How Does Environmentally Induced Decoherence Affect Quantum State Reduction?

[naima]

In the no-hiding theorem (conservation of information) AK Pati says that in the conditions of the theorem the unknown qbit to be manipulated can be found unchanged somewhere else in the environment. Is this related to the general process of decoherence.

 

[StevieTNZ]

Ouch. I'm confused again. Could you please define your terms for this explanation? Specifically, what precisely is designated by "environment", "apparatus" and "system", with respect to the cat in the box scenario, and the "observable" that is the sum of those three?

Sure.

The system is the cat.

The apparatus is what measures whether the radioactive material has decayed or not.

The environment is the rest of the universe, entangled to the apparatus.

For more info on the observable, I can refer you to chapter 8 of 'Quantum Mechanics and Experience' by David Albert.

Before it is brought up, the 'classical probabilities' you get from decoherence don't refer to states that actually exist (pg 209 of 'Quantum Enigma' by Bruce Rosenblum and Fred Kuttner). For example, if I have a coin that has, existing, heads and tails, then I have 50% chance of getting heads if I flip the coin -- if I get heads I know it existed on the coin before I flipped it etc. Quantum mechanically, I'd still have 50% probability of getting heads, but if that coin has undergone decoherence there is no heads (or tails for that matter) actually on the coin until measured.

 

[Zafa Pi]

Sure.

The system is the cat.

The apparatus is what measures whether the radioactive material has decayed or not.

The environment is the rest of the universe, entangled to the apparatus.

For more info on the observable, I can refer you to chapter 8 of 'Quantum Mechanics and Experience' by David Albert.

Before it is brought up, the 'classical probabilities' you get from decoherence don't refer to states that actually exist (pg 209 of 'Quantum Enigma' by Bruce Rosenblum and Fred Kuttner). For example, if I have a coin that has, existing, heads and tails, then I have 50% chance of getting heads if I flip the coin -- if I get heads I know it existed on the coin before I flipped it etc. Quantum mechanically, I'd still have 50% probability of getting heads, but if that coin has undergone decoherence there is no heads (or tails for that matter) actually on the coin until measured.

So according to your three part decomposition, the original atom in a superposition of decay and not decay must be part of the environment. I would have never have guessed.

 

[StevieTNZ]

Sure.

The system is the cat.

The apparatus is what measures whether the radioactive material has decayed or not.

The environment is the rest of the universe, entangled to the apparatus.

So according to your three part decomposition, the original atom in a superposition of decay and not decay must be part of the environment. I would have never have guessed.

No, that is certainly not what I am saying. The systems in this instance are the cat, the radioactive material, and the poison.

 

[bhobba]

Just to be absolutely clear: decoherence does not result in the cat being either dead or alive.

Sorry - but it does.

Decoherence results in the cat, like all classical objects, being in a mixed state of definite position so it can't be alive and dead ie in a superposition. The issue with decoherence is as I have said many times it only explains apparent collapse. One does not have to place the quantum classical cut just after decoherene (which is exactly the same as interpreting the mixed state as a proper one) but you are free to do so if you desire - there is no way to tell the difference. Placing it there leads to the conclusion the cat can't be in a superposition of alive and dead. Since there is no way to tell the difference it is true regardless.

Thanks
Bill

 

[Zafa Pi]

No, that is certainly not what I am saying. The systems in this instance are the cat, the radioactive material, and the poison.

First you said the system was the cat. This is precisely why I would like to see a minimal model: a few photons, a PA or two, the resulting density matrix and how it evolves. The cat and poison are not necessary merely the detector. Without such a model the level of abstraction is too high and vague, the disagreements will go on for ever.

 

[bhobba]

Without such a model the level of abstraction is too high and vague, the disagreements will go on for ever.

You are way over complicating it.

A cat is a classical object and has definite position so can't be in a superposition of alive and dead. Why does it have definite position - see:
https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20

But that's the gory technical detail - understanding it is not required to resolve this issue.

Thanks
Bill

 

[Feeble Wonk]

Sure.

The system is the cat.

The apparatus is what measures whether the radioactive material has decayed or not.

The environment is the rest of the universe, entangled to the apparatus.

For more info on the observable, I can refer you to chapter 8 of 'Quantum Mechanics and Experience' by David Albert.

OK. Thanks. I'll check out your reference.
But for now, there are a couple of points I'm confused by. First, I'm not sure what you're saying about the result of decoherence.
Working with density matrices is beyond my abilities, but my previous understanding was that decoherence "cancels out" (apologies for poor terminology) observable superposition in macroscopic systems. So I'm confused by what you mean when you say that decoherence doesn't result in the cat being dead or alive. I suppose, from a universal perspective, I can see that both states (dead or live) exist in potentiality. Does that technically constitute "superposition"? Even if so, aren't the potential states in superposition dead OR alive, and never dead AND alive? And isn't that limitation due to decoherence?

