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:
- The premise of treating the measuring device as part a quantum system that interacts with the thing being measured is faulty.
- 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.