There is no “measurement problem” in Quantum Mechanics

22Oct22

The purpose of this post is to show that there is no mystery in the measurement process from the perspective of standard quantum mechanics. It is based on my (rather vocal) comment under the recent ycombinator news story which again spreads popular, but misguided views about quantum mechanics.

The community

Many people I know seem to think about quantum mechanics as this ultimate complicated theory that is impossible to learn. This is of course in some aspects a fair statement, but not surprising given the amount of wrong things one can read about it.
Many physicists I know, just go along with it and apply it. It was even the advice given by my second quantum mechanics lecturer – which for me was the ultimate incentive for learning it deeply.
Small random (in the sense of academic achievement) subgroup of my fellow physicist, when asked in depth, agreed that doing quantum mechanics (QFT etc.) leaves them “feeling” differently than when applying the rest of the theories in physics (including General Relativity), as if there’s some missing “enjoyment” in doing quantum mechanics. I certainly agree with this point and I have some suspicions about the answer to why it is so, and no, it’s not only about its probabilistic nature, but I will leave the topic for another post.
There’s also Jarek Duda, who, like some others, sometimes older physicists, once at peace with QM, suddenly can no longer accept its truths and are looking for an alternative theory. I have respect and sentiment towards Jarek’s and other attempts, because it represents the true quest for truth, even if many of the attempts fail. Besides, he clearly deserves to be given the time to pursue his scientific itches given his outside-the-box thinking skills.
Finally, there’s one last group: The Patchers. The group is mainly composed of new-comers, but sometimes people with academic title who just got a little lost. It might have been due to the fast-paced academic curses and the general attitude of just following the maths. The existence of the older folks in the previous group, the Rejecters, and the media (even journal article headlines!) certainly accelerate the enthusiasm of newcomers – who, unequipped with enough understanding of the existing framework to build something entirely new, try to patch places which not only do not require patching – the patches won’t “stick to it” even in principle.

Not surprisingly, many of them skims through the fundamentals and goes straight to the Schrödinger Cat like paradox. After all, that’s the ultimate thought experiment that tries to reconcile the quantum framework with macroscopic observations. And no, naming it “the measurement problem” doesn’t make any difference. It’s still the same old “insight” as it was in 1930.

In what follows, I’ll try to present how standard quantum mechanics gets out of the Schrödinger Cat paradox and convince you that just supplementing or patching the Born rule isn’t an option. I will not consider interpretations of quantum mechanics as they do not bring much to the table for a physicist. I personally believe that the reasons quantum mechanics is the way it is can and will be found, but an interpretation is just a shoot in the dark. There’s no way to verify it.

The gist

The reasoning below comes from John von Neumann and from Wojciech Żurek. I’m fairly certain that von Neumann was aware of the details even if he didn’t spell out all of them.

To get the argument across, let’s assume that $\psi$ describing our particle is a superposition of two eigenstates:
$|\psi \rangle = c_1 |1 \rangle + c_2 |2\rangle$ , where $|c_1|^2 + |c_2|^2 = 1$ .
Without loss of generality we can pick:
$c_1 = x$ and $c_2 = \sqrt{1-x^2}e^{i \phi}$ , where $x <1$ is a real number.
The density matrix of this pure state can then be written as $\rho = |\psi \rangle \langle \psi|$ and
one by writing the explicit form of this density matrix one can see that the diagonal terms are:
$x^2$ and $1-x^2$ , while the non-diagonal terms are:
$x \sqrt{1-x^2} e^{i \phi}$ and $x \sqrt{1-x^2} e^{-i \phi}$ .

In the most complete scenario of a measurement, the density matrix of the system can change in many ways, including the diagonal terms of the density matrix. However, in this simplistic example, a measurement will, from construction, bring only the non-diagonal terms to zero (I hope most of the interested readers will have enough background to understand why).

Now, the measuring device, as a macroscopic object, will have the number of degrees of freedom far greater than the simple particle of which state we’re about to measure. This number will be the order of the Avogadro number (~ 10^23) – even the smallest human visible indicator will be this big. The measurement, by necessity, includes an interaction of our small system with the enormous measuring device.

