Is information lost in wavefunction collapse?

[Ian J Miller]

Then surely, taking your electron spin as an example, prior to observation, which is argued to determine the spin, because the spin is not determined the Universe does not know what will be determined, so information is created. Alternatively, the Universe could be argued to know that it has spin that will be determined one way or the other, but it doesn't know which so after determination it still knows there is spin, but it knows which. In this, "know" does not imply some sort of God; it is just I can't think of a better word. The question is, what is determined prior to observation as opposed to after.

Of course if the debate ends up with what does the formalism say, then I agree I am wrong and bow out.

 

[stevendaryl]

Then surely, taking your electron spin as an example, prior to observation, which is argued to determine the spin, because the spin is not determined the Universe does not know what will be determined, so information is created.

Yes, but more information is lost.

 

[PeterDonis]

prior to observation, which is argued to determine the spin, because the spin is not determined the Universe does not know what will be determined, so information is created.

This assumes that only one result happens, but that is interpretation dependent. In the many worlds interpretation, all results happen (each result for the measured system is correlated with the corresponding state of the measuring device) and the time evolution is always unitary, so no information is created or destroyed.

 

[bhobba]

Think of the cat paradox. You can assert that the cat is in a superposition of quantum states, but you cannot know that.

There are a number of ways of resolving the so called Schrodinger's Cat paradox, but I think the simplest is to realize that because a cat is a macro object interacting with it's environment it has for all practical purposes an exact position. If you consider the cat to be made up of a large number of small parts - not so small they are quantum but large enough they can be considered classical then they to have an exact position. Now consider the constituent parts of a live and dead cat - they have entirely different positions eg the live cat has a beating heart, expanding lungs etc the dead cat just sits there - dead. Sine they have for all practical purposes exact positions of those small parts they cannot be in a superposition. In other words you can't have eg superposition of a live and dead cat. There is another argument based on the fact the cat is entangled with the atomic source. Now if you chug through the math of entanglement you find the cat acts as it it. With a certain probability, is alive or dead - its what is called a mixed state without going into what that exactly is. But it is not, and never is in a superposition.

Thanks
Bill

 

[Stephen Tashi]

. What they are saying is that there is information in the initial state of a system that is lost when you make a measurement. If you have an electron that is in a superposition $\alpha |u\rangle + \beta |d\rangle$, there is information in the coefficients $\alpha$ and $\beta$ which is (apparently) lost forever if you measure the spin.

My interpretation of that example is that the "information" being discussed depends on how complicated the state of a system is. So a state whose description needs two complex numbers has more "information" than state that needs only one bit to describe it.

So if Nature allows a system to transform from a complicated state to a simpler state, then Nature has lost information -with respect to that particular system.

However, is there some quantitative definition of "information" that implements this concept? Does a state whose description requires four complex numbers have twice the information as a state whose description requires only two complex numbers? Must two distinct descriptions of the state of the same physical system, have the same amount of information?

 

[stevendaryl]

However, is there some quantitative definition of "information" that implements this concept? Does a state whose description requires four complex numbers have twice the information as a state whose description requires only two complex numbers? Must two distinct descriptions of the state of the same physical system, have the same amount of information?

Well, there's the Shannon definition of information, which is the number of bits necessary to specify a parameter. A real (or complex) number has an infinite amount of information, while spin-up/spin-down has 1 bit of information.

 

[Lord Jestocost]

H. Dieter Zeh in "Roots and Fruits of Decoherence" (https://arxiv.org/abs/quant-ph/0512078):

"The collapse of the wave function (without observing the outcome) or any other indeterministic process would represent a dynamical information loss, since a pure state is transformed into an ensemble of possible states (described by a proper mixture, for example). The dislocalization of quantum mechanical superpositions, on the other hand, leads to an apparent information loss, since the relevant phase relations merely become irrelevant for all practical purposes of local observers."

 

[.Scott]

This assumes that only one result happens, but that is interpretation dependent. In the many worlds interpretation, all results happen (each result for the measured system is correlated with the corresponding state of the measuring device) and the time evolution is always unitary, so no information is created or destroyed.

