I Is information lost in wavefunction collapse?

[.Scott]

There are no "splits" in the MWI in the sense you mean. There is only one wave function and its time evolution is unitary.

Of course, MWI is always described as "splitting". But I am now rereading exactly what Everett was claiming. I guess he was not claiming that this "splitting" created new versions of the universe that acted independently of all other versions. He was just using it as an accounting tool to track the developing wave function. Is that right?

Which, in the MWI, is not the case. In the MWI, there is no unpredictability. The time evolution is always unitary. All of the measurement results happen; each one is appropriately correlated with the appropriate state of the measuring device. All of this is unitary and does not create or destroy any information.

To me, unitary denotes many possibilities that add up to 100% - no more than that. If these possibilities live out independently, then the problem I saw was that it would create each possibility as a new independent starting point.

But I think you're saying that these are not independent. Each continues to have its influence on all the others.

 

[PeterDonis]

I guess he was not claiming that this "splitting" created new versions of the universe that acted independently of all other versions. He was just using it as an accounting tool to track the developing wave function. Is that right?

I think that's a reasonable way of looking at it, yes.

To me, unitary denotes many possibilities that add up to 100% - no more than that.

Unitary means that the inner product between all pairs of vectors in the Hilbert space is preserved. One consequence of that is that probabilities always have to add to 100%, but it's by no means the only consequence; unitarity is a much stronger condition than just that.

I think you're saying that these are not independent. Each continues to have its influence on all the others.

That's possible, but it's not required for the time evolution to be unitary. Decoherence indicates that in practice the "branches" of the wave function do not influence each other after the decoherence time.

 

[.Scott]

That's possible, but it's not required for the time evolution to be unitary. Decoherence indicates that in practice the "branches" of the wave function do not influence each other after the decoherence time.

So, at that point, "in practice", there are independent time lines? But, not in theory? Do these other branches still have a chance at resurrection?

 

[PeterDonis]

So, at that point, "in practice", there are independent time lines?

There are terms in the wave function, written in a particular basis, that do not interfere with each other.

But, not in theory?

No, the theory says the same thing as above.

Do these other branches still have a chance at resurrection?

I don't know what you mean by "resurrection". All of the branches are there. Nothing happens to them. They don't go away. They just don't interfere with each other after the decoherence time.

 

[bhobba]

I don't know what you mean by "resurrection". All of the branches are there. Nothing happens to them. They don't go away. They just don't interfere with each other after the decoherence time.

And after decoherence its in a mixed state so superposition isn't really applicable anyway.

That's basically what's going on - after decoherence each element of the mixed state is interpreted as a world.

There is more to it - a couple of issues:
1. Decoherence requires the Born Rule. How to you prove it from just the concept of state.
2. The modern version doesn't do it quite that way - it uses the concept of history - which is simply a sequence of projections. That way you can speak about something before the Born Rue is even derived. Decoherent histories, which many say is Copenhagen done right, does the same thing. Here instead of a history being a world the theory is a stochastic theory about histories. This is the reason Gell- Mann says in a certain sense the difference between MW and DH is just semantic.

This precisely defining an observation is an issue for all interpretations (including the one I tend to like - the ensemble interpretation). Working with histories is an attempt to fix that. Whether it requires fixing is debatable - but requires another thread.

Thanks
Bill

 

[Stephen Tashi]

And after decoherence its in a mixed state so superposition isn't really applicable anyway.

That's basically what's going on - after decoherence each element of the mixed state is interpreted as a world.

In a non-MWI interpretation of QM, is being in a "mixed" state a meaningful property for a single particle or "system"? Or is "mixed state" only a property of a population of particles or systems? ( For example, in a non-QM setting, when we speak of "the probability that a person is over 6 ft tall", we have in mind picking individuals at random from a population and measuring their height once rather than picking an individual from a population and measuring his/her height at 100 randomly selected times during the day. )

 

[stevendaryl]

In a non-MWI interpretation of QM, is being in a "mixed" state a meaningful property for a single particle or "system"? Or is "mixed state" only a property of a population of particles or systems? ( For example, in a non-QM setting, when we speak of "the probability that a person is over 6 ft tall", we have in mind picking individuals at random from a population and measuring their height once rather than picking an individual from a population and measuring his/her height at 100 randomly selected times during the day. )

Being in a mixed state is not (in my opinion) an objective fact about a system, but is a fact about our model of the system. You can describe an isolated system, such as a single hydrogen atom that is not interacting with anything else, as a pure state. But if a system interacts strongly with the rest of the universe, then you have really two options:

1. Go the MW route, and try to describe the entire universe using quantum mechanics.
2. Describe the system of interest as a mixed state.

I consider it a consequence of how we draw the boundary of what the system of interest is.

 

[stevendaryl]

Being in a mixed state is not (in my opinion) an objective fact about a system, but is a fact about our model of the system. You can describe an isolated system, such as a single hydrogen atom that is not interacting with anything else, as a pure state. But if a system interacts strongly with the rest of the universe, then you have really two options:

1. Go the MW route, and try to describe the entire universe using quantum mechanics.
2. Describe the system of interest as a mixed state.

I consider it a consequence of how we draw the boundary of what the system of interest is.

It's sort of similar to the modeling choices in statistical mechanics. If a system is isolated, you can model it as having definite values of quantities such as pressure, volume, total energy, total number of particles. If the system is in contact with an environment, then those quantities are not constants, so you have to talk about average values for them. You can enlarge the system of interest to include the environment, as well, and then volume and energy and number of particles becomes constants again.

 

[Stephen Tashi]

It's sort of similar to the modeling choices in statistical mechanics. If a system is isolated, you can model it as having definite values of quantities such as pressure, volume, total energy, total number of particles. If the system is in contact with an environment, then those quantities are not constants, so you have to talk about average values for them.

Presentations of statistical mechanics are often unclear about what is meant by an "average" value. To define an expected value precisely, it must be an expectation of a specific random variable. There can be averages with respect to randomly selected times, averages with respectd to a randomly selected container of gas, averages with repsect to a randomly selected point of space, etc. What kind of "average" is involved in a mixed state?

 

[stevendaryl]

Presentations of statistical mechanics are often unclear about what is meant by an "average" value. To define an expected value precisely, it must be an expectation of a specific random variable. There can be averages with respect to randomly selected times, averages with respectd to a randomly selected container of gas, averages with repsect to a randomly selected point of space, etc. What kind of "average" is involved in a mixed state?

Technically, if you know the wave function for a composite state (system of interest + environment), then you can get a corresponding mixed state by

• Forming the composite density matrix.
• "Tracing out" the degrees of freedom that you're not interested in.

It's a kind of average, in the sense that the resulting density matrix can be written in the form:

$\sum_j p_j |\psi_j\rangle \langle \psi|$

which can be sort of thought of as a weighted average of different pure state density matrices $|\psi_j\rangle \langle \psi_j|$.