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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.

Solarflare0
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2 Answers2

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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.

Rick
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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.

Jagerber48
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