No.
If you start out knowing that you have a collection of possible states, and then the result is that collection, then the collection as a whole does not represent either and information loss or gain.
But each individual state has more information than what was started with. Each "world" will have "which world" information.
In my murderer example, we can designate the SSN of the murderer as 012-34-?, which is the same information as {012-34-0000, 012-34-0001, ... 012-34-9999}. Every member of that set taken alone has more information than the whole set. So a cluster of worlds can have less information than anyone of its parts. That extra information comes from the designation or selection of that world's necessary uniqueness - which you get when you are in it.

So the problem you have with MWI, and the reason that it is not fully an "interpretation", is that it requires an exponentially large number of unique universes as time progresses. Say we start out with a universe at time zero with only one bit of information, say a "1". And let's say that every "collapse" (or whatever), splits the universe in two. So at the end of the first QM cycle, we have universe 10 and 11. Now let's say that after every cycle, each bit in the universe meets up with another decision and "splits". So at the end of the second cycle "10" has split into "1000", "1001", "1010", and "1011" and "11" has split in a similar fashion. Given the initial "1", we need 3 bits in each universe if they are to be unique universes.
After the third cycle, that jumps to 7 bits; 4th: 15 bits; 5th: 31 bits.

So how long is a cycle? As long as it takes for a collapse. Many per second. How many seconds can go by before a universe of our size is unable to hold the information? In no time, we would have all possible instances of a universe of our size.

 

[PeterDonis]

If you start out knowing that you have a collection of possible states, and then the result is that collection

This is irrelevant to the MWI since the MWI does not say this is what's happening. The MWI says that the state of the entire universe is a pure state with a unitary time evolution. There is no "collection of possible states"; there is just one state.

The problem with even talking about the MWI is that ordinary words don't have their usual referents, so it's very easy to get confused about what you're actually saying. For example: what is the referent of the word "you" in the sentence quoted above, according to the MWI?

 

[Stephen Tashi]

Well, there's the Shannon definition of information, which is the number of bits necessary to specify a parameter.

Is that the Shannon definition of "information"? I thought the Shannon definition of information required a probability model. We'd need some concept of a probability distribution for the possibles values of $\alpha,\beta$. Bits can be used as measure of Shannon information with some assumptions about the behavior of a communication channel.

A real (or complex) number has an infinite amount of information, while spin-up/spin-down has 1 bit of information.

I understand the general idea, but technically the measure of information isn't given by the number of bits required to represent a number unless we are in a a specific communication scenario. For example, if the only possible messages are $\pi, 2\pi, 3\pi$ then the message $\pi$ doesn't contain an infinite amount of Shannon information. If we assume a scenario where a message can be any real number in an infinite set of real numbers, then I agree that representing a specific message requires an infinite number of bits.

If we are counting bits to measure information then how is it that a mixture of states may contain less information that a superposition of states? Don't we end up claiming that one infinity is larger than another infinity?

 

[john taylor]

Do you think that this experiment could solve the measurement problem:" One of the main problems with solving the measurement problem, is the problem of quantum decoherence. This is a problem because it makes it difficult to distinguish between whether it was the decoherence effect or the act of measurement which caused the wave nature of particles to disapear, in experiments such as the quantum eraser experiment. However an experiment has been concieved which distinguishes between the two, in order to determine the cause of wave function collapse. It does so by controlling the process of decoherence(as best as possible), and then observing the wave nature of the decohered system, by virtue of diffraction, and then carrying out a measurement to see whether the wave nature disapears or not. The exact experiment is a modification of the davisson germer experiment. At the start of the experiment, there will be a vacuum chamber containing a single proton, and an electron gun which will fire electrons slowly into the system in order to decohere the proton. It will be bombarded with about 5 electrons in order to decohere it, and once it has decohered, an anode shall be switched on with a hole in the middle of it and the whole object shall be fired towards a nickel plate, which leads to scattering in various directions. The nickel target can also be rotated, in which electrons can be deflected towards a detector on a mounted arc which could be rotated in a circular motion. The detector, which would be used during the experiment is a faraday cup. When the particle touches the nickle plate in order to test whether measurement causes collapse, the location of the proton shall be measured. There will be two groups. The first group will not have the location of the proton measured on contact of the nickle plate, whereas the second group will have it's location measured on contact with the nickle plate, by virtue of a detector. Because the location of the proton has been measured in group 2, it could affect the scattering of the decohered particles, because the wave function has collapsed for that individual particle(it would be different to those not measured), and so the measurement problem could be solved by being able to see whether the act of measurement has any affect on the scattering of the decohered particles, and distinguish between whether it was decoherence which caused it to behave classically because it has already decohered and therefore the experiment would be testing the causality of measurement on wave function collapse because we are able to measure the wave nature of the decohered system and so any change upon measurement would be down to the act of measurement not decoherence because it is being measured via the diffraction of the particles. "

