About nature of superposition of states

[HighPhy]

The key point to take away from my posts is to stop using the term "collapse of the wave function" without explicitly defining exactly what you mean by it.

I was trying to assume as closely as possible the interpretation you had provided in the thread "Why don't we bury Schrodinger's Cat?". Did you mean something particular?

But at this point it seems I need to make some sort of mathematical statement explicit so as not to be ambiguous.

 

[PeterDonis]

at this point it seems I need to make some sort of mathematical statement explicit so as not to be ambiguous.

Exactly.

 

[PeterDonis]

I was trying to assume as closely as possible the interpretation you had provided in the thread "Why don't we bury Schrodinger's Cat?"

Since you didn't provide an actual link in what you quoted, I don't know what the context was of the quote you gave.

 

[HighPhy]

Since you didn't provide an actual link in what you quoted, I don't know what the context was of the quote you gave.

I provided the link in post #63.

 

[PeterDonis]

I provided the link in post #63.

The post of mine that you linked to there (which is not a post that you quoted explicitly anywhere in this thread--when you explicitly quoted posts of mine from that thread here, you did not do so with links) says:

Even on interpretations that make this claim, the collapse occurs when the cat decoheres inside the box, not when the box is opened. Opening the box still doesn't change anything about the cat.

This makes it clear that I am talking about collapse interpretations. Note also the last sentence.

 

[Lord Jestocost]

A clarification would take away much of my confusion.

The superposition state ##\left|\Psi\right>=a\left|alive\right>+b\left|dead\right>## with ##\left|a^2\right| + \left|b^2\right|=1## does not describe two cats, a dead one and an alive one, overlapping each other in some way. It describes a cat that is known to the observer to be either alive, with probability ##\left|a^2\right|##, or dead, with probability ##\left|b^2\right|##.

Quantum mechanics differs from classical physics because the assumption that one of the answers (dead/alive, in this case) is "objectively" realized in between the measurement at a certain time is simply impossible.

 

[HighPhy]

As far as the basic math of QM is concerned, once decoherence has happened, the measurement has a result. You might not know what the result is until you open the box, but that doesn't mean the result doesn't happen until you open the box. It happens as soon as decoherence happens (and the decoherence time for an object like a cat is very, very short).

What different interpretations of QM disagree on is what "the measurement has a result" means. Collapse interpretations say it means the wave function has actually, physically collapsed to a single result: i.e., the two terms in the decoherent entangled superposition of dead and alive have become one, either the dead term or the alive term.

No collapse interpretations, such as the MWI, say that both results happen, just in different branches of the wave function. What happens when you open the box in these interpretations is that you now become part of the entangled superposition; a ket representing your observation becomes part of each term.

I was re-reading this message of yours and some doubts arose.
Here you are saying that the result is the combination of decoherence plus interpretation, right? But AFAIK decoherence is mathematically well defined, but the interpretation is not: as far as I know there are no mathematically well defined interpretations; even many worlds has a problem since probabilities are hard to calculate using it.
What's wrong with my reasoning above?

This makes it clear that I am talking about collapse interpretations. Note also the last sentence.

OK. But I'm really confused. In post #45, you say:

the entangled superposition goes away as soon as decoherence happens, because a collapse interpretation says that the actual, physical collapse happens when decoherence happens.

But in post #66, you say:

It is pointless to say "well, the wave function collapsed when decoherence happened, even though I didn't know the result until I opened the box".

What is the subtle difference that makes these two statements not contradict each other? There must be something still eluding me about this point. I can't figure out what.

The superposition state ##\left|\Psi\right>=a\left|alive\right>+b\left|dead\right>## with ##\left|a^2\right| + \left|b^2\right|=1## does not describe two cats, a dead one and an alive one, overlapping each other in some way. It describes a cat that is known to the observer to be either alive, with probability ##\left|a^2\right|##, or dead, with probability ##\left|b^2\right|##.

It's a good point. I, too, had said this a few posts ago. But @Nugatory also said:

That last statement - "either alive or dead" - does fairly describe a mixed state, one that we now understand will be reached by decoherence as the wave function evolves. That resolves Schrodinger's challenge to the 1920's-vintage Copenhagen interpretation but it doesn't do anything for the popular understanding of superimposed states - it's not the result of addition in the wave function.

They sound like contradictory statements to me, although there must be some subtle reason why they are not. And again, I can't figure out which one.

 

[gentzen]

This makes it clear that I am talking about collapse interpretations. Note also the last sentence.

OK. But I'm really confused. In post #45, you say:

What you are doing here is not funny. You quoted PeterDonis in a misleading way, he even appologized, because he believed you that his post was as misleading as your quote suggested:

Sorry for the confusion.

Then later he found out that it was actually the way you quoted him which was misleading.

What is your goal here, what do you want to achieve for yourself?

But AFAIK decoherence is mathematically well defined, but the interpretation is not: as far as I know there are no mathematically well defined interpretations; even many worlds has a problem ...

Where are you getting this from? Just because there are different possible interpretations doesn't mean that the interpretations are not mathematically well defined. For example, which part of Bohmian mechanics is mathematically not well defined? And just because not all orthodox interpretations clearly subscribe to either Heisenberg's or Bohr's views doesn't mean that they are not mathematically well defined either.

