistic measurement settings. In the classical case I don't know any work on this question. In the quantum case, the work by Allahverdyan, Balian and Nieuwenhuizen reviewed here and discussed here gives a reasonably convincing answer.

 

[Prathyush]

But according to post #1 which defines the topic of this thread, we do not want to analyze this question; traditionally classical mechanics simply assumes an underlying objective reality. The goal of the thread is to analyze the question ''What is a classical (or quantum) measurement and how can it be that we can infer from something read from an apparatus an exact property of a single particle?''.

If you concern is one of a practical nature, In the classical case, I don't think that is an interesting question. I don't think that it will have a sufficiently general answer either. One can always say shine weak light on an object and look at it and describe what you see. One can ask questions about how weak can I make a light beam, the answer is indefinitely. One can also ask how weak a light beam can I measure, the answer is again indefinitely. I don't think there will be general principles governing it. Unless you give me a strong reason to think that there are general principles, I am inclined to think that there are none. So without a context it is not reasonable to analyse this question.

 

[A. Neumaier]

If your concern is one of a practical nature

... then problems of foundations do not matter at all, neither in classical mechanics nor in quantum mechanics.

These questions are interesting only if one wants to understand the foundations for their own sake.

One can always say shine weak light on an object and look at it and describe what you see.

This doesn't answer my question how the information about the position of a microscopic particle is transferred with a certain accuracy to the pointer upon we shine the light so that we can see it. The latter is a question of classical statistical mechanics. Reading the pointer is not the problem; this is a macroscopic process at which nothing of real interest happens.

So without a context it is not reasonable to analyse this question.

The question is well-defined and nontrivial to answer, but surely answerable, with an amount of work not exceeding that of a good PhD thesis). To give a very specific scenario (i.e., the full context that you asked for):

Describe in terms of classical $N$-particle mechanics (only) mechanics a classical apparatus that measures in a fixed lab frame the position and momentum of a single prepared classical particle to an accuracy exceeding the accuracy limit imposed by the Heisenberg uncertainty relation (which is of course not limiting the classical accuracy). The lab frame is specified by the centers of mass of 4 marks (means of the corresponding particle positions making up the marks) defining
the origin and the endpoints of the basic unit vectors of the coordinate system. The beam is assumed to be aligned to the z-axis, so that position and momentum are given by two real numbers. Deduce from the classical equations of motion (with a reasonable potential of your choice) that the two pointers indeed produce the position and momentum of the particle at the time it leaves the preparation device.

To settle the problem in this setting probably requires a very good PhD student; but various simplifications can be considered that makes the analysi less formidable.

 

[Prathyush]

... then problems of foundations do not matter at all, neither in classical mechanics nor in quantum mechanics.

What I am saying is in classical mechanics it is not an issue about foundations, it is a practical one. Given a limited amount of resources how can one most accurately measure a particle. The foundations of classical mechanics are very clear since Newton.

This doesn't answer my question how the information about the position of a microscopic particle is transferred with a certain accuracy to the pointer upon we shine the light so that we can see it.

Given we can make and measure light that is weak enough not to significantly affect the particle. Have a collection of mirrors at far away from the particle, we can measure the energy transferred, measure the momentum momentum transferred, and from this we can find out the location of the particle. We can also measure, its wavelength to find out the velocity, these measurements in classical mechanics can be made independently.

What is the real problem here beyond being pedantic?

 

[A. Neumaier]

Given we can make and measure light that is weak enough not to significantly affect the particle. Have a collection of mirrors at far away from the particle, we can measure the energy transferred, measure the momentum momentum transferred, and from this we can find out the location of the particle. We can also measure, its wavelength to find out the velocity, these measurements in classical mechanics can be made independently.

What is the real problem here beyond being pedantic?

Well, measurement always looks trivial unless one is pedantic.

