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Is there any fundamental demonstration that result of an experiment cannot have been determined before the measurement, yet according to the probabilistic rules of quantum mechanics?

I understand that Bell's theorem demonstrates that the outcome cannot be a result of a deterministic hidden system, but is there any problem that the states are defined by quantum probabilistic rules before the measurement is made?

To be more clear:

I am not trying to question if quantum probability is similar to classical probability. I am indeed questioning in terms of the interpretations of quantum mechanics.

As I understand, the most accepted interpretation is that the collapse of the wave function occurs at the instant of measurement. This raises the problematic of information being apparently transferred at speeds faster than light and so on and the other features of entanglement phenomena.

The violations of Bell's inequality, if I understand it correctly, demonstrate that the properties of the entangled particles being probed (spins for instance) could not result from an intricate classical probability dependent on hidden variables. This is ruled out, clearly, such that the outcome of the measurement is effectively probabilistic in the quantum mechanical sense.

However, I could not find a discussion on whether the definition of the particle properties could have occurred BEFORE the measurement, perhaps at a random moment, yet respecting quantum probabilities.

For example, if we perform two measurements of the properties, one after the other. The first measurement supposedly collapses the wave function and provides a specific set of properties for the entangled particles. The second measurement of the same properties would simply confirm the the first measurement. What if the author of the second measurement didn't know about the first measurement? How one should interpret his/her observation? Is that any difference from the second author thinking that he/she was the first one to measure and interpret that the wave function collapsed at the instant of his/her measurement?

If, from the point of view of the second measurement (without knowledge of the first measurement) the result is identical to thinking that it was the first measurement at all, could we think (with no observable differences) that the wave function collapsed at any time between moment the system was produced and the instant of the measurement?

And if so, even more speculatively: could vacuum fluctuations be the reason for the perturbation and ultimate probabilistic collapse of any wave function, thus decoupling the progress of reality from the observations in a philosophical way?

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  • $\begingroup$ Could you give an example of what you mean? Otherwise I'm pretty unclear what you mean. $\endgroup$ – Jon Custer Jul 31 at 15:45
  • $\begingroup$ I will edit the question to be more clear. $\endgroup$ – leandro Jul 31 at 16:20
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Your question is expressing a muddle about the measurement problem in quantum mechanics. You ask "is there any fundamental demonstration that the state of the system cannot have been determined before the measurement, yet according to the probabilistic rules of quantum mechanics". This is a bit unclearly asked, I think, but I take it you want to know if quantum behaviour is like classical probability, where one gets probablistic outcomes too. The answer is "yes and no". The quantum predictions for the outcomes of a given complete scenario, including whatever measurement apparatus is involved, give perfect ordinary probabilities which add to $1$ and behave in all respects in the way we expect for probability. On the other hand, these predictions are not statements about a quantum system alone; they are statements about the outcome of an interaction between the quantum system under discussion and whatever larger apparatus is involved---the apparatus commonly called "measuring apparatus".

Let's unpack this a little.

The standard way to formulate quantum mechanics asserts that any given system evolves according the Schrodinger's equation. This means that if some system is prepared and then evolves, then the state at some later time is fully determined by the theory, before any measurements are carried out, but this does not resolve all the issues to do with measurement.

By solving Schrodinger's equation one finds that before a measurement the state of the system is $$ |{\psi(t)}\rangle = {\cal T} \exp\left( -i \int_0^t \hat{H} {\rm d}t/\hbar \right) |\psi(0)\rangle $$ where $|\psi(0)\rangle$ is the state prepared at some initial time and $\cal T$ is an operator called time-ordering operator, or Dyson time-ordering operator, which is required when the Hamiltonian is time-dependent (to be precise, when the Hamiltonian at one time does not commute with the Hamiltonian at another).

I have written this admittedly abstract answer in order to make it clear that the state of the system is completely determined before any measurement is made.