 

[HighPhy]

What you are doing here is not funny.

I apologize very much if I offended you or anyone else with my comment. That was not my intention.

You quoted PeterDonis in a misleading way, he even appologized, because he believed you that his post was as misleading as your quote suggested:

OK, I've learned the lesson. But I never claimed to be right. In fact, I repeatedly asked to be corrected where I was wrong.

With this last message of mine I in no way meant to stir up discord. I simply wanted to understand why two statements that seem contradictory to me actually are not, and where I am wrong.

That said, I feel very sorry that I have provoked anger toward you. I have always tried to learn from experts by placing myself in a constant attitude of expectation. I am not a bad person.

Please accept my apologies.

 

[gentzen]

I have always tried to learn from experts by placing myself in a constant attitude of expectation.

But if you don't know what you want to achieve for yourself, the experts have very little chance to help you in that endeavor.

I am not a bad person.

You should realize that "OK. But I'm really confused." is much more related with you yourself than with what an of the "experts" said or wrote. If you don't take responsibility yourself for your goals, nobody will be able to help you with them.

 

[PeterDonis]

Here you are saying that the result is the combination of decoherence plus interpretation, right?

No. A physical result can't be the product of a human interpretation.

decoherence is mathematically well defined, but the interpretation is not

That's correct. Now take a step back and think carefully about the implications.

I'm really confused.

That's because you keep reading my words out of context (and even quoting them out of context) and getting the different contexts mixed up.

I have already explained that there are multiple meanings for terms like "collapse". That means that, when you read anything that includes that word, you can't just assume it means what you would like it to mean, or that it always means the same thing. You have to read the words carefully, in context, and figure out what the words mean in that context.

Go do that.

 

[HighPhy]

I have already explained that there are multiple meanings for terms like "collapse". That means that, when you read anything that includes that word, you can't just assume it means what you would like it to mean, or that it always means the same thing. You have to read the words carefully, in context, and figure out what the words mean in that context.

Go do that.

OK. I'll try.

There are two different notions of "wave function collapse" in "standard textbook quantum theory":

1) One of these is that when a system whose quantum state is initially pure, becomes entangled with a larger environment, its state must now be described as mixed, if one wants to exclude the environment. That means instead of a single quantum vector wave function, we must use a density operator, mathematically. In terms of conceptualization, pure states are extremal points; mixed states are in the middle.

2) The second of these is the von Neumann-Lueders "collapse postulate", which introduces a new primitive undefined term, "measurement", or "observation", into the quantum theory, in which a random replacement of one wave function with another occurs, representing a single observational outcome.

The question is what, if any, relationship there is between these two things, and what is the significance of the undefined term "measurement" or "observation". In the classical limit of quantum theory, the von Neumann-Lueders collapse looks like the "gain of information" in Bayesian probability, and moreover becomes indispensible to make sense of what we're seeing as "really being classical mechanics", and thus I think it makes sense that this interpretation should be retained in the non-classical regime as well, because the structure of the mathematical formalisms are identical; the only difference is whether ##\hat{x}## and ##\hat{p}## commute or not or, equivalently, if ##\hbar## is or isn't zero.

Here you say:

Collapse interpretations say it means the wave function has actually, physically collapsed to a single result: i.e., the two terms in the decoherent entangled superposition of dead and alive have become one, either the dead term or the alive term.

On a collapse interpretation ... the entangled superposition goes away as soon as decoherence happens, because a collapse interpretation says that the actual, physical collapse happens when decoherence happens.

It seems to me that you're saying that on a collapse interpretation the physical process (whatever It means) of "collapse of the wave function" happens corresponding to decoherence: in Schroedinger's Cat, it happens long before the box is opened.
The collapse of the wave function happens whenever the quantum system initially described by the wave function becomes entangled with environment — the part of the Universe that wasn't tracked by the wave function. For example, when the particle's state evolves to the point at which it has a significant amplitude in the vicinity of the detector, the counter clicks and there is the collapse of the wavefunction.
In fact, an observation is certain physical effect that the observer has on the measured object. E.g., the Landau & Lifshitz textbook specifies that any macroscopic object can be an observer.

The theorem of von Neumann (it can be found on "Mathematical Foundations of Quantum Mechanics") says that it doesn't make a bit of difference whether you model the cat (or anything else along the causal chain between closing the box and opening it to observe the cat) as capable of collapsing the wave function. You'll make exactly the same testable predictions no matter where along the way you place the collapse.

Indeed, as you say:

In the basic math of QM, "collapse" means the mathematical process of updating the wave function you are going to use to predict the results of future measurements, once you know the results of the measurement that was already made. In the case of Schrodinger's cat, you don't know the result until you open the box. That is the case even though decoherence happens well before you open the box. It is pointless to say "well, the wave function collapsed when decoherence happened, even though I didn't know the result until I opened the box". In the basic math of QM, "collapse" is not something that happens in the world at some particular time; it's something that happens in your model when you get new information.

This is a "mostly subjective" viewpoint of quantum mechanics as it is really, despite looking at all the alternatives, the only one that fits the closest to the mathematics of the theory as given with no other adulterations.

