QM Measurement: Randomness Explained

[McLaren Rulez]

Hi,

In QM, when we make a measurement, our wavefunction collapses into one of the eigenstates of the operator. This process is supposed to be random for any single measurement but obeys some statistics if we make a large number of measurements. Could someone explain how we know that it is indeed random for each measurement?

I guess this should have been the first question I asked when I started QM but somehow, I never did. At first, this sounded similar to saying that there is a hidden variable which determines which eigenstate it collapses into, so I thought that Bell's inequalities deal with it. But everything I can find about Bell's inequalities is talking about entanglement and non-locality and other things that I don't really know much about.

Thank you.

 

[bapowell]

Bell's Theorem imposes a limit on the amount of correlation that entangled states can exhibit. Are you familiar with the EPR paradox? EPR was the original thought experiment that suggested the incompleteness of QM, and tacitly advocated for additional ingredients (e.g. hidden variables.) It does have to do with wave function collapse, but that's not the spooky part. The spooky part has to do with wave function collapse of entangled states: say, two electrons arising as decay products with zero total momentum. A measurement of the z-component of the spin of one of the electrons will not only collapse the wave function of the measured electron, but it will also determine the state of the unmeasured electron -- instantaneously. This is non-locality, and it appears to be a necessary, if not unsettling, ingredient of QM. Bell's theorem showed that, if there are indeed hidden variables, then they must be non-local in order to explain the high degree of correlation seen in EPR-type experiments.

With regards to your first question, we know the collapse is random because of experiments, e.g. Stern-Gerlach. A single electron has a 50% of passing through the selector.

 

[McLaren Rulez]

Thank you for your reply bapowell.

I know a little bit about the EPR paradox and the way entagled states have correlated statistics. But I don't see the connection between that and my question.

You mention that we know it is random because of experiments like the SG experiment. How so? To claim that it is random means that we have to prove that there is nothing we can possibly do or know to determine the outcome of each individual measurement. This has nothing to do with locality or entaglement as far as I can tell, its just basic measurement in QM.

EPR was the original thought experiment that suggested the incompleteness of QM, and tacitly advocated for additional ingredients (e.g. hidden variables.) It does have to do with wave function collapse, but that's not the spooky part.

Could you explain how Bell's theorem proved that EPR is wrong if it advocates the presence of a hidden variable that controls the eigenstate into which the wavefunction collapses?

Thank you.

 

[bapowell]

I know a little bit about the EPR paradox and the way entagled states have correlated statistics. But I don't see the connection between that and my question.

The connection is in your comment "At first, this sounded similar to saying that there is a hidden variable which determines which eigenstate it collapses into, so I thought that Bell's inequalities deal with it." I was trying to explain how Bells' inequality fits into the picture of wave function collapse by describing its connection with EPR.

You mention that we know it is random because of experiments like the SG experiment. How so? To claim that it is random means that we have to prove that there is nothing we can possibly do or know to determine the outcome of each individual measurement.

So you are asking about whether there are hidden variables that control the state of the particle vs. whether wave function collapse determines the reality. In either case, though, a measurement like SG will give you tried and true random results because 1) either hidden variables are true and, well ,they're hidden -- we cannot know how they affect the outcome, or 2) wave function collapse is the true mechanism and the particle's state really isn't determined until the act of measurement is made. In order to tell the difference between hidden variables and the Copenhagen interpretation, statistical measurements of correlated particles are necessary. These experiments show us that the correlations are too large to be explained by a local hidden variables theory, via Bell's inequality.

Could you explain how Bell's theorem proved that EPR is wrong if it advocates the presence of a hidden variable that controls the eigenstate into which the wavefunction collapses?

Bell's Theorem doesn't advocate hidden variables. It provides a prediction for the amount of correlation expected between entangled particles under the assumption of local hidden variables. It didn't show that EPR is wrong -- read the paper to see why. EPR can still be satisfied with non-local hidden variables, but many find these just about as unpalatable as Copenhagen QM is.

EDIT: Also, your language is confusing. Hidden variables don't control wave function collapse -- they replace it as a concept. Wave function collapse implies that the quantum system does not exist in a prescribed state until the act of measurement. Hidden variable theories, on the other hand, propose that the state is fully determined from the get go, and that the statistical nature of measurement is instead due to our lack of knowledge of the dynamics controlled by the hidden variables.

