A Teaching ideal quantum measurement, from dynamics to interpretation

[A. Neumaier]

TL;DR Summary

A recent paper by Allahverdyan et al. discusses in detail the dynamics of measurement.

A.E. Allahverdyan, R. Balian, and T.M. Nieuwenhuizen,
Teaching ideal quantum measurement, from dynamics to interpretation,
Comptes Rendus Physique 25 (2024), 251--287.

https://www.researchgate.net/publication/380862732

 

[bhobba]

I liked the paper.

I especially liked:

'In fact, what we call “quantum state”, whether pure or not, is generally an abstract mathematical object that allows us to make probabilistic predictions about the outcomes of any experiment that will test the object under study for arbitrary observables. It encodes everything that we can say about this object and is related only indirectly to its physical properties. It is thus irreducibly probabilistic: even the above pure state |ψ1> is a probabilistic mathematical object with regard to observables other than s, as their measurement on different samples all described by |ψ1> will yield different answers, the probabilities of which are determined by |ψ1. Thus, within the “frequentist” interpretation of probabilities, a quantum state refers to a large set of similar systems. In particular, the density operator D(t) of S+A associated with an ideal quantum measurement process does not describe only a single operation, but a large set E of similarly prepared individual run'

Of course, it's true, but for some reason, it is not often found in books.

If it was I think a lot of confusion about QM would simply 'disappear'

Thanks
Bill

 

[A. Neumaier]

In particular, the density operator D(t) of S+A associated with an ideal quantum measurement process does not describe only a single operation, but a large set E of similarly prepared individual run'

Of course, it's true, but for some reason, it is not often found in books.

If it was I think a lot of confusion about QM would simply 'disappear'

It is true in certain applications but not always. A single piece of ice, say, is quite well described by an equilibrium density operator, and very few measurements done on this single system completely reveal all its properties and hence its state. A single ion in an ion trap is also described by a time-dependent density operator satisfying a Lindblad equation, which describes the stochastic properties of the complete set of observations done on the single ion.

 

[gentzen]

In particular, the density operator D(t) of S+A associated with an ideal quantum measurement process does not describe only a single operation, but a large set E of similarly prepared individual run'

Of course, it's true, but for some reason, it is not often found in books.

If it was I think a lot of confusion about QM would simply 'disappear'

It is true in certain applications but not always. A single piece of ice, say, is quite well described by an equilibrium density operator, ...A single ion in an ion trap is also described by a time-dependent density operator ...

On the one hand, whether a certain density operator is a good description depends on what I intent to measure. On the other hand, the preparation for these two examples is quite different: The single ion is prepared with good control over the time development, while the piece of ice is not. This explains why "a time-dependent" description was appropriate in one case, but "an equilibrium" in the other.

If you think that "If it was I think a lot of confusion about QM would simply 'disappear'" if it were "often found in books", would you be willing to to also admit that "a good description depends on what I intent to measure"? I once had discussion with vanhees71 in the thread "Statistical ensemble interpretation done right", where he refused to admit this:

Given the interpretation of the state as formally describing a preparation of a single system, this doesn't say anything about what I intend to measure on this system.

The degrees of freedom and the Hilbert space you choose to describe them says something about what you intent to measure on the system.

I'm unhappy about your formulation

The degrees of freedom and the Hilbert space you choose to describe them says something about what you intent to measure on the system.

because it seems to state a very common misconception about quantum mechanics. It seems as if you have an interpretation of the Heisenberg uncertainty relations in mind as if it would prevent from meausring the one or the other observable with arbitrary precision. ...

... One thing I could do is to explain the concrete example I had in mind:
When you describe a Stern-Gerlach experiment, your magnet may be described as fixed such that the magnetic field (except for its inhomogenity) points in z-direction, or you may describe it such that the magnet can be rotated around the particle beam. In the second case, you probably need to allow density matrices to describe the actual state of the incoming particles. But most introductory QM textbooks have not yet introduced density matrices at the point where they describe and analyse SG, so they typically go with the description using a fixed magnet. Independent of this, the silver atoms have more degrees of freedom in their quantum state than just the spin of the unpaired electron in the outer shell. But we won't describe those in our Hilbert-space for analyzing SG, because we don't intent to measure anything for which they would be relevant.