On a subjective account, the wave function belongs to you, the one outside of the box. It models, your information or knowledge about the state of affairs in the box. The transition from "live cat" to "live or dead cat" to "dead cat" starting from the initial state is just showing how your best knowledge, without looking into the box, that you can predict from that initial state, changes. All we can say about the "superposition" at the in-between point is that it means the predicted information about the answer to the question "is the cat alive?" is less than one bit.

As a subject, the cat may be assigned a wave function talking about the information it has regarding the contraption that is going to kill it. Of course, soon after that one "collapses" then there won't be wave function any more because this subject, the information-bearer, has been terminated.

Hence, from that point of view, it makes no sense to ask ourselves when physical collapse is going to occur, because the wave function models your knowledge, not the cat's. The cat can't do anything to that.

In this sense:

As far as the basic math of QM goes, where "collapse" is just a mathematical procedure where you update the wave function once you know the result of a measurement, obviously you don't know the result of the measurement until you open the box, so you wouldn't actually make the update to the wave function until then. But that says nothing whatever about any physical process that does or does not happen at any particular time.

In this sense:

It is pointless to say "well, the wave function collapsed when decoherence happened, even though I didn't know the result until I opened the box".

Is my explanation reasonable? Where did I go wrong?

 

[PeterDonis]

1) One of these is that when a system whose quantum state is initially pure, becomes entangled with a larger environment, its state must now be described as mixed, if one wants to exclude the environment. That means instead of a single quantum vector wave function, we must use a density operator

What does this have to do with wave function collapse? Do you have a reference?

The second of these is the von Neumann-Lueders "collapse postulate", which introduces a new primitive undefined term, "measurement", or "observation", into the quantum theory, in which a random replacement of one wave function with another occurs, representing a single observational outcome.

This is a postulate of basic QM, yes. It is what I referred to before as an information update which is part of the basic math.

However, you have left out another meaning of "collapse", namely, as an actual physical process by which the wave function of a system changes in a way that is not in accord with the Schrodinger Equation. In some versions of this, what I called "collapse interpretations" of QM, it is simply asserted that the mathematical postulate referred to above corresponds to an actual physical process. In other versions, which are actually different theories from standard QM, such as the GRW stochastic collapse model, the dynamics are explicitly changed to include "collapse" as well as standard unitary evolution, without any mention of "measurement".

 

[PeterDonis]

It seems to me that you're saying that on a collapse interpretation the physical process (whatever It means) of "collapse of the wave function" happens corresponding to decoherence: in Schroedinger's Cat, it happens long before the box is opened.

More precisely, in "collapse interpretations" as I described that term in post #83 just now, the actual physical process is said to occur when decoherence occurs. (At least, that's where such interpretations have ended up since the development of decoherence theory--see further comments below.) Which means it does not occur when a human observer opens the box and looks inside (and so the actual physical process of "collapse" does not necessarily happen at the same time as the mathematical update of the wave function in our human models, which is what "collapse" refers to in the basic math of QM). In other words, the undefined and problematic terms "measurement" and "observation" are removed from the interpretation, and instead the well-defined physical process of "decoherence" becomes what triggers the collapse.

The collapse of the wave function happens whenever the quantum system initially described by the wave function becomes entangled with environment — the part of the Universe that wasn't tracked by the wave function.

To the extent that "becomes entangled with the environment" is the same as "decoherence", yes. But note that in the Schrodinger's Cat experiment, there does not have to be any entanglement with an "environment" in order for the cat to decohere; the cat can decohere itself just fine even if it doesn't interact with anything else. Of course a real cat will be interacting with, say, the air inside the box as it breathes. But that interaction is not required for the cat to decohere--and indeed the cat's decoherence time is much shorter than the time scale on which it breathes. That is why "decoherence" is the proper criterion, and "entanglement with the environment" is best viewed as one possible mechanism by which decoherence can occur.

The theorem of von Neumann (it can be found on "Mathematical Foundations of Quantum Mechanics") says that it doesn't make a bit of difference whether you model the cat (or anything else along the causal chain between closing the box and opening it to observe the cat) as capable of collapsing the wave function. You'll make exactly the same testable predictions no matter where along the way you place the collapse.

Yes. And before decoherence theory was developed, there were indeed different versions of "collapse interpretations" which put the collapse in different places. And all of them suffered from the same problem, that the placement of the collapse was arbitrarily chosen, and there was no physical basis for preferring any particular choice.

The development of decoherence theory was important in that it gave a physical basis for a particular choice of "when the collapse occurs" for interpretations that claimed that collapse was a physical process. But of course there is still no testable prediction that distinguishes this particular choice from other possible ones. Decoherence does not invalidate the von Neumann theorem you refer to.

On a subjective account, the wave function belongs to you, the one outside of the box.

Yes. And there is a family of QM interpretations in which this is explicitly stated and made the basis of the interpretation: that the wave function represents our knowledge about the quantum system, not the system itself. On these interpretations, "collapse" is just the update of our knowledge, and that is all it is. But in these interpretations there is no story to tell at all about what "actually happens" with the quantum system itself; all we can talk about is what we know about it.