 

[bhobba]

So I'm confused by what you mean when you say that decoherence doesn't result in the cat being be dead or alive.

He is referring to the well known Von Neumann regress (see page14)
http://arxiv.org/pdf/quant-ph/0209123.pdf

What he didn't take into account is the flip side to this. If you assume it happens after decoherene there is no way to tell the difference. That leads to the conclusion the cat is not in a superposition of alive or dead. But since you can't tell the difference that is true even if you don't assume that.

Thanks
Bill

 

[StevieTNZ]

If the environment, which is suppose to entangle with the system, is quantum mechanical (in principle), then by nature it is described by the fundamental Schrödinger equation. Thus, it is in a superposition. Entangle a superposed environment with a superposed system = superposition. A density matrix to describe the system (or environment) is useful because we don't have FULL information about it.

As I've stated elsewhere, if you get one outcome from decoherence, then no one needs to fuss about the measurement problem. So no, the cat is not either dead or alive. It is in a superposition of both. I can start quoting from all this literature I have, but right now I am occupied with other things.

 

[bhobba]

It is in a superposition of both. I can start quoting from all this literature I have,

You can quote from the literature all you like. That does not change the facts.

You can place the Von Neumann cut anywhere - as was proved by the man himself - there is no way to tell the difference. Place it just after decoherence and the cat can't be in a superposition of position. But since there is no way to tell the difference you must get the same result even if you placed somewhere else. The cat can never, ever, ever, be in a superposition of alive and dead - its impossible, utterly impossible and follows from what a cat is - it must interact with the environment.

There is a lot of misconceptions about this but that does not change what's going on. Classical objects like a cat simply can't display quantum weirdness by virtue of what they are. Some objects can - but not cats.

Thanks
Bill

 

[StevieTNZ]

From post #209: https://www.physicsforums.com/threads/decoherence-clarification.828712/page-11#post-5434769

"There is an observable of environment + apparatus + system that, if measured, would tell an observer whether all three are in a superposition or not - my guess is QM predicts you'd get the value indicating they are in superposition."

I then referred the member who queried about this observable to chapter 8 of David Albert's 'Quantum Mechanics and Experience'.

 

[naima]

In the no-hiding theorem (conservation of information) AK Pati says that in the conditions of the theorem the unknown qbit to be manipulated can be found unchanged somewhere else in the environment. Is this related to the general process of decoherence.

Do you know the no-hiding theorem? Do you think it has no place here?

 

[Demystifier]

So I am asking what is the simplest model that could elucidate decoherence?

I am not sure what is the simplest model, but one quite simple model is described in my paper
http://arxiv.org/abs/1406.3221
Sec. 4.

 

[bhobba]

"There is an observable of environment + apparatus + system that, if measured, would tell an observer whether all three are in a superposition or not - my guess is QM predicts you'd get the value indicating they are in superposition."

You are shifting context here.

If more than one system are in superposition ie entangled then you can't speak about individual systems being in a superposition ie in a pure state. You can observe each of those systems and you find they are in a mixed state - in fact that's how decoherence works. I have posted the math many times eg see post 22:
https://www.physicsforums.com/threads/is-the-cat-alive-dead-both-or-unknown.819497/page-2

The whole Schroedinger's Cat set-up can be in a pure state theoretically - it makes no difference for the cat - its entangled with all sorts of stuff - the air, the stool its sitting on, the sweat that it evaporates - all sorts of things. That's why it can't be in a superposition - its not in a pure state - its entangled.

Thanks
Bill

 

[zonde]

There is no meaningful definition for classical system in QM.
Let's illustrate it with simplest example. Diatomic molecule has one normal vibration mode. Molecule can be in ground state or first excited state for example. So we can speak about superposition of ground state and first excited state. But for one state classical system includes quanta of energy (excited state) but for other state quanta of energy is part of environment (say as a free photon). So it turns out that superposition of two states means as well superposition of system/environment cut.

 

[Feeble Wonk]

I am not sure what is the simplest model, but one quite simple model is described in my paper
http://arxiv.org/abs/1406.3221
Sec. 4.

I particularly like section 5 (your discussion and conclusion). It's a great explanation in normal language for those of us that are mathematically challenged.[emoji106]

 

[Feeble Wonk]

I am not sure what is the simplest model, but one quite simple model is described in my paper
http://arxiv.org/abs/1406.3221
Sec. 4.