I think that the confusion won't 'disappear' if one insists on such a refusal.

 

[A. Neumaier]

would you be willing to to also admit that "a good description depends on what I intent to measure"?

It depends on the level of precision in the concept definition.

In algebraic quantum physics, the definition of a classical or quantum system requires the specification of a Lie *-algebra L of quantities with a distinguished family of generators with a physical meaning. This Lie algebra L then defines what is 'observable in principle' - namely all well-defined expressions in these generators. An example is the Heisenberg algebra generated by Hermitian position and momentum coordinates, and angular momentum, say, is observable since it is definable in terms of positions and momenta.

Whether one actually intents to measure is here completely irrelevent. No observer or intender is involved at all; the system speaks for itself.

The actual state of a classical or quantum system is given by a positive linear functional on L, possiblly with additional conditions, and describes the properties realized in particular individual systems.

 

[gentzen]

In algebraic quantum physics, the definition of a classical or quantum system requires the specification of a Lie *-algebra L of quantities with a distinguished family of generators with a physical meaning. ....

Whether one actually intents to measure is here completely irrelevent. No observer or intender is involved at all; the system speaks for itself.

The initial argument I gave to vanhees71 still applies:

The degrees of freedom and the Hilbert space you choose to describe them says something about what you intent to measure on the system.

You just need to replace "degrees of freedom and the Hilbert space" by "Lie *-algebra L of ... physical meaning".

The same is true for your example with the "piece of ice". Whether non-relativistic QM, QED or QCD is a good description depends on what I intent to measure on that piece of ice.

 

[A. Neumaier]

The initial argument I gave to vanhees71 still applies:

You just need to replace "degrees of freedom and the Hilbert space" by "Lie *-algebra L of ... physical meaning".

The same is true for your example with the "piece of ice". Whether non-relativistic QM, QED or QCD is a good description depends on what I intent to measure on that piece of ice.

The Lie algebra together with the dynamics is the description that defines the system. Whether you intend to measure anything it is immaterial. I never intended to measure anything but made use in my work of a lot of different systems.

But suppose that you intend to measure a system and compare the measurement with predicitions from QM. In this case you need of course to use a description in which at least the things you want to measure can be represented. It is not necessary but prudent to use the simplest description that is accurate enough, since the amount of computational work needed to extract the predictions is then minimal.

 

[gentzen]

The Lie algebra together with the dynamics is the description that defines the system. Whether you intend to measure anything it is immaterial. I never intended to measure anything but made use in my work of a lot of different systems.

Not "to intent to measure" is also an intention. In fact, what I mean by the something in "says something about what you intent to measure on the system" is mostly this, namely all the infinite possibilities of potential measurements that have been excluded.

But suppose that you intend to measure a system and compare the measurement with predicitions from QM. In this case you need of course to use a description in which at least the things you want to measure can be represented. It is not necessary but prudent to use the simplest description that is accurate enough, since the amount of computational work needed to extract the predictions is then minimal.

I had read "But suppose that you intent" as "But I suppose that you intent" and thought: You bet I do! You can't imagine how far I went to get on the one hand "physical measurement data", and on the other hand "simulation models", in cases where I had only one, but not both.

But your actual point is even more valid: I am very focused on the simplest description (even to the point that it is often no longer accurate enough). I really want to use the "intended potential measurements" for forming the equivalence classes of identically prepared states, to the point where even states with different mathematical descriptions can be equivalent, if only they are equivalent with respect to the intended potential measurements. The reason is that I want to make sense of the "caricature version" of the statistical ensemble interpretation, i.e. the version with perfectly sharp equivalence classes. And also for comparison to experimental data, I want to maintain the "caricature version", by excluding measurement data which is too strongly affected by any of the effects I omitted in my oversimplified model:

Even so an inconsistency between observations and the theory for the state might also be the fault of the model, I want to limit the "falsification" here explicitly to the state. Operationally this could mean to exclude observations which seem to be caused by effects not included in the model, like cosmic rays, radioactive decay, or human errors of the operator making the observations.