 

[HighPhy]

More precisely, in "collapse interpretations" as I described that term in post #83 just now, the actual physical process is said to occur when decoherence occurs. (At least, that's where such interpretations have ended up since the development of decoherence theory--see further comments below.) Which means it does not occur when a human observer opens the box and looks inside (and so the actual physical process of "collapse" does not necessarily happen at the same time as the mathematical update of the wave function in our human models, which is what "collapse" refers to in the basic math of QM). In other words, the undefined and problematic terms "measurement" and "observation" are removed from the interpretation, and instead the well-defined physical process of "decoherence" becomes what triggers the collapse.

This was the crux of the matter, the source of my initial confusion. Many thanks!

the wave function represents our knowledge about the quantum system, not the system itself.

About this I have a question: is it wrong (or at any rate controversial) to say that the wave function describes --in general-- the behavior of a particle?

 

[PeterDonis]

About this I have a question: is it wrong (or at any rate controversial) to say that the wave function describes --in general-- the behavior of a particle?

It's interpretation dependent.

 

[HighPhy]

What does this have to do with wave function collapse? Do you have a reference?

I used a convoluted turn of phrase to say that this "collapse" is actually not a discrete event: it's something that can happen gradually because the evolution from the non-entangled to entangled configuration is fully continuous.
Is it correct?

It's interpretation dependent.

Could you give me some references from which I can verify the interpretations that support this thesis and those that do not?

 

[PeterDonis]

I used a convoluted turn of phrase to say that this "collapse" is actually not a discrete event: it's something that can happen gradually because the evolution from the non-entangled to entangled configuration is fully continuous.

The evolution from a non-entangled to an entangled configuration is not collapse, in any sense of the term.

Collapse, mathematically, is the change from the entangled configuration you describe, where you have a system, a measuring device, and potentially the environment (if you want to include it in your analysis) in an entangled superposition, to a non entangled state that has just one of the superposed terms. For example, in the Schrodinger's cat scenario, it would be a change from this entangled superposition (ignoring normalization and the environment)...

$$
\ket{\text{atom decayed}} \ket{\text{cat dead}} + \ket{\text{atom not decayed}} \ket{\text{cat alive}}
$$

...to either...

$$
\ket{\text{atom decayed}} \ket{\text{cat dead}}
$$

...or...

$$
\ket{\text{atom not decayed}} \ket{\text{cat alive}}
$$

Neither of those last two are entangled states.

Collapse interpretations claim that the mathematical change I just described corresponds to an actual physical process.

 

[PeterDonis]

Could you give me some references from which I can verify the interpretations that support this thesis and those that do not?

For an example of an interpretation that treats the wave function as physically real, i.e., it directly describes the physical state of the quantum system, look up the Many Worlds interpretation.

For an example of an interpretation that does not treat the wave function as directly describing the physical state of a quantum system, you could try Ballentine, which uses the ensemble interpretation and discusses how that interpretation works at some length.

 

[HighPhy]

The evolution from a non-entangled to an entangled configuration is not collapse, in any sense of the term.

Yes, I reversed "non-entangled" with "entangled." Actually, I meant to say:

I used a convoluted turn of phrase to say that this "collapse" is actually not a discrete event: it's something that can happen gradually because the evolution from the entangled to non-entangled configuration is fully continuous.

How stupid I was.

 

[PeterDonis]

I reversed "non-entangled" with "entangled."

Ah, ok. Then I would say that we don't actually have any mathematical model in which the "collapse" evolution from the entangled superposition to one of its terms (the non-entangled result) is continuous. The basic math of QM just does the mathematical update; it does not even try to model it as a continuous evolving process. Collapse interpretations of QM don't change that at all. And the only different models I'm aware of, like the GRW stochastic collapse model, don't model it as a continuous process but a stochastic jump.

 

[HighPhy]

Let me summarize according to what I've learned.

At the time of Schroedinger (1935) the concept of quantum decoherence was not known, so the concept of observation was posited to correspond to the opening of the box.

The box contains, in addition to the cat, a hammer, a vial of poison and a radioactive atom.
If the radioactive atom emitted radiation, then the hammer would fall on the vial of poison, which would leak out and kill the cat (one of many formulations).
But until we open the box, we cannot know whether the cat is dead or alive.

This mechanism can only be triggered by an event on a microscopic scale, namely, the decay of the atom. And it is the nucleus that is in an entangled superposition of states until it is observed by human beings (decoherence was not known according to the view of the time).

This state of entangled superposition is expanded and propagated to the macroscopic world (via the poison reflected on the cat, for example).
Consequently, the macroscopic world would also be in an entangled superposition of states, which would be absurd according to Schroedinger.

Is this how Schroedinger wanted to criticize the old Copenhagen interpretation and prove that QM had to be an incomplete theory (with various implications on the wave function, etc.), when decoherence had not yet been formulated?

Schrodinger isn’t around to ask, but it seems likely from the context (we’re doing history of science here, not science - they’re different disciplines with different objectives) that he considered that the dead/live status of the cat immediately before the box is opened ought to be fairly close to the dead/live status immediately after the box is opened.

Could you please explain this statement?

 

[PeterDonis]

This mechanism can only be triggered by an event on a microscopic scale, namely, the decay of the atom.