The problem with your analysis is that in $N$-particle mechanics, a mirror is a very complex object, to be described by (in your case classical) classical statistical mechanics. You need to find out how given the $N$-particle dynamics alone you can make a subsystem called a mirror having approximately the properties of a real mirror. This is already a nontrivial problem since you need to find interactions that will do the job. Then you need to find out how much accuracy you can prove from the $N$-particle description of your mirror. For this you have to show that the dynamics of the total $N+1$ particle system implies that the particle actually imparts a definite amount of energy and momentum to the mirror and leaves the mirror with a sufficiently known position and momentum so that subsequent measurements (''we can also measure'') on it are possible and still give information about the original particle momentum.

Then you need to figure out how the change of energy and momentum of the mirror is measured - otherwise you have no information gained. For this you need other equipment, which makes the total system even bigger, and the behavior of this other equipment must also be deduced from a classical multiparticle dynamics. You can stop the chain only when you have reached something that is observable with the naked eye - i.e., is big enough and permanent enough that a human being can read it reliably.

In each step you incur inaccuracies that must be estimated as a function of the size of the mirrors and the other pieces of equipment. Since the total size of the equipment is limited by the number of particles in your lab (much bigger mirrors etc are unlikely to behave properly unless you take into account additional control mechanisms as used in modern astronomical observatories) you find out in this way a definite limit of the accuracy with which you can do the experiment. I wonder whether this would be enough to beat the Heisenberg uncertainty relation classically.

The above is the classical version of the problem solved on the quantum level for a spin by A/B/N in the papers mentioned above. They start with the quantum interaction and end with the detector being in an equilibrium state from which one can read off a binary answer. More work of the same kind would be needed to turn their approach into something that works for the measurement of continuous variables. (Some work in this direction is reported here and discussed here (and posts #28 and #83 there.)

 

[Prathyush]

Well, measurement always looks trivial unless one is pedantic.

Fair enough, I don't think I will be thinking about this question beyond this post.

You need to find out how given the N-particle dynamics alone you can make a subsystem called a mirror having approximately the properties of a real mirror. This

I doubt this can be done within the context of N particle dynamics(of charged particles), because all matter at microscopic level obeys quantum mechanical laws. The well known problem of classical stability of the atom would apply.

You can stop the chain only when you have reached something that is observable with the naked eye - i.e., is big enough and permanent enough that a human being can read it reliably.

Its probably much easier to do if you work with hard potentials and not running into all kinds of problems to do with infinite potentials in classical electromagnetism. You can always construct a domino like effect, with a ball on top of a potential rolling down knocking out other which are heavier etc. But lot of details about how it can be constructed which I am not thinking about.

I can also construct another apparatus which can measure a energy of probe particle. Consider a gas at temperature T, with force that is non zero in a small region around the particle to keep the situation ergodic. Assuming V is large U ~ NKT. Walls exist without any explanation, and our probe particle can penetrate the walls. When your probe particle interacts with the gas molecules, it will eventually equilibrate so long as the energy of the probe particle ~NKT (Or some other number) there will be a measurable pressure difference.(which can be measured using another system of similar system as a reference ). The time scale for equilibriation etc could be relevant.

It is likely you will find lots of problems with the incomplete description of these apparatus. But that is ok with me. I don't want to analyse what a wall is constructively etc, I am happy to take them as granted.

 

[A. Neumaier]

The well known problem of classical stability of the atom would apply.

Only if you model atoms as composite. But one can model atoms as point particles with effective interactions.

Its probably much easier to do if you work with hard potentials

Yes, but then it is not microscopic. The challenge in solving the quantum mechanical measurement problem also goes away if you don't model the detectors as multiparticle systems. Nothing worth doing remains once you take an apparatus as a black box with simplified laws - whether exact reflection classically or some form of collapse quantum mechanically. The foundational challenge is to show how these assumptions are compatible with an underlying microscopic dynamical law. The work by Allahverdyan, Balian and Nieuwenhuizen is relevant only if one is interested in taking up this challenge.

You can always construct a domino like effect, with a ball on top of a potential rolling down knocking out other which are heavier etc.

There are indeed papers that address this quantum mechanically, using multiparticle models. Doing the same with classical multiparticle models is an interesting challenge.