Many measurements are not covered by Born's rule

[vanhees71]

You repeat again and again this example about the energy of an atom and just don't realize that you cannot measure it in the way you imply. Of course you cannot measure things that are not observable in principle. You can measure the energy differences by spectroscopy or infer them from scattering experiments, including the corresponding transition probabilities (relative intensities of spectral lines) or cross section, but you cannot measure absolute energy levels (neither within QT nor classical mechanics).

 

[A. Neumaier]

Arnold, what do you think of this experiment

https://physics.aps.org/featured-article-pdf/10.1103/PhysRevLett.110.213001

If you open another thread, we can discuss it; it is an interesting experiment. But starting tonight I am on travel for a few days and cannot reply before next Monday or Tuesday. So please create then a new thread, and mention me (i.e., place @A.Neumaier in the text so that I am notified).

 

[A. Neumaier]

You repeat again and again this example about the energy of an atom and just don't realize that you cannot measure it in the way you imply.

You mean, in the way Born's rule says! I imply nothing else than what Born's rule says about measuring observables. And there is no doubt that the total energy is one of the most important observables in quantum mechanics.

 

[A. Neumaier]

Of course you cannot measure things that are not observable in principle.

So you want to claim that total energy (normalized as commonly done to ground state energy zero) is unobservable in principle?

Then why is $H$ called an observable?

And why can it be observed in the bulk, which is done routinely in thermodynamics?

How big must a quantum system be before its total energy becomes observable in principle?

 

[Paul Colby]

How big must a quantum system be before its total energy becomes observable in principle?

This depends almost entirely on how accurately the observable is to be determined. The more accurate, the larger the system or the larger the sample set. Even in the case where theory states the answer is certainty, one single measurement would not suffice to determine the answer experimentally.

 

[A. Neumaier]

This depends almost entirely on how accurately the observable is to be determined.

Born's rule says (in its textbook version) that measuring any observable gives as results an exact eigenvalue.

Thus Born's rule is only an approximate rule?

The more accurate, the larger the system or the larger the sample set.

The sample set has size 1 in all my arguments. Single measurements give a property of the single system.

Born's rule says something definite about the possible results of each single measurement. How can it be that the result for a small system is inaccurate if only a discrete set of possibilities are allowed?

 

[vanhees71]

You mean, in the way Born's rule says! I imply nothing else than what Born's rule says about measuring observables. And there is no doubt that the total energy is one of the most important observables in quantum mechanics.

You do not even try to understand what I'm saying. Once more: The energy of a system is defined only up to an additive constant, and you can only measure energy differences, and that's also valid for an atom. We can in fact very accurately measure the energy levels of an atom by doing spectroscopy, and that's described, of course, by Born's rule. If that was not the case, nobody would ever have studied QT as discovered by Heisenberg, Born, Schrödinger, and Dirac in 1925/26! For the most simple atom, the hydrogen atom, it's among the most accurate measurements ever done, and the predictions of QT (in this case QED) are among the best confirmed in the history of physics.

 

[Paul Colby]

Born's rule says (in its textbook version) that measuring any observable gives as results an exact eigenvalue.

Thus Born's rule is only an approximate rule?

In cases where the observable spectrum is a continuum (like the energy transitions in a hydrogen atom interacting with the EM field) the Born rule states only one of a set of energies will result in a single measurement but not which one of the continuum will result. In addition it makes an exact (theoretical) statement about the frequency of observed values. Experimentally verifying this frequency or probability to infinite accuracy is for ever beyond ones capabilities which seems completely reasonable to me since exact measurement is never possible even in principle.