 

[stevendaryl]

I understand the general idea, but technically the measure of information isn't given by the number of bits required to represent a number unless we are in a a specific communication scenario. For example, if the only possible messages are $\pi, 2\pi, 3\pi$ then the message $\pi$ doesn't contain an infinite amount of Shannon information. If we assume a scenario where a message can be any real number in an infinite set of real numbers, then I agree that representing a specific message requires an infinite number of bits.

Yes, you're right. Counting bits only gives an upper bound to the information content. But quantum mechanics certainly doesn't place any restrictions on the values of the coefficients of a superposition.

 

[stevendaryl]

If we are counting bits to measure information then how is it that a mixture of states may contain less information that a superposition of states? Don't we end up claiming that one infinity is larger than another infinity?

I'm thinking that exploring the subject of information loss might require a lot of work.

However, I don't think that mixed states count as information in the way you are talking about them.

If I check to see if an electron is spin-up, and then later I forget what the answer was, I can describe things using the mixed state:

$\rho = \frac{1}{2} |u\rangle \langle u| + \frac{1}{2} |d\rangle \langle d|$

However we compute information, there's got to be less information in such a mixed state than there is in the pure state spin-up.

I realize that I'm being a little inconsistent, if I consider the amplitude of a pure state to be information, but I don't consider the coefficients of a mixed state. I guess that betrays an interpretation bias on my part: I'm assuming that a mixed state reflects subjective uncertainty, while a pure state is objective. I guess you could consider amplitudes to be subjective, although it's harder for me to see how two people could both assign a pure state to the same particle, but assign different pure states. Maybe someone could come up with a scenario for that?

In contrast, it's easy to come up with scenarios in which people assign different mixed states to the same particle. So it seems more subjective.

 

[.Scott]

No.
If you start out knowing that you have a collection of possible states, and then the result is that collection, then the collection as a whole does not represent either [an] information loss or gain.

This is irrelevant to the MWI since the MWI does not say this is what's happening. The MWI says that the state of the entire universe is a pure state with a unitary time evolution. There is no "collection of possible states"; there is just one state.

I didn't say it was. I was just laying the groundwork for what will and will not result in an increase in information.
As long as you have a wave function that describes the possible measurement results, you have not increased the amount of information. As soon as you allow that the actual (unpredictable) result is arbitrary, then you (the Phycisist or investigator) have created a model that involves the synthesis of information when "measurements" are made. For MWI, this makes it difficult to avoid a steady increase in the information content of the universe. The only way for information to be reduced would be for most of the "splits" to result in identical results - effectively causing "joins".

The counter to MWI is a model which says that the full result of a measurement is contained in a single universe.

 

[PeterDonis]

As soon as you allow that the actual (unpredictable) result is arbitrary,

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.

For MWI, this makes it difficult to avoid a steady increase in the information content of the universe.

It does no such thing. See above.

The only way for information to be reduced would be for most of the "splits" to result in identical results - effectively causing "joins".

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

The counter to MWI is a model which says that the full result of a measurement is contained in a single universe.

I'm not sure what you mean by this, but it sounds like you are describing the actual MWI, not any "counter" to it.

 

[.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 $|\psi_j\rangle \langle \psi_j|$.