I Many measurements are not covered by Born's rule

[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.