 

[vanhees71]

Also if the operator $\hat{A}$ representing the observable $A$ has a continuous spectrum (or if part of its spectrum is continuous), then Born's rule gives probability distributions (not probabilities) since the generalized eigenvectors for these spectral values are distributions not in the Hilbert space but in the dual of the nuclear space of the operator. E.g., for the position operator $\hat{x}$ the spectrum is entire $\mathbb{R}$, and for a normalized $|\psi \rangle$ representing the state $|\psi \rangle \langle \psi|$ the expression $P(x)=|\langle x|\psi \rangle|^2=\langle x|\hat{\rho} |x \rangle$ is the probability distribution, i.e., the probability to find the particle in an interval $[x-\epsilon,x+\epsilon]$ is given by $\int_{x-\epsilon}^{x+\epsilon} \mathrm{d} x' P(x')$. All this is no argument against Born's rule.

The more I follow Arnold's arguments the less I understand what his criticism against Born's rule is about :-(.

 

[Paul Colby]

The sample set has size 1 in all my arguments. Single measurements give a property of the single system.

Born's rule says something definite about the possible results of each single measurement. How can it be that the result for a small system is inaccurate if only a discrete set of possibilities are allowed?

This is only half of the Born rule which is why it's never applied this way in practice or theory. Case in point, the Stanford Magnetic Monopole. A single measurement is insufficient to establish a result. You should know this.

 

[A. Neumaier]

The energy of a system is defined only up to an additive constant, and you can only measure energy differences

The total energy of a system when normalized (as I always emphasized) such that the ground state energy $E_0$ is zero is fully defined, and each of its eigenvalues $\lambda_i$ is an energy level with a physical meaning; $\lambda_i=E_i-E_0$ is an energy difference, and hence measurable as you just state. Nevertheless its measurement has no resemblance at all to what Born's rule claims for the measurment of $H$.

You do not even try to understand what I'm saying.

I can say the same of you.

This is only half of the Born rule ...

If half the rule is already proved deficient in some case, the full rule with both parts is deficient in this case, too. Other cases don't change this.

... which is why it's never applied this way in practice or theory.

If it is never used in theory or practice, why is it part of the rule? A good rule never contains completely useless parts!

 

[A. Neumaier]

In cases where the observable spectrum is a continuum ...

But we are talking about energy levels, i.e., the discrete part of the spectrum.

... the Born rule states only one of a set of energies will result in a single measurement but not which one

The same is claimed in Born's rule for the discrete part. And it is claimed to hold exactly. If it were to hold holds only approximately, any of the many theoretical conclusions traditionally drawn from it would be slightly erroneous, too.

exact measurement is never possible even in principle.

But then Born's rule is valid only approximately, and cannot be fundamental.

The more I follow Arnold's arguments the less I understand what his criticism against Born's rule is about :-(.

You haven't even started to follow it. You react against things I never claimed and repeat things that have nothing to do with my arguments.

 

[dextercioby]

Many PF threads with debates in the QM section end up with "let's agree to disagree". This and the other current one will share no different fate.

 

[A. Neumaier]

Many PF threads with debates in the QM section end up with "let's agree to disagree". This and the other current one will share no different fate.

In my view, the point of such a discussion is not to agree but to express all the relevant facts and arguments so that the readers (not the combatants) can judge for themselves. Usually I contribute as long as new aspects come up, and a little longer just in case. And it is good to have sharp adversaries, since this forces one to put one's statement into the most clear and expressive form.

In the present case, I learned only through these discussions and my background studies caused by them what a can of worms Born's rule is if one takes if fully at face value and compares with real measurements. This made the discussion worthwhile for me, not whether vanhees71 or anyone else agrees or disagrees.

 

[dextercioby]

I am mostly interested in the mathematical formulation of physical theories, but reading through these two threads really made me learn something about the things I was in turn taught 13 years ago. That it is a very bold statement to claim that an axiom from a mathematical/logical construct/framework/model makes an unbreakable link to real life laboratory experiments. So the question remains? What is a quantum observable?

 

[Paul Colby]

That it is a very bold statement to claim that an axiom from a mathematical/logical construct/framework/model makes an unbreakable link to real life laboratory experiments.

If I understand your statement, why are you surprised? The situation where theory is devoid of a mandatory discussion of what's measurable is actually more shocking than the reverse. In classical mechanics one could get away with such abstraction. I find it very comforting that QM as a theory includes such an intimate connection with the world we interact with. In my opinion theory is just what we make up to fit phenomena we can measure and nothing more.

Also, my interpretation of the discussion in this thread is just a disagreement on what the rules "say" about nature. Some of it bordering on wrong or not even wrong.

 

[A. Neumaier]

a disagreement on what the rules "say" about nature.

Born's rule says nothing about nature (which doesn't know measurements), it is about measurements, which are not part of nature but of our scientific culture.

 

[Paul Colby]

Born's rule says nothing about nature (which doesn't know measurements), it is about measurements, which are not part of nature but of our scientific culture.

Yes, in my world view your statement is simply wrong but there is likely no way to make that point.

One parting shot. How can one claim that a "rule" that defines the probability of experimental observation doesn't "know" anything about said observation.

 

[A. Neumaier]

How can one claim that a "rule" that defines the probability of experimental observation doesn't "know" anything about said observation.

The knowing is in those using the rule and applying it very liberally to all sorts of situations without bothering about what the rule actually says, according to the traditional meaning of the terms used in expressing it.

I want to have foundations (and believe they can be clearly formulated) where the words match the meaning.

 

[Strilanc]

Sorry for quoting from way back at the start, but...

1. Within the framework of a Hilbert space for an atom one cannot find an observable in the sense of ''self-adjoint Hermitian operator'' that would describe the measurement of the frequency of a spectral line of the atom. For the latter is given by the differences of two eigenvalues of the Hamiltonian, not by an eigenvalue itself, as Born's rule would require.

Anytime you see measurement in the middle of some quantum mechanical model, you can add ancillae, replace that measurement with the relevant information leaking into the ancillae, and defer measuring the ancillae until the experiment finishes. That's the deferred measurement principle (...it's really more of a theorem). For example, instead of measuring then doing computations classically you can do the computations under superposition on a quantum computer before measuring.

The measurements you are describing in your examples are very complicated. We use computation to deal with those complications. In practice, for obvious pragmatic reasons, we do those computations classically and think about them classically. But, speaking theoretically, there's no fundamental obstacle to doing the computations under superposition on a quantum computer.

Doing all the statistics on a quantum computer, and only measuring at the very end, allows you to defer applying Born's rule until the final outcome has been computed. This fixes the theoretical problem you're pointing out.

Yes, you can apply Born's rule halfway through your experiment and then do some classical stuff outside of the regime that Born's rule is meant to apply to. But that doesn't mean it's impossible to translate what you are doing back into a regime that Born's rule does apply to. That translation may be ridiculously inconvenient, but it's still possible.

 

[A. Neumaier]

Doing all the statistics on a quantum computer, and only measuring at the very end, allows you to defer applying Born's rule until the final outcome has been computed. This fixes the theoretical problem you're pointing out.

No. Born's rule claims (in the context you had quoted) that the outcome is one energy level for each measurement done, while after all computations are done one has in addition to the energy levels all energy level differences. This is independent of anything related to ancillas and deferred interpretations. How did these measurements materialize? Certainly not through Born's rule!

One needs a modified Born rule with an appendix stating ''but when measuring $H$, the possible measurement results are approximate energy differences. They are obtained simultaneously in the measurement, and one cannot say at all which one of these is the actual measurement value of the energy of any of the atoms that contributed to the spectrum.''

In German there is a saying ''no rule without exception''. Born's rule has many such exceptions...

 

[Jilang]

Born's rule says nothing about nature (which doesn't know measurements), it is about measurements, which are not part of nature but of our scientific culture.

I would suggest it says something about the measurement and the thing being measured.

 

[Strilanc]

No. Born's rule claims (in the context you had quoted) that the outcome is one energy level for each measurement done, while after all computations are done one has in addition to the energy levels all energy level differences.