More precisely, it needs to be triggered by something that involves quantum uncertainty in a binary yes/no phenomenon. Schrodinger chose the decay of a radioactive atom, but it could just as easily have been a measurement of spin on a spin-1/2 particle (with spin up, say, triggering the release of the poison), or sending a photon through a polarizer (with the photon being passed through instead of absorbed triggering the release of the poison).

it is the nucleus that is in an entangled superposition of states until it is observed by human beings

The nucleus and the detector of the radiation that triggers the hammer, and the vial of poison, and the cat are in an entangled superposition of states. The nucleus can't be entangled with itself; it has to interact with something else to become entangled. It first interacts with the radiation detector and becomes entangled with that, and then the entanglement spreads to the other things. All of that was clear even without decoherence theory, so it was clear to Schrodinger in 1935.

This state of entangled superposition is expanded and propagated to the macroscopic world (via the poison reflected on the cat, for example).

The entangled superposition already encompasses the macroscopic world before it reaches the cat. The detector of the radiation from the atomic decay, that triggers the hammer, is already macroscopic.

Consequently, the macroscopic world would also be in an entangled superposition of states, which would be absurd according to Schroedinger.

I don't think "macroscopic" was the condition Schrodinger had in mind. As noted above, the detector of the radiation from the atomic decay is already macroscopic. So if "macroscopic" is where the entangled superposition becomes absurd, it's already absurd when the radiation is detected (or not). You don't even need to bring in the cat.

I think Schrodinger brought in the cat because the cat is, at least in some sense, "conscious", or "sentient", or "able to have experiences" (I can't remember what specific phrase Schrodinger used), and that was where he thought an entangled superposition became absurd.

 

[HighPhy]

This is a pretty reasonable point of view that I hadn't really thought about.

I mentioned "macroscopic" because I thought I could extrapolate from Schroedinger's paper. But, of course, I'm not at all sure what I said and most likely I was wrong.

What I was able to deduce from the paper (I don't know if correctly) is the following:

The cat paradox is presented as part of the argument that granting reality to the wave function, "blurring" the real, as Schroedinger puts it, is absurd when applied to macroscopic objects, a rhetorical reductio of that idea. Since the Copenhagen interpretation did not treat the wave function this way (it was rather the most complete description an observer can have) this particular argument was not directed against it:

The other alternative consisted of granting reality only to the momentarily sharp determining parts - or in more general terms to each variable a sort of realization just corresponding to the quantum mechanical statistics of this variable at the relevant moment. That it is in fact not impossible to express the degree and kind of blurring of all variables in one perfectly clear concept follows at once from the fact that Q.M. as a matter of fact has and uses such an instrument, the so-called wave function or ##\psi##-function, also called system vector. Much more is to be said about it further on. That it is an abstract, unintuitive mathematical construct is a scruple that almost always surfaces against new aids to thought and that carries no great message.

That is the part where he, like Einstein, opposes Copenhagen. However, as I said, I could only deduce that his target with the cat is not Copenhagen itself, but rather a combination of completeness claim with a naive realist view of the wave function (which Copenhagen says is a complete description). It is similar to Einstein's point in EPR that Copenhagen's view is incompatible with local realism that he advocated.

From some assumptions, completeness and realism in a classical observer-removed sense that Copenhagen gave up, Schroedinger arrives to the conclusion, that quantum mechanics is incomplete.
Einstein and Schroedinger thought that parting with it was too high a price, hence the hope for "completion".

[...] But serious misgivings arise if one notices that the uncertainty affects macroscopically tangible and visible things, for which the term "blurring" seems simply wrong... One can even set up quite ridiculous cases. A cat is penned up in a steel chamber along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has
decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts. It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so naively accepting as valid a "blurred model" for representing reality".

The part in bold (emphasis mine) is the one from which I inferred that Schroedinger regarded the existence of mixed states/superposition of states/entangled superposition of states for macroscopic objects like cats to be unacceptable.
I used three expressions by not choosing one because Schroedinger's words are far from clear (in fact, he says "pardon the expression") and uses vocabulary such as "living and dead cat" and "mixed or smeared out in equal parts."
What do you think about the "macroscopic issue"? And what did Schroedinger mean by "living and dead cat" for the purposes of his experiment?

Indeed, there are two possibility, and I can't figure out which one is correct.

1) The cat is a macroscopic object obeying the classical mechanics equations. It cannot be described by a quantum wave function, so any observation will show whether the cat is alive or dead with no ambiguity.

2) The cat does obey quantum mechanics, and the consequence of doing so is that it is either alive or dead. The mistake would be to assume interpretations of superposition at a microscopic and apply them in a directly equivalent way at a macroscopic level, which results in the cat supposedly being alive and dead at the same time, which is either clearly nonsense or at least a misleading use of the words 'alive and dead at the same time'.

But in any case: how is this thought experiment designed to critique the view of QM existing in the 1920s/'30s supposed to be a paradox?

Side note. How will decoherence collapse the cat into either dead or alive state? AFAIK it will just stop the cat forming an entangled state with its environment, so it explains why we don't see entangled states in macroscopic systems. But how does it explain why we don't see superpositions in macroscopic realm?

