I About nature of superposition of states

[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.

 

[Nugatory]

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?

Just about any random statement is "sort of" reasonable if you can get those words to do enough work - and you're asking "sort of" to do a lot of work here.
I know that a tossed coin will eventually land either heads or tails, so I can say that while it is spinning through the air it is sort of assuming two states at once - but it makes way more sense to say that while it is spinning through the air its state is neither "heads" nor "tails", but instead "spinning through the air".

Three pages into the thread and you are still treating natural-language statements couched in vague and imprecise language ("sort of" may be a new extreme of vagueness and imprecision) as if they are statements of fact. It won't work.

 

[HighPhy]

Just about any random statement is "sort of" reasonable if you can get those words to do enough work - and you're asking "sort of" to do a lot of work here.
I know that a tossed coin will eventually land either heads or tails, so I can say that while it is spinning through the air it is sort of assuming two states at once - but it makes way more sense to say that while it is spinning through the air its state is neither "heads" nor "tails", but instead "spinning through the air".

Three pages into the thread and you are still treating natural-language statements couched in vague and imprecise language ("sort of" may be a new extreme of vagueness and imprecision) as if they are statements of fact. It won't work.

Yes, you're right.

IMO, this happens because I still don't have clear in my mind the distinction between the states.

If a system has $50\%$ chance to be in state $\left|\psi_1\right>$ and $50\%$ to be in state $\left|\psi_2\right>$, then this is a mixed state. Both states exist, and as we said, this is not the case with the cat in the box. Right?

Now, consider the state
$$\left|\Psi\right>=\frac{\left|\psi_1\right>+\left|\psi_2\right>}{\sqrt{2}},$$ which is a superposition of the states $\left|\psi_1\right>$ and $\left|\psi_2\right>$.
It is a pure state. Meaning, there's not a 50% chance the system is in the state $|\psi_1\rangle$ and a 50% it is in the state $|\psi_2\rangle$. There is a 0% chance that the system is in either of those states, and a 100% chance the system is in the state $|\Psi\rangle$. Right?

This is consistent with what PeterDonis says:

(if you agree with Schrodinger) ... 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.

But in the course of the thread, it was mentioned that the term "superposition of states" is also incorrect. But is it incorrect only because the term "superposition" does not inherently provide for the notion of "entanglement"?
Or is there some more subtle reason?

Also, what I want to ask after pointing out this difference between pure state and mixed state is: is the problem in pop science (and also in my previous response) the confusion between mixed state and superposition of states?

 

[Nugatory]

is the problem in pop science (and also in my previous response) the confusion between mixed state and superposition of states?

Not the problem but a problem, one of many. The solution to all of them is to stop wasting one’s time with pop-sci sources.

 

[PeterDonis]

If a system has $50\%$ chance to be in state $\left|\psi_1\right>$ and $50\%$ to be in state $\left|\psi_2\right>$, then this is a mixed state.

More precisely, this is one situation in which we model the system using a mixed state. (The other situation is modeling a system which is entangled with other systems and therefore does not have a well-defined state of its own; we can trace over the other systems and obtain a mixed state for the system we are interested in.)

Both states exist

No. A mixed state in the scenario you describe reflects our lack of knowledge of how the system was prepared. It does not reflect any quantum superposition with regard to the system itself.

this is not the case with the cat in the box. Right?

If you mean the cat in the box is not modeled using a mixed state, that is correct. We have full knowledge of how the system was prepared, and we are not interested in tracing out other subsystems besides the cat, so there is no need to use a mixed state; we can model everything using pure states.

Now, consider the state
$$\left|\Psi\right>=\frac{\left|\psi_1\right>+\left|\psi_2\right>}{\sqrt{2}},$$ which is a superposition of the states $\left|\psi_1\right>$ and $\left|\psi_2\right>$.
It is a pure state. Meaning, there's not a 50% chance the system is in the state $|\psi_1\rangle$ and a 50% it is in the state $|\psi_2\rangle$. There is a 0% chance that the system is in either of those states, and a 100% chance the system is in the state $|\Psi\rangle$. Right?

No.

First, this state is not entangled, so it is not the same as the entangled superposition cat states we have been looking at.

Second, because the state is not entangled, calling it a "superposition" is basis dependent. If you are representing the state as a superposition of $\psi_1$ and $\psi_2$, that implies that you intend to measure the system and that $\psi_1$ and $\psi_2$ are eigenstates representing possible measurement results. And if you are doing that, then the relative amplitudes for $\psi_1$ and $\psi_2$ give you the probability of getting the measurement results corresponding to $\psi_1$ and $\psi_2$. In the state you wrote down, these probabilities are 50% each. But you could choose to make some different measurement which would lead you to write down the state in a different basis and give different probabilities. You could even choose a measurement for which $\Psi$ itself is an eigenstate, in which case the probability would be 100% that you would get the corresponding measurement result.

Third, what the state represents apart from probabