How Does Environmentally Induced Decoherence Affect Quantum State Reduction?

[Demystifier]

I cannot explain it with words only. So let me give an explanation in terms of pure states without density matrices.

Consider the Schrödinger cat together with the unstable atom. If the atom decays then the cat dies, in which case the full state is $|decay\rangle |dead\rangle$. Likewise, if the atom does not decay then the cat lives, in which case the full state is $|not\; decay\rangle |live\rangle$. But we don't know which of the two possibilities is realized, so the full state is the superposition
$$|decay\rangle |dead\rangle + |not\; decay\rangle |live\rangle$$
A state which is in a superposition is not in a mixture.

So we know the state of the full system (the superposition above), but what is the state of the cat alone? Someone's first guess might be the superposition $|dead\rangle + |live\rangle$. But why $+$? Why not $|dead\rangle - |live\rangle$? Or why not $|dead\rangle + i|live\rangle$? Since we cannot decide which of those superpositions would be the correct one, we must decide that neither is correct. We cannot write the state of the cat alone as a superposition. So the state of the cat is only a mixed state (dead OR alive).

Is it a proper or improper mixture? It is improper mixture. Why? Because mixture is an artefact of looking only at a subsystem (the cat) and not on the the full system (cat + atom). In the full system we still have the superposition above with a definite $+$ sign, so the full system is not mixed. Hence the mixture is improper.

On the other hand, the proper mixture would takes place if there was no bigger system that made the whole system not mixed. For instance, if the atom somehow disappeared from the universe (without giving its information to something else), then the cat would be in the proper mixture. But as far as we know such a thing does not happen, so the cat must be in the improper mixture.

Does it make sense to you?

 

[Feeble Wonk]

Why not $|dead\rangle - |live\rangle$? Or why not $|dead\rangle + i|live\rangle$? Since we cannot decide which of those superpositions would be the correct one, we must decide that neither is correct. We cannot write the state of the cat alone as a superposition. So the state of the cat is only a mixed state (dead OR alive).

This is the decoherence part... Right?

Does it make sense to you?

I think this is going to work for me. Let me roll it around in my head for a bit. Thank you.

 

[Demystifier]

This is the decoherence part... Right?

Right.

 

[Feeble Wonk]

It is improper mixture. Why? Because mixture is an artefact of looking only at a subsystem (the cat) and not on the the full system (cat + atom). In the full system we still have the superposition above with a definite $+$ sign, so the full system is not mixed. Hence the mixture is improper.
>>>
Does it make sense to you?

This is where things still get fuzzy for me.
The "cat in the box" seems to be a great mental tool to consider the decoherence process because there is such a definitive prohibition of information exchange between the "external" system and the "internal" systems.
But this hard delineation still creates confusion for me when I try to consider the extended system to include an external observer and the unopened box? The external observer does not know the state of the unstable atom (and resultant state of the cat). So, from the external observer's perspective, you might think the |DECAY>|DEAD + |NOT DECAY>|LIVE system is still in superposition. However, now the state of the "atom+cat" is not the "full" system, but a subsystem of "atom+cat+observer", and therefore "atom+cat" becomes a mixture relative to the extended system including the external observer. Yet, is that actually the case BEFORE the observer opens the box (before information exchange occurs)?
I strongly suspect that this is where my mathematical incompetence and inability to deal with density matrices bites my backside again. It still seems logical to me that, regardless of the external observer's lack of knowledge with respect to the unstable atom, there is no possible "informational state" of the cat that can represent both dead AND alive at the same time, but I'm trying to understand it using the guidelines of your explanation.

 

[Demystifier]

This is where things still get fuzzy for me.
The "cat in the box" seems to be a great mental tool to consider the decoherence process because there is such a definitive limit in information exchanged between the "external" system and the "internal" system.
But this hard delineation still creates confusion for me when I try to consider the extended system to include external observer and the unopened box? The external observer does not know the state of the unstable atom (and resultant state of the cat). So, from the external observer's perspective, you might think the |DECAY>|DEAD + |NOT DECAY>|LIVE system is still in superposition. However, now the state of the "atom+cat" is not the "full" system, but a subsystem of "atom+cat+observer" and therefore a mixture. Yet, is that actually the case BEFORE the observer opens the box?
I strongly suspect that this is where my mathematical incompetence and inability to deal with density matrices bites my backside again. It still seems logical to me that, regardless of the external observers knowledge with respect to the unstable atom, there is no possible "informational state" of the cat that can represent both dead AND alive at the same time, but I'm trying to understand it using the guidelines of your explanation.

There should be no additional confusion when external observer is added. Let the possible states of the observer be
|not look>, |see dead cat> and |see alive cat>
Then before opening the box the full state is
|DECAY>|DEAD>|not look> + |NOT DECAY>|ALIVE>|not look>
After opening the box it is
|DECAY>|DEAD>|see dead cat>+|NOT DECAY>|ALIVE>|see alive cat>
No new mathematics is needed.

One philosophical comment is in order. Here the states of the observer are the states of his brain. How the brain creates a mind is an unresolved question, but QM is probably not essential for that.

 

[Feeble Wonk]

One philosophical comment is in order. Here the states of the observer are the states of his brain. How the brain creates a mind is an unresolved question, but QM is probably not essential for that.

Fair enough. We'll leave that aside for now. But it seems clear to me that there should be a logical correlation between the the "brain state" of the observer and the state of the opened box being observed, regardless of the potential philosophical issues.

 

[Demystifier]

But it seems clear to me that there should be a logical correlation between the the "brain state" of the observer and the state of the opened box being observed

Of course, and that correlation is encoded in the last state above that I have written. Rougly speaking, the quantum state
|DECAY>|DEAD>|see dead cat>+|NOT DECAY>|ALIVE>|see alive cat>
can be translated into a logical expression
(DECAY and DEAD and see dead cat) OR (NOT DECAY and ALIVE and see alive cat).

But one should be careful, because the translation is not reversible. In the reverse direction, an expression like
A OR B
gets translated into
a|A>+b|B>
where a and b are unknown coefficients.

 

[Feeble Wonk]

After opening the box it is
|DECAY>|DEAD>|see dead cat>+|NOT DECAY>|ALIVE>|see alive cat>

This seems to be somewhat of a slippery slope for me. When considering the observer+cat+atom in the "post-observation" status, would this be thought of as a "full" system... meaning a "pure" system in superposition... if we imagine that the universe consists of only these physical elements.

 

[Demystifier]

This seems to be somewhat of a slippery slope for me. When considering the observer+cat+atom in the "post-observation" status, would this be thought of as a "full" system... meaning a "pure" system in superposition... if we imagine that the universe consists of only these physical elements.

Yes. If you are now going to ask why do we not see a superposition, I will tell you that the answer depends on the interpretation.

 

[Feeble Wonk]

Yes. If you are now going to ask why do we not see a superposition, I will tell you that the answer depends on the interpretation.

[emoji39] You saw that coming a mile away. I've got some more thinking to do, and then I'd like to ask you more about how this relates to the SHV interpretation if that would be OK.

 

[Demystifier]

[emoji39] You saw that coming a mile away. I've got some more thinking to do, and then I'd like to ask you more about how this relates to the SHV interpretation if that would be OK.

Of course.