Is all my reasoning reasonable, or are there underlying errors?

For an example of an interpretation that treats the wave function as physically real, i.e., it directly describes the physical state of the quantum system, look up the Many Worlds interpretation.

For an example of an interpretation that does not treat the wave function as directly describing the physical state of a quantum system, you could try Ballentine, which uses the ensemble interpretation and discusses how that interpretation works at some length.

One question: does the old Copenhagen interpretation treat the wave function as describing the behavior of a particle?

 

[PeterDonis]

does the old Copenhagen interpretation treat the wave function as describing the behavior of a particle?

I would say no, although a better response might be that there is no single "Copenhagen interpretation", either "old" or otherwise, because that term has become associated with so many different statements in the literature, many of them by Bohr, and a fair number of which are at least apparently mutually inconsistent. But I think the general gist of a "Copenhagen interpretation" is that there is no way to describe what is "actually going on" at the level of quantum particles; the best we can do is to describe what we know about the possible results of measurements on them, and that is what the wave function does.

 

[PeterDonis]

The part in bold (emphasis mine) is the one from which I inferred that Schroedinger regarded the existence of mixed states/superposition of states/entangled superposition of states for macroscopic objects like cats to be unacceptable.

But it would have to be much more general than cats if "macroscopic" is to be the criterion. As I have already pointed out, everything in the cat scenario except the radioactive atom itself is "macroscopic", so if there is a problem with anything "macroscopic" being in a state that is indeterminate in a classical sense, that problem starts as soon as we get to the detector that detects the decay of the atom and triggers the hammer. But Schrodinger did not discuss that in his paper, whether because he simply didn't think of it, or because thought the point would be made more sharply by focusing on the cat, I don't know. That's the problem with trying to figure out what an author who is no longer around to be asked was thinking when they wrote a paper published decades ago: if the text leaves the question unanswered, there is no other place to go to get an answer.

vocabulary such as "living and dead cat" and "mixed or smeared out in equal parts."

I think by expressions like these he simply means that the wave function appears to be saying that the cat (or any other macroscopic object) is in a state that is classically indeterminate. The cat is not living or dead, which are the classical determinate states, but in some kind of combination of the two which would never occur classically.

there are two possibility, and I can't figure out which one is correct.

1) The cat is a macroscopic object obeying the classical mechanics equations. It cannot be described by a quantum wave function, so any observation will show whether the cat is alive or dead with no ambiguity.

This is the possibility that I think Schrodinger intended (but it would apply, as I have pointed out, just as well to every other object in the scenario except the radioactive atom itself).

2) The cat does obey quantum mechanics, and the consequence of doing so is that it is either alive or dead. The mistake would be to assume interpretations of superposition at a microscopic and apply them in a directly equivalent way at a macroscopic level, which results in the cat supposedly being alive and dead at the same time, which is either clearly nonsense or at least a misleading use of the words 'alive and dead at the same time'.

I don't think Schrodinger intended this possibility, because he clearly considers the kind of cat state described by the entangled superposition wave function to be absurd; he thought the actual outcome of the experiment would have to be either the "alive" state of the cat or the "dead" state of the cat, not an entangled superposition containing both. But he was working out the implications of the equation named after him, which cannot produce either the "alive" state or the "dead" state of the cat from the starting point given in the experiment; the only kind of wave function it can produce is the kind we have already written down in this thread, where both "alive" and "dead" terms appear (in entangled superposition with corresponding states of other systems in the scenario). So his conclusion was that QM must be incomplete, and cannot describe the actual dynamics of a cat.

As far as I can tell, he never considered an interpretation of QM in which "collapse of the wave function" would be an actual physical process, but such an interpretation is the only way to "obey quantum mechanics" and end up with either the "alive" or the "dead" state of the cat rather than an entangled superposition. (But, as I have pointed out, this kind of interpretation only "obeys quantum mechanics" in a formal sense, by simply declaring by fiat that the entangled superposition changes into just one of its terms, without giving any actual dynamics of how this happens.)

 

[HighPhy]

I think by expressions like these he simply means that the wave function appears to be saying that the cat (or any other macroscopic object) is in a state that is classically indeterminate. The cat is not living or dead, which are the classical determinate states, but in some kind of combination of the two which would never occur classically.

I'm confused with respect to this statement.
In some posts in this thread, it has been said that "either dead or alive" is the best expression in common parlance to represent an "entangled superposition of dead and alive," despite the fact that you yourself have rightly said that no phrase is really suitable to represent it.
Perhaps you are implicitly saying that this is further evidence that, in this case, ordinary language increases confusion?

However, I don't see a paradox in Schroedinger's thought experiment.

But it would have to be much more general than cats if "macroscopic" is to be the criterion. As I have already pointed out, everything in the cat scenario except the radioactive atom itself is "macroscopic", so if there is a problem with anything "macroscopic" being in a state that is indeterminate in a classical sense, that problem starts as soon as we get to the detector that detects the decay of the atom and triggers the hammer. But Schrodinger did not discuss that in his paper, whether because he simply didn't think of it, or because thought the point would be made more sharply by focusing on the cat, I don't know. That's the problem with trying to figure out what an author who is no longer around to be asked was thinking when they wrote a paper published decades ago: if the text leaves the question unanswered, there is no other place to go to get an answer.