I'd like to clarify what is meant by this passage in the discussion...
"Now the answer to the question is very simple. In order for interference to be seen by the external observer, the box must be almost perfectly isolated from the influence of the external environment, so that the conscious being in the box cannot have any information about the external world. Therefore, even though the wave function of the conscious being would travel along both paths (as would be demonstrated by the measurement of interference), the conscious being would experience nothing unusual because she would not even know that she travels along some of the paths. Hence, being a Schr ̈odinger cat would be like being an ordinary cat living in a box without any information about the world external to the box."
Are you saying that the cat is experiencing a reduced state "internally" (inside the box), but to the external observer (external environment) the cat (the system inside of the box) remains in superposition of all possible states?

 

[Demystifier]

Are you saying that the cat is experiencing a reduced state "internally" (inside the box), but to the external observer (external environment) the cat (the system inside of the box) remains in superposition of all possible states?

Yes, but note that the external observer only observes the position ${\bf X}$ of the box-with-cat as a whole. The external observer does not observe any internal details of the cat in the box.

 

[Feeble Wonk]

Yes, but note that the external observer only observes the position ${\bf X}$ of the box-with-cat as a whole. The external observer does not observe any internal details of the cat in the box.

OK. I think I'm following you so far.
But this leads us back to the "internal" decoherence issue, and the question of WHICH states are still in superposition to the external observer.
I suspect that this is going to be a very semantically sensitive question, but...
Is it meaningful to assert that, to the external observer, the cat is in a superposition of dead AND alive (as opposed to dead OR alive)... not in terms of "probabilities", but in terms of its actual quantum state?
I also suspect that some will object to how the question is phrased, contending that the quantum state is itself a mathematical object, and cannot be conceptually differentiated from the "probabilistic" expression. But still...?

 

[StevieTNZ]

That's why it can't be in a superposition - its not in a pure state - its entangled.

I agree that the system is entangled with air molecules etc (i.e. the rest of the universe). But the system + environment is in a pure state. No, they cannot be considered separate systems. I need to be careful with terminology because what I wish to convey may be construed differently to others.

Let us consider this simple bell-state:
|H>|V> - |V>|H>

What I mean by the system and environment entangled and in a pure state is (and not expressed as a density matrix):
|system:catalive>|environment> - (or even +) |system:catdead>|environment>

 

[Zafa Pi]

the pure state was a superposition of observable eigenstates, the mixed state is a superposition of pure states.

A mixed state is a probability density over pure states not a superposition of pure states

 

[bhobba]

But this leads us back to the "internal" decoherence issue, and the question of WHICH states are still in superposition to the external observer.

After decoherence its not in superposition - its in a mixed state. The difference can only be explained in math.

Its a general characteristic of entanglement - see post 22:
https://www.physicsforums.com/threads/is-the-cat-alive-dead-both-or-unknown.819497/page-2

Superposition only applies to pure states because they can be mapped to a complex vector space - it expresses the general property of a complex vector space that given any two elements |a> and |b> then c1||a> + c2|b> where c1 and c2 are complex numbers is also an element. But mixed states are not elements of that vector space - they are positive operators so superposition does not apply to them. When systems are entangled with other systems they are no longer in a pure state - in fact the concept of state doesn't really make any sense - but they act like they are in a mixed state.

Thanks
Bill

 

[Demystifier]

OK. I think I'm following you so far.
But this leads us back to the "internal" decoherence issue, and the question of WHICH states are still in superposition to the external observer.
I suspect that this is going to be a very semantically sensitive question, but...
Is it meaningful to assert that, to the external observer, the cat is in a superposition of dead AND alive (as opposed to dead OR alive)... not in terms of "probabilities", but in terms of its actual quantum state?
I also suspect that some will object to how the question is phrased, contending that the quantum state is itself a mathematical object, and cannot be conceptually differentiated from the "probabilistic" expression. But still...?

As you said it is semantically sensitive, or to put it in my own words - interpretation dependent. In the paper I refrained from using any specific interpretation, so such questions remained unanswered there. Here I can answer such questions as well, but only if we first agree on which interpretation to use. If such questions confuse you, that's only because you didn't yet pick up your favored interpretation. Once you do that, no confusion will remain.

 

[Feeble Wonk]

After decoherence its not in superposition - its in a mixed state. The difference can only be explained in math.
Its a general characteristic of entanglement - see post 22:
https://www.physicsforums.com/threads/is-the-cat-alive-dead-both-or-unknown.819497/page-2
Superposition only applies to pure states because they can be mapped to a complex vector space - it expresses the general property of a complex vector space that given any two elements |a> and |b> then c1||a> + c2|b> where c1 and c2 are complex numbers is also an element. But mixed states are not elements of that vector space - they are positive operators so superposition does not apply to them. When systems are entangled with other systems they are no longer in a pure state - in fact the concept of state doesn't really make any sense - but they act like they are in a mixed state.

