In many interpretations of quantum mechanics, the result of a measurement is regarded as non-deterministic. However, my question is: is the time at which a measurement occurs deterministic? To be precise, given an initial state $|\psi\rangle$ of a system-observer Hilbert space, and the unitary evolution $U(t)$, can I predict the time $t_0$ at which the next measurement will occur?

It seems that, to predict when a measurement will occur, one must have "macroscopic/classical" knowledge (e.g.: "at the double slits, a camera will take a picture of the electron"). But, it seems that one cannot predict that a measurement will occur from "wavefunction-level" knowledge (e.g. "as the electron passes through the slits, photons will entangle with the atoms in the camera").

My understanding is that, in the mathematical formulation of a quantum measurement, one must regard the measuring device as classical to avoid contradictions, meaning that it does not necessarily have to evolve unitarily. Does this have something to do with the answer to my question?

Edit: In response to a comment, here I clarify what I mean by "one must regard the measuring device as classical". The idea here is to try to eliminate the measurement problem by describing the measuring device as a quantum system that interacts with the system being measured, and obtain a contradiction. Suppose the system being measured is a qubit with basis states $|0\rangle, |1\rangle$, and the macroscopic measuring device, which I regard as a quantum system initially in the state $|\text{no measurement}\rangle$, goes to state $|\text{measured 0}\rangle$ if it measures $0$, and goes to the state $|\text{measured 1}\rangle$ if it measures $1$. One can regard these as macroscopic states, like a screen that presents the measurement results to the experimenter. Then

  • If the initial state of the system-device Hilbert space is $|0\rangle|\text{no measurement}\rangle$, then, at $t_0$, the state is $|0\rangle|\text{measured 0}\rangle$.
  • If the initial state of the system-device Hilbert space is $|1\rangle|\text{no measurement}\rangle$, then, at $t_0$, the state is $|1\rangle|\text{measured 1}\rangle$.
  • Thus, if the initial state of the system-device Hilbert space is $(\alpha|0\rangle + \beta|1\rangle)|\text{no measurement}\rangle$, then, at $t_0$, the state is $\alpha|0\rangle|\text{measured 0}\rangle + \beta|1\rangle|\text{measured 1}\rangle.$

In the last example above, now the macroscopic measuring device is in an entangled quantum state, which generally disagrees with experiment. To remedy this, there are two arguments one can make:

  1. The premise of treating the measuring device as part a quantum system that interacts with the thing being measured is faulty.
  2. In reality, there is another measuring device, that measures the system-device joint Hilbert space. This in practice is the same as 1., since this second measuring device is not treated quantum mechanically, lest you run into the same contradiction as above.

This is where the notion of a Heisenburg cut comes in, since one must make a "cut" that separates the system being measured, which is described by a wavefunction, and the observer, which is treated classically.

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    $\begingroup$ The calculation of decoherence timescales is probably the closest thing to an answer to the question you are asking springer.com/gp/book/9783540357735 $\endgroup$ Aug 14 '21 at 1:07
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    $\begingroup$ Regarding the idea that we must regard the measuring device as classical: If "classical" means "its observables are all commutative," which is the usual meaning of that word among most quantum physicists, then we cannot regard the measuring device as classical. If we tried to do that, then measurement could never happen, because the measuring device's observables would all commute with the Hamiltonian, so the thing being measured would be unable to influence the measuring device. Sometimes people use the word "classical" as a synonym for "macroscopic," but that makes the idea a tautology. $\endgroup$ Aug 14 '21 at 3:27
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    $\begingroup$ Independently of my other comments, I have a question to help clarify what you're asking: if a Geiger counter clicks at an unpredictable time because a radioactive atom decays at an unpredictable time, would you consider that to be an example of an unpredictable time-of-measurement? I mean, are you asking if quantum theory can tell us when a measurement will occur in some situations, even if it can't in other situations? $\endgroup$ Aug 14 '21 at 3:31
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    $\begingroup$ Maybe I'm taking your question too literally, but I would expect your supervisor to tell you when to perform the next measurement. $\endgroup$
    – D. Halsey
    Aug 14 '21 at 15:29
  • $\begingroup$ @ChiralAnomaly I've edited the post to explain what I mean by "classical". In response to your second comment: I am referring to the case where one has full knowledge of how the system evolves, and how the measuring device evolves quantum mechanically. In general, assuming this is true, can quantum mechanics tell us when the unitary evolution stops and the measurement occurs? Perhaps radioactive decay wouldn't fall under this purview, since one cannot know in principle when radioactive decay will occur. $\endgroup$ Aug 14 '21 at 19:26

the measurement problem in quantum mechanics is still an open problem, and there is no final consensus regarding what consists a measurement. One of the few things we can say about measuring a system is that if you apply an interaction that changes the system by a lot, you expect this to behave like a measurement.