I don't see how this is a problem. You just encode the energy level differences of the sub-system into the energy of the whole system. This is an easy task for quantum computation.

I can give a concrete example.

A common quantum computation primitive is phase estimation, which takes an input vector and an operation and, if the vector is an eigenvector, tells you how much the vector is being phased by. The result is stored in binary: if you have a 10-qubit register, and the resulting value is 1011011100, then the eigenvalue is probably pretty close to $exp(i 2 \pi \cdot 732/1024)$. If the input vector is not an eigenvector, you instead get a superposition of results based on decomposing the input vector into the eigenbasis and the corresponding eigenvalues.

Phase estimation is our analogy for "measuring the energy level". It looks like this:

Now, instead of doing phase estimation once, do it twice. This produces two registers, each storing a superposition of estimated eigenvalues. To get the difference in energy levels, just subtract one register out of the other:

For the example operation $U$ that I chose, the register size shown is too small. All the differences are getting smeared over each other when subtracting. I initially chose a simpler operation, but then the spectrum was too boring. To make the process more accurate, I'd add more qubits to the phase estimation register.

Quantum computation is Turing complete. You can do more than just subtract; you can build up statistics. And although the individual operations we are applying may have their own eigenvalues, we can arrange things so that the eigenvalues of the circuit as a whole can tell you many separate things about the eigenvalues of the sub-operations.

Does that make it clearer?

 

[Prathyush]

The more I follow Arnold's arguments the less I understand what his criticism against Born's rule is about :-(.

I agree :-(

You mean, in the way Born's rule says! I imply nothing else than what Born's rule says about measuring observables. And there is no doubt that the total energy is one of the most important observables in quantum mechanics.

Quantum mechanics allows us to calculate amplitudes for processes. And Born's rule tells us that probability is amplitude square. In the context of measurable quantities, one needs to construct a suitable apparatus, and show that that appratus indeed measures energy. We we do in practice is we measure scattering amplitudes, which require Born's rule to interpret. Show me an experiment where the sense in which I define the Born's rule is ambiguious.

 

[vanhees71]

Many PF threads with debates in the QM section end up with "let's agree to disagree". This and the other current one will share no different fate.

Yes, obviously Arnold and I are unable to communicate our points of view to each other. For one last time I want to emphasize that I strongly disagree with him that Born's rule (both "parts" of it as it seems to be understood by the majority in this forum, i.e., that eigenvalues of the self-adjoint operators representing observables are the possible outcomes of precisely measuring them and the usual probabilistic meaning of the states) is in any way disproven. Would that be the case, it would mean a scientific revolution in physics, only paralleled by the discovery of QT in 1925/26 itself. With that said, I don't participate in this discussion anymore.

 

[Prathyush]

Yes, obviously Arnold and I are unable to communicate our points of view to each other.

I can follow Arnold when he says that in the usual spectroscopy experiments we are not infact measuring <H> but transition amplitudes(and hence probabilities). But that is not a problem and is not in contradiction with the Born's rule, the founders understood how to use quantum mechanics and correctly apply it. Born's rule is probability is amplitude modulus square. Without Born's rule we have no natural way to convert amplitudes, which the quantum formalism allows us to calculate into probabilities which are measured.

 

[atyy]

Yes, obviously Arnold and I are unable to communicate our points of view to each other. For one last time I want to emphasize that I strongly disagree with him that Born's rule (both "parts" of it as it seems to be understood by the majority in this forum, i.e., that eigenvalues of the self-adjoint operators representing observables are the possible outcomes of precisely measuring them and the usual probabilistic meaning of the states) is in any way disproven. Would that be the case, it would mean a scientific revolution in physics, only paralleled by the discovery of QT in 1925/26 itself. With that said, I don't participate in this discussion anymore.

I agree with vanhees71 totally on this point.