However, the interaction commonly called 'measurement' typically involves an interaction of the system with a complex apparatus, typically involving some sort of permanent change such as the avalanche in a photodetector or something like that. Tracking the details of such an avalanche leads one into the metaphysical puzzles surrounding measurement and interpretation of quantum theory. But for most results of interest we can avoid those metaphysical puzzles by asserting that the outcome of such an interaction is captured by a collapse postulate. So we note which are the states $| M_n \rangle$ which form the basis which a given type of measurement distinguishes in a stable way, and we just assert that the system state changes during the measurement so as to go to one of those states, with a probability given by $$ \mbox{Prob[obtain $M_n$, given state $\psi(t)$]} = \left| \langle M_n | \psi(t) \rangle \right|^2 $$

This brings out the issues I mentioned in my opening paragraph. For, the state $|\psi(t)\rangle$ is determined, before the measurement, but this does not suffice to determine the result of each run of such an experiment, except in that it furnishes the above probabilities.

The essence of the issue here can be expressed as saying that, given a fixed type of measuring apparatus---the apparatus that is physically present in any given scenario---the quantum state $|\psi(t)\rangle$ is not so much the "state of the system" as a convenient way to furnish the probabilities. It is those probabilities which are the actual prediction of the theory, and the basis states accompanying them.

The projection $| \psi(t) \rangle \rightarrow | M_n \rangle$ is often presented in discussions as if it is a physical process (called 'collapse') undergone by the system, but one does not absolutely have to see it that way. Rather one can say that the physical world evolves between states that are the stable outcomes of irreversible processes, and the mathematical apparatus of quantum theory yields the probabilities for moving between these states. (This type of interpretation is close to the one called Copenhagen Interpretation, but it tries to be specific about what constitutes measurement by making an appeal to irreversibility).

If one sees $| \psi(t) \rangle \rightarrow | M_n \rangle$ as a physical process then one can assert that it takes place at any time between preparation and measurement---that is, at any moment during the time interval when ordinarily we say the evolution is unitary. To be precise, one would assert $$ |{\psi(t_c)}\rangle \rightarrow {\cal T} \exp\left( -i \int_{t_m}^{t_c} \hat{H} {\rm d}t/\hbar \right) |M_n\rangle $$ where $t_c$ is the moment assigned (rather arbitrarily) to this collapse, and $t_m$ is the time later on when one deems that the whole process is over. Thus in this way of looking at it, the system is deemed to collapse to whatever state is the one that evolves into the one observed later on as the outcome of the measurement. Quantum mechanics is not normally presented this way, but since it has the same outcomes and same probabilities it is equivalent to the more common interpretation where we suppose the collapse happens at the last minute. (But again, one need not regard the collapse as a physical process at all; see above).

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  • $\begingroup$ Thanks Andrew for your answer, but I think it is not exactly what I was expecting. I have edited the question in order to be more clear. $\endgroup$ – leandro Jul 31 at 16:19
  • $\begingroup$ @leandro ok; I added a paragraph which I think deals with what you had in mind. See also 'weak measurement' for some further physics related to this. Sometimes it is helpful to specify a problem not in terms of initial conditions, nor final conditions, but a combination of both. $\endgroup$ – Andrew Steane Jul 31 at 16:44
  • $\begingroup$ Thank you. Indeed, QM is generally presented such that "collapse" occurs at the last minute. That would imply a reality which is completely dependent, at every instant, and somehow subjectively, on the observations. If the alternative, collapse occurring at any prior moment, is indistinguishable, "reality" exists independently of the measurement, and that is much more acceptable, I think, and transposing that to classical behavior seems much simpler. I have not found many discussions about that. $\endgroup$ – leandro Jul 31 at 16:52
  • $\begingroup$ Add on: I understand that the discussion does not make much sense if we think, for example, of the position or spin of an electron in an atom, for example, which "collapses" ONLY when measured, and due to the interaction with the measuring apparatus. But when we deal with these entanglement phenomenon in which the particles are far from each other, even if the result is the same from an observational point of view, the implications for the interpretation are important, I think. $\endgroup$ – leandro Jul 31 at 16:54

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