That being said, you are right when you claim that measurements are not unitary. In fact, there is no unitary operator that can correspond to a measurement, because it necessarily involves a collapse, which is associated with a projector. Regarding the time at which the measurement happens, this is something that is modelled by the physicist that is modelling the system, and thus depends on what will be called "a measurement". That is, you must prescribe the time at which the measurement (and thus the state update) happens to the system. After having prescribed it, everything goes normally.

A general example is the following: consider an initial state $|\psi(0)\rangle$ that will go through a measurement device at $t = t_0$, then the state that will be measured will be $|\psi(t_0)\rangle = U(t_0)|\psi(0)\rangle$. Depending of the outcome of the measurement, we must then project the state $|\psi(t_0)\rangle$ on the eigenspace corresponding to the measured observable (in the case of a projective measurement). In particular, measurements that involve collapse in quantum mechanics are usually modelled as "instantaneous", and the effect of this in relativity is a recent topic of discussion (see e.g. https://arxiv.org/abs/2108.02794).

Other ways to model the measurement in quantum mechanics are usually associated with coupling an external system to the target system, and tracing over the external one. However, a selective measurement must happen at some stage, and this will be instantaneous. The stage at which this measurement happens is usually called the Heisenberg cut.


You've hit on the biggest flaw with the ill-defined and incomplete copenhagen interpretation.

Without "measurement" or "collapse" the unitary theory of quantum mechanics is fully deterministic in the sense that, given initially conditions (This includes an initial wavefunction $|\psi(t=0)\rangle$ and a Hamiltonian $\hat{H}$), it is possible to predict the wavefunction $|\psi(t)\rangle$ for all later times.

However measurement breaks this unitary evolution as you point out. Copenhagen relies on the concept of a measurement, but does not, in a physically rigorous way, define what is meant by a measurement. This means that the theory is incomplete because, even given a complete description of reality (say the wavefunction of all particles), it is not possible to use the copenhagen interpretation to determine when a measurement will happen.

However, there are a number of interpretation or alternative theories of quantum mechanics which are in the vein of the Copenhagen interpretation and which try to make the concept of measurement or collapse more rigorously defined. These can be googled by looking for things like spontaneous collapse theories (notably Ghirardi-Rimini-Weber or GRW interpretation) or non-linear extensions to the Schrodinger equation.

These theories explore the possibility that collapse occurs when a "system" of particles reaches a certain mass, spatial extent, or entanglement participation or something.

They key takeaway I would like the OP and all other readers of this question to take away is the following: the copenhagen intepretation/theory gives us an incomplete picture of the physical world because it does not tell us in physically explicit terms when a measurement should be expected to occur. It's true that we have an intuitive idea for when a measurement should occur, but a physical theory needs to give us a physically rigorous criteria or description.

  • $\begingroup$ I think you should add that it is your opinion that the Copenhagen interpretation is ill-defined and incomplete, whatever this means exactly. What about the other interpretations? The Copenhagen interpretation makes the same predictions as any other interpretation (by definition of interpretation) within in the framework of quantum mechanics. Each interpretation might come with some advantages and disadvantages with regard to certain aspects of QM (like measurement), but all make the same predictions. $\endgroup$ Aug 14 '21 at 15:09
  • $\begingroup$ @Jakob No, the Copenhagen interpretation does NOT make the same predictions as other theories. For example Copenhagen and many-worlds may disagree on the result of a Wigner's friend experiment. I say they "may" disagree because it depends on when collapse happens in the Wigner's friend experiment. Does it occur when Wigner's friend performs a measurement or Wigner? Copenhagen interpretation gives different results depending on the answer. Yes, other interpretations have shortcomings too. This answer doesn't deny that, but it doesn't address or need to address that either. $\endgroup$
    – Jagerber48
    Aug 14 '21 at 16:02
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    $\begingroup$ If an 'interpretation' predicts things which are contradicted by experiments, it is falsified as a theory. I don't see where the Copenhagen interpretation (which uses the exact mathematical formalism of QM as the other interpretations) would disagree with experiment. Could you provide a source? $\endgroup$ Aug 14 '21 at 17:46
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    $\begingroup$ Do you mean things like these discussed here? I just think it is not appropriate to say that it is generally accepted that the CI is ill-defined or incomplete (what I mean is that you state it as a fact); after all, it is an interpretation, which agrees (up to now) with all experimental evidence, i.e. is NOT falsified. Hence my first comment: It is your opinion. The CI was neither proven 'correct' nor 'wrong'; same holds of course true for the MWI. $\endgroup$ Aug 14 '21 at 18:38
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    $\begingroup$ @Jakob see my answer here for a discussion of the term "interpretation". physics.stackexchange.com/questions/651576/…. For me different interpretations need not make the same prediction, but that is simply a semantic point. If you prefer I can call CI and MWI different theories since I'm entertaining the possibility that they make different predictions. But just because they make different predictions doesn't mean we've falsified one or the other. $\endgroup$
    – Jagerber48
    Aug 14 '21 at 20:09

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