 

[Ken G]

To clarify, hidden variables are not ruled out by experiments involving Bell's inequality, what is ruled out are hidden variables that "come along with the particles themselves", i.e. hidden variables that exhibit "local realism." The deBroglie-Bohm interpretation of quantum mechanics goes through considerable contortions to be able to restore a concept of hidden variables, and allow the "collapse" to not be truly random. However, like bapowell said, they are still completely hidden from us, and they are not "carried along by" the particles themselves, so it's not really clear if deBroglie-Bohm is really accomplishing anything other than allowing us to cling to a predisposition toward wanting things to be deterministic. If I detect a similar desire behind the OP question, then you may be reassured that determinism is restored by deBroglie-Bohm, but it is determinism of the nonconstructive and untestable (some might therefore say unscientific) variety.

 

[McLaren Rulez]

Thank you for the replies.

I realize my wording might have been confusing but Ken G is talking about what I wanted to know.

So how do we know that the variable that determines the outcome of each measurement is not carried along by the particle itself? That is, suppose one claims that when we measure the spin in an SG experiment, the electron has some parameter that the experimenters don't know which tells it to go through or be rejected. How do Bell's inequalities show this parameter does not exist?

Thank you.

 

[Ken G]

So how do we know that the variable that determines the outcome of each measurement is not carried along by the particle itself?

That's the violation of Bell's inequality. There has to be some "holistic" character to an entangled system, which deBroglie-Bohm encodes in a holistic "probability current." The trick they employ is that they assert the uncertainty in the probability current traces back to uncertainty in the initial condition, rather than being a fundamental uncertainty. So they still get the algebra of the Heisenberg uncertainty principle, but they interpret that as an algebra of our initial uncertainties propagate forward-- they don't propagate forward like the classical concept of a probability, but rather like the quantum mechanical concept of a probability amplitude. Plus, the probability currents encode correlations within entangled systems. By allowing probabilities to include phase information, they can get all the same predictions, while still maintaining that the uncertainties we encounter in experiment stem from the way initial uncertainties propagate. The basic problem as I see it as that neither "determinism" nor "random" are attributes of our models, they are attributes of interpretation of our models (in particular, interpretations of the meaning of an "initial condition"), and can often be interchanged rather casually.

That is, suppose one claims that when we measure the spin in an SG experiment, the electron has some parameter that the experimenters don't know which tells it to go through or be rejected. How do Bell's inequalities show this parameter does not exist?

You can't tell by looking at one electron, but you can tell by looking at entangled electrons. Their correlations cannot be explained by hidden variables that "come along with the electrons", it requires the holistic character of a joint wave function.

 

[McLaren Rulez]

Okay, I was clearly under the mistaken impression that we could somehow disprove that a single electron came with its own hidden variable. Looking at entangled states makes more sense.

Thank you very much, Ken and bapowell.

 

[Ken G]

I can offer a very simple counterargument to any unconstrained claim that hidden variables are impossible. Just say "God did it." I don't mean this as a claim on any religious viewpoint, merely that the ultimate "hidden variable" is some kind of equivalent to a "mind of God." If we assert that it is the "mind of God" to follow the statistical distributions predicted by quantum mechanics, but that each individual instance reflects a purposeful decision (or even, requires such a decision), then the decision-maker is a "hidden variable." I point this out simply to show that any no-go theorem about hidden variables must spend some time elucidating just exactly what a hidden variable actually is. Bell's inequality rules out an important class of such hidden variables-- the class that is carried locally along with the particle it governs, so includes no holistic content that sees "over the trees" if you will. But to distinguish these types of hidden variables, it is necessary to have something over those trees-- a single particle by itself could never be decisive there.

However, you don't necessarily need entangled particles-- you could consider an ensemble of unentangled particles that are all prepared in the same state. But this is kind of a trivial case, because you can't tell if these particles are entangled or not-- entangled indistinguishable particles that are all in the same state look just like unentangled particles that are all in the same state. The key for me is there is still an "over the trees" element-- these many particles will be off interacting with measuring devices, and the failure of various classes of hidden particle descriptions will have to involve combining those outcomes and seeing how it all hangs together. The Bell inequality is one way to do that, which involves nontrivial entanglements, but it's not the only way to address the issue.