This is helpful. Thanks Bill. Nice blend of the minimal mathematics with the conceptual description that I think I could actually follow.
OK. So, INSIDE the box, I can clearly see that decoherence should occur following the either/or decay event which forks the potential systems into distinctly different macroscopic states. Hence, the state of the cat should always be thought of as dead OR alive. As Nugatory said in your sited PF thread, the state is UNKNOWN, but not really in superposition. A technical difference, but I think I've got a better handle on that now. It makes sense to me in terms of the isolated system's state WITHIN the box. Yet, where things still get dicey for me is the state of the "internal system" from the "external" perspective, as Demystifier describes it.

Yes, but note that the external observer only observes the position ${\bf X}$ of the box-with-cat as a whole. The external observer does not observe any internal details of the cat in the box.

If I'm understanding you correctly Dr D, this comes down to an interpretational question.

As you said it is semantically sensitive, or to put it in my own words - interpretation dependent. In the paper I refrained from using any specific interpretation, so such questions remained unanswered there. Here I can answer such questions as well, but only if we first agree on which interpretation to use. If such questions confuse you, that's only because you didn't yet pick up your favored interpretation. Once you do that, no confusion will remain.

If I may, I'd appreciate it if we could look at it from a variety of interpretational perspectives to see where the differences lie. I suppose we might initially consider the ensemble perspective as preferred by Bill. This interpretation would suggest that the cat is not in a true "undetermined" superposition (pure state), but in an "unknown" selection of possible mixed states. Have I got that right?

 

[Demystifier]

If I may, I'd appreciate it if we could look at it from a variety of interpretational perspectives to see where the differences lie. I suppose we might initially consider the ensemble perspective as preferred by Bill. This interpretation would suggest that the cat is not in a true "undetermined" superposition (pure state), but in an "unknown" selection of possible mixed states. Have I got that right?

From the ensemble perspective that feels right, but your wording seems strange. I would not say that it is in an "unknown selection of possible mixed states". It is in one pure state. That pure state is unknown because it is an unknown member a collection of possible pure states. The collection of pure states (with the corresponding probabilities of each) can be described as a known mixed state.

 

[Feeble Wonk]

From the ensemble perspective that feels right, but your wording seems strange. I would not say that it is in an "unknown selection of possible mixed states". It is in one pure state. That pure state is unknown because it is an unknown member a collection of possible pure states. The collection of pure states (with the corresponding probabilities of each) can be described as a known mixed state.

Apologies. I got the "unknown" reference from the previous thread, but phrased it poorly. I suppose the primary point is that, in the ensemble interpretation, decoherence results in true state reduction within the (internal) isolated system. From the external perspective you are left with an "unidentified" pure state, which can be described stochastically as a collective in terms of being a ("proper?") mixed state (mixture of the potential pure states). But conceptually, from the external perspective, you shouldn't think of that mixed state as being a wholistic entity in superposition (regardless of ontology). It is merely a statistical prediction of which of the potential unidentified pure states one would find when the box is opened. And even before the box is opened, we would know that none of those states predicted refer to a cat that is both dead AND alive.
Is that better?

 

[Demystifier]

It's better.

 

[Feeble Wonk]

It's better.

Great. Good enough for now anyway.
Now, if we consider the "cat in the box" scenario from an Everett style MW interpretation, how does that change things? My first impression would be that it doesn't change much. Decoherence would still occur as before, and the "potential world forks" would only include cats that are dead OR alive... even when considered from the "external" perspective... Right?

 

[Demystifier]

Great. Good enough for now anyway.
Now, if we consider the "cat in the box" scenario from an Everett style MW interpretation, how does that change things? My first impression would be that it doesn't change much. Decoherence would still occur as before, and the "potential world forks" would only include cats that are dead OR alive... even when considered from the "external" perspective... Right?

Well, from MW perspective there are two cats: one dead and one alive.

In a "potential-world" language I guess one would say that the cat has (a potential for both being dead) and (a potential for being alive). However, due to decoherence it does not have a potential for being (dead and alive). What's the difference? Let me give a non-quantum analog from my own experience. When I was a child I have showed some talent for physics and also some talent for arts. But I had to choose only one profession. So I had (a potential to be a physicist) and (a potential to be an artist), but I did not have a potential to be (a physicist and an artist). As you may guess, in actuality I have chosen the former (and never regretted ).

(BTW, does anybody know how to attach a jpg picture which would demonstrate my child artistic talents? I have a scanned version of the picture on my computer, but I don't have an URL.)