Before the interaction, the whole system (the particle and the measuring device) can be written as a tensor product of the two wave-functions:

$|\Omega_{before}\rangle = |\psi\rangle \otimes |\xi\rangle = ( c_1 |1\rangle + c_2 |2\rangle ) \otimes |\xi\rangle$

where $|\xi\rangle$ represents the wave-function of the measuring device and everything it interacts with before the measurement. When the interaction occurs the state of our measuring device changes unitarily (as everything in nature) according to the full Hamiltonian of the system, and with some regrouping of the terms, we can write the state after the interaction as:

$|\Omega_{after}\rangle = c_1 |1\rangle \otimes |\xi_1\rangle + c_2 |2\rangle \otimes |\xi_2\rangle$

This is the true state of the system as unitarily evolved by nature.
The individual subsystems are no longer in pure states, but the whole system $|\Omega_{after} \rangle$ (even if we’re unable to completely describe it) – is.

Now, comes the final part, which some call the “collapse”, but in reality it is just “an average” over all possible states of the bigger (measurement) system which we declared a priori not be our system of interest and states of which we are not able to follow because we measure with it:

${Tr}_{\text{device's degrees of freedom }} |\Omega_{after}\rangle \langle \Omega_{after}|$

In result, we obtain a reduced matrix with only diagonal elements $x^2$ and $1-x^2$ , containing “classical” information about the possible results of measurements. We could use this information to again resume our experiment from a “classical checkpoint”. Note that we have just lost the off-diagonal terms, in this case phases.

Why are off-diagonal terms zero?
Let’s inspect one of the non-diagonal terms over which the above trace is taken:
$x*\sqrt{1-x^2} e^{i \phi}$

It is effectively zero, because the trace over the degrees of freedom of the measuring device is a multiple integral, again of the multiplicity of order of Avogadro’s number and a similar number functions which change in various ways. It is enough that only a fraction of such integrals will have a value lesser than 1 to guarantee that the product will be equal to a practical zero.

And this is all. I hope that it now becomes clear that the measurement procedure has a well-established place in quantum mechanics and is built on top of previous assumptions about its probabilistic nature. The very unfamiliar features (to a classical mind) of quantum mechanics occur at the very start, where we are made to accept superpositions: not when we convert the information to the macro environment. Hence, any attempt to change “patch” or “reformulate” the measuring process would need to reject quantum mechanics completely, because probability calculus is at the very heart of it.

Bonus aid #1

Simple truths about quantum mechanics:

1) The wave-function is only the DESCRIPTION of the underlying phenomena.
2) Within this description, everything, and I mean everything, evolves unitarily. No exceptions ever.
3) Whenever you decide to measure, i.e., probe the microscopic system with an object that is not within your quantum description, i.e., have no details about all the phase/amplitude information you’re destined to average/trace over the unknown states (apply the born rule), as was done above. That’s always what we’re left with in case of a large system outside of our description interacting with a small system within our description.

On the other hand, if you put a small quantum system, with another small quantum system (say two particles), there’s no need to trace/average/apply the born rule immediately because your description can be complete both in principle and in practice: you can just unitarily evolve the system for as long as you wish/can compute for.

However, sooner or later you’ll want to measure, because ultimately that’s what physics is all about – verifying your predictions with experiment – and you go back again to small vs big, because that’s the only way we, humans, can perceive the microscopic reality. The result will be completely analogous to the one before, the only change being that you’ll now be able to predict probabilities of a two-particle system.

Do you feel uneasy with QM wave-functions which have the status of descriptions of reality?
Go and study classical field theory in which the fields are to be thought of as real physical entities.
go step deeper and you’re in quantum field theory in which you deal with descriptions again.
Would a theory in which we deal with “real physical entities” be better than that of “descriptions”?

Bonus aid #2

There are some parallels between electrodynamics and QM that might help a little in making QM less unfamiliar.

One example is that you can have your complete description with the four-potential, about which you know, from the Aharonov-Bohm effect, carries more information than electric/magnetic fields alone, although fields are the only thing we measure – not the electric/magnetic potentials. The potentials were the side product of the formalism that turned out to have real consequences. In a similar matter, we learned the importance of the wave-function/phases in the description, even though we only measure probabilities.

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