Yes, you're right. It seems very difficult to really understand what Schroedinger's intent was in detail.

As far as I can tell, he never considered an interpretation of QM in which "collapse of the wave function" would be an actual physical process, but such an interpretation is the only way to "obey quantum mechanics" and end up with either the "alive" or the "dead" state of the cat rather than an entangled superposition.

Sorry, I didn't understand this statement.
Why is an interpretation of QM in which "collapse of the wave function" would be an actual physical process, the only way to "obey quantum mechanics" and end up with either the "alive" or the "dead" state of the cat rather than an entangled superposition?
In other words: why does an interpretation in which "collapse of the wave function" would not be an actual physical process, not make any of this possible?

Most likely, I cannot understand this argument because I cannot provide an answer to the following question:

How will decoherence collapse the cat into either dead or alive state? AFAIK it will just stop the cat forming an entangled state with its environment, so it explains why we don't see entangled states in macroscopic systems. But how does it explain why we don't see superpositions in macroscopic realm?

 

[PeterDonis]

Perhaps you are implicitly saying that this is further evidence that, in this case, ordinary language increases confusion?

If you mean that, in trying to express what the entangled superposition wave function is describing in ordinary language, Schrodinger was bound to fail since there is no ordinary language that can do it, yes, I think that's true. Whether Schrodinger realized it was true is a different question.

I don't see a paradox in Schroedinger's thought experiment.

There isn't a "paradox" unless you agree with Schrodinger that whatever the entangled superposition wave function describes is not compatible with the fact that we observe cats to be either alive or dead.

Why is an interpretation of QM in which "collapse of the wave function" would be an actual physical process, the only wayto "obey quantum mechanics" and end up with either the "alive" or the "dead" state of the cat rather than an entangled superposition?
In other words: why does an interpretation in which "collapse of the wave function" would not be an actual physical process, not make any of this possible?

Because if "collapse of the wave function" is not an actual physical process, then there is no way in basic QM to go from the entangled superposition wave function to either the "alive" or "dead" non-entangled states. The only other dynamics in basic QM is the Schrodinger Equation, which can't do that.

Note, though, that I made that comment in the context of Schrodinger treating the wave function as describing the physical state of the individual system. So he also did not consider interpretations where that is not the case, such as an ensemble interpretation. In such interpretations no claim is made that any individual cat is in the entangled superposition state; that state is only used to describe an abstract ensemble of cats (and of the other stuff inside the box). That would remove the issue Schrodinger is concerned about. However, I don't know that ensemble interpretations were known or understood in 1935.

How will decoherence collapse the cat into either dead or alive state?

It doesn't. As a matter of dynamics, decoherence is just the unitary Schrodinger Equation, which cannot collapse an entangled superposition into just one of its terms.

AFAIK it will just stop the cat forming an entangled state with its environment

No, it won't. In fact decoherence includes spreading the entanglement to the environment.

so it explains why we don't see entangled states in macroscopic systems.

No, it doesn't. See above.

What decoherence does explain is why we don't see interference between, for example, the "alive" and "dead" states of the cat. For collapse interpretations, this is helpful because it justifies using decoherence as the trigger for collapse. For no collapse interpretations like the MWI, it is crucial in order to explain why each branch of the wave function can't observe or interact with the others.

 

[HighPhy]

Whether Schrodinger realized it was true is a different question.

I have the impression that this question does not have an answer, doesn't it?

There isn't a "paradox" unless you agree with Schrodinger that whatever the entangled superposition wave function describes is not compatible with the fact that we observe cats to be either alive or dead.

I thought the label "paradox" was attributed to this thought experiment for another reason: the entangled superposition of states, peculiar to the quantum laws of the microscopic world, is reflected on a sentient being like the cat, which behaves classically, and therefore this would not be possible.
In what sense is there disagreement between entangled superposition wavefunction description and "cat either dead or alive"?
Sorry if I can't grasp this argument.

It doesn't. As a matter of dynamics, decoherence is just the unitary Schrodinger Equation, which cannot collapse an entangled superposition into just one of its terms.

Excuse me, I don't understand.
In post #45, you said:

As far as the basic math of QM is concerned, once decoherence has happened, the measurement has a result. You might not know what the result is until you open the box, but that doesn't mean the result doesn't happen until you open the box. It happens as soon as decoherence happens (and the decoherence time for an object like a cat is very, very short).

Doesn't this mean that once decoherence has occurred, the cat is either dead or alive?
What am I missing and where am I going wrong?

 

[PeterDonis]

I have the impression that this question does not have an answer, doesn't it?

What question? The question of what Schrodinger realized? Yes, that's unanswerable if it's not contained in what writings of his we have.

the entangled superposition of states, peculiar to the quantum laws of the microscopic world, is reflected on a sentient being like the cat, which behaves classically, and therefore this would not be possible.

Isn't that what I said?

In what sense is there disagreement between entangled superposition wavefunction description and "cat either dead or alive"?

In the sense (if you agree with Schrodinger) that the entangled superposition wave function does not describe a cat that is dead or alive, we just don't know which. It describes a cat which is in some different state that, whatever it is, is not the state of a cat that is dead or a cat that is alive.

If you don't agree with Schrodinger, then you don't need to say there is any disagreement (unless you are using some other interpretation that says there is one).

Doesn't this mean that once decoherence has occurred, the cat is either dead or alive?

No. Remember that decoherence is not interpretation dependent. So decoherence is compatible with the MWI, in which "the measurement has a result" means that it has a result in all branches of the wave function. In this case, that would be both the "dead" branch and the "alive" branch. So you can't say the cat is "either dead or alive" in the MWI, because both of those results occur, not just one. Or, to put it another way, the MWI says that the entangled superposition wave function is the physically real state of the overall system. Decoherence, in the MWI, explains why each branch of the wave function involves a single result, without any interference between them (i.e., there is a "dead" branch and an "alive" branch in the case of the cat).

 

[HighPhy]

Isn't that what I said?

Sorry, I hadn't grasp that.

So saying that

the entangled superposition wave function does not describe a cat that is dead or alive, we just don't know which. It describes a cat which is in some different state that, whatever it is, is not the state of a cat that is dead or a cat that is alive

is equivalent to saying that

the entangled superposition of states, peculiar to the quantum laws of the microscopic world, is reflected on a sentient being like the cat, which behaves classically, and therefore this would not be possible.

? Are these two equal concepts that characterize the paradox as Schroedinger designed it?

the entangled superposition wave function does not describe a cat that is dead or alive, we just don't know which. It describes a cat which is in some different state that, whatever it is, is not the state of a cat that is dead or a cat that is alive.

Is this compatible with the fact that a particle (or the cat, in the paradox) can assume sort of two states at the same time? And that, according to the original formulation of the Copenhagen interpretation, the observer looking at the box after opening collapses the wave function into a single state (cat either alive or dead), creating a contradiction and characterizing the paradox?

(Of course, I'm assuming Schroedinger's original formulation without introducing the concept of decoherence that we have already discussed)

No. Remember that decoherence is not interpretation dependent. So decoherence is compatible with the MWI, in which "the measurement has a result" means that it has a result in all branches of the wave function. In this case, that would be both the "dead" branch and the "alive" branch. So you can't say the cat is "either dead or alive" in the MWI, because both of those results occur, not just one. Or, to put it another way, the MWI says that the entangled superposition wave function is the physically real state of the overall system. Decoherence, in the MWI, explains why each branch of the wave function involves a single result, without any interference between them (i.e., there is a "dead" branch and an "alive" branch in the case of the cat).

OK, this was the step I missed.

So, let me see if I understand.
On a collapse interpretation, after decoherence occurs, the cat is either alive or dead.
On a no collapse interpretation such as MWI, after decoherence occurs, the cat is both alive and dead in two different branches.
But in general, since decoherence is not interpretation dependent, it cannot be said that decoherence collapses the cat into "alive" or "dead" state. Yes?

 

[PeterDonis]

Are these two equal concepts that characterize the paradox as Schroedinger designed it?

I believe that is how Schrodinger intended his scenario to be interpreted, yes.

Is this compatible with the fact that a particle (or the cat, in the paradox) can assume sort of two states at the same time?

There is no such things as "two states at the same time". That is a notorious pop science misconception.

The overall system has only one quantum state. A subsystem that is entangled, like the cat in the state we are considering, has no well-defined state of its own, unless you want to treat the mixed state you get by tracing over all of the other subsystems to be a valid "state". But in that case there is only one such mixed state, not two.

On a collapse interpretation, after decoherence occurs, the cat is either alive or dead.

Yes.

On a no collapse interpretation such as MWI, after decoherence occurs, the cat is both alive and dead in two different branches.

Yes.

But in general, since decoherence is not interpretation dependent, it cannot be said that decoherence collapses the cat into "alive" or "dead" state. Yes?

Yes.

 

[HighPhy]

I believe that is how Schrodinger intended his scenario to be interpreted, yes.

So could you please explain to me why these two concepts are equivalent, and not one subsequent to the other?

There is no such things as "two states at the same time". That is a notorious pop science misconception.

The overall system has only one quantum state. A subsystem that is entangled, like the cat in the state we are considering, has no well-defined state of its own, unless you want to treat the mixed state you get by tracing over all of the other subsystems to be a valid "state". But in that case there is only one such mixed state, not two.

I apologize for the mistake. Unfortunately, after criticizing it, I fell into the trap of bad pop science.

Thanks for the corrections and for all the insightful responses.

 

[PeterDonis]

could you please explain to me why these two concepts are equivalent

No. I have never claimed that the two concepts are equivalent. All I have said is that I think your statements are a reasonable description of how Schrodinger intended his scenario to be interpreted.

At this point we are way too far into the weeds of trying to use ordinary language to describe things that can't be described in ordinary language.

 

[HighPhy]

No. I have never claimed that the two concepts are equivalent. All I have said is that I think your statements are a reasonable description of how Schrodinger intended his scenario to be interpreted.

At this point we are way too far into the weeds of trying to use ordinary language to describe things that can't be described in ordinary language.

Then I had misunderstood your words. It was just a misunderstanding.

Honored to have received such in-depth help.