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Some context:

An ideal quantum measurement (in Von Neumann's sense) is described by the following situation:

There is a hilbert space $\mathcal{H_S}$ with basis $|\sigma_i\rangle_{1\leq i \leq n}$ describing the system and a hilbert space $\mathcal{H_M}$ with basis $|\mu_i\rangle_{1\leq i \leq n}$ describing the measurement apparatus.

We want the measurement apparatus to tell us in which state the system is. We also assume that the measurement apparatus takes a state $|M_0\rangle$ as a pre-measurement state. Lastly, we assume that the measurement apparatus doesn't backact on the system - that the measurement interaction doesn't change the state of the system.

The system-measurement apparatus interaction is then described by $$|\sigma_i\rangle_{\mathcal{S}}\otimes|M_0\rangle_{\mathcal{M}}\rightarrow |\sigma_i\rangle_{\mathcal{S}}\otimes|\mu_i\rangle_{\mathcal{M}}.\tag{1}$$

Above characterization is taken from p.51 ff. out of the following book: 978-3-540-35773-4.

The question itself:

Now the question for me is how "no backaction" or "the measurement interaction doesn't change the state of the system" is the correct description for whatever measurement is represented formally. Obviously there is no change in the state of the system ONLY if the system is in one of the states $|\sigma_i\rangle_{1\leq i \leq n}$ prior to the measurement. This can be seen from (1). In all other cases system and measurement apparatus will end up in an entangled state after the measurement - which very well is a change in the state of the system due to the measurement interaction.

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2 Answers 2

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von Neuman's mathematics is simply the solution theory of the Schroedinger equation. It doesn't tell you anything about the physics of the measurement process. From an educational point of view that is not even necessary because you have been told about it already at the high school level (at least in countries in which the photoelectric effect is part of the high school curriculum) and for certain in your introductory atomic physics course.

An actual measurement in physics is an irreversible energy transfer. In the case of quantum mechanics this is usually done by measuring a quantum of light (a photon) or one of the massive "particles" (electrons, muons, neutrinos, mesons, protons, neutrons, nuclei) etc.. Each of these quanta carries an amount of energy, momentum, angular momentum and a number of charges (electric, lepton number etc.) that are conserved or nearly conserved. The "information" about the quantum system is in those physical properties.

So... the "real" measurement process is what we are talking about in atomic physics (optical spectroscopy) and high energy/nuclear physics (spectroscopy with particle detectors). The experimental physics textbooks give you the recipe for how to "translate" that into the solution theory language that dominates the theoretical quantum mechanics textbooks.

And this also answers the question about the "backreaction". In order to make a measurement on a quantum system, we have to exchange a quantum of energy/momentum/angular momentum/charges with it. Since these are conserved quantities, what the quantum system loses in energy etc. is gained by the measurement system and vice versa.

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  • $\begingroup$ So to put it short: In practise, measurement apparatuses can only detect those $|\sigma_i\rangle$ states which is why they are the only ones to be considered when looking at "wheter the interaction has backaction on the system"? Is that a way to put it, without going into detail of the exact physical context? $\endgroup$
    – manuel459
    Commented Apr 4, 2023 at 20:45
  • $\begingroup$ @manuel459 A measurement detects a quantum of energy. That's not a state. A state is an ensemble property made of an infinite number of such quanta. The solution theory says absolutely nothing about the single quantum. It doesn't tell us where it will be measured, when, or what energy, momentum or angular momentum it will have. It only tells us about the statistical distribution of such quanta. The physical context is that quantum mechanics is an ensemble theory. It is not a theory of individual outcomes. What is "disorienting" is that there is no underlying theory for the individual outcome. $\endgroup$ Commented Apr 4, 2023 at 21:07
  • $\begingroup$ I think we are talking past each other. In an abstract way of describing it (e.g. in quantum informational systems when talking about qubits) one usually talks about quantum states. $\endgroup$
    – manuel459
    Commented Apr 4, 2023 at 21:15
  • $\begingroup$ @manuel459 One can talk about a lot of things in physics all day long, but you were asking about the measurement process. That is always an interaction that involves an amount of energy. There is a pretty trivial explanation for that: because of the third law of thermodynamics the (classical) measurement system has an effective temperature T, which means that any amount of energy <kT is indistinguishable from thermal noise. This means that we need to extract an amount of energy >kT from the quantum system to learn something about it. $\endgroup$ Commented Apr 4, 2023 at 21:18
  • $\begingroup$ @manuel459 What is "ingenious" about von Neumann is that he managed to abstract that ugly reality away completely. What is problematic about it for teaching purposes is that he managed to abstract it away completely. Now students who learn about "states" in quantum mechanics have a hard time correlating that level of mathematics to reality. I had a hard time, too... until I got to build experiments. At that point it all becomes about measuring small amounts of energy, which clarifies what really happens in quantum systems. $\endgroup$ Commented Apr 4, 2023 at 21:21
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After further research I finally found a somewhat satisfactory answer for my question. One could define "ideal measurement" as a measurement that, if repeated shortly after its first realization, would lead to the same result. This implies the fact, that eigenvalues of the observable are not changed by the measurement. Source: "Quantum philosophy" by Roland Omnès, p.232 (ISBN: 0-691-02787-0).

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    $\begingroup$ All measurements do that. That's called the quantum Zeno effect. If you take the energy out of an atom by detecting a photon and then repeat the measurement, where in the world would new energy to get the atom excited again come from? If Omnes has written such nonsense, then he really doesn't understand how nature works. I mean... that's a trivial "insight" at the level of high school physics. $\endgroup$ Commented Apr 5, 2023 at 3:57
  • $\begingroup$ Maybe from the measurement device? I don't know. All I know is that the notion of an ideal measurement (in contrary to "real" measurements) the system shouldn't change the state of the system (or rather the eigenstates of observable). That's what (in all literature I use) makes an ideal measurement ideal. $\endgroup$
    – manuel459
    Commented Apr 5, 2023 at 11:21
  • $\begingroup$ There are no "ideal" measurements in nature. There are only real measurements. That's part of the problem with the way we teach the theory of quantum mechanics. It idealizes so much that people stop distinguishing between theory and reality. All real measurements are removing energy from the system. Copenhagen describes that properly, by the way, by distinguishing between free dynamics and Born rule. Whenever we apply the Born rule, we model the process that takes energy out of the system. $\endgroup$ Commented Apr 5, 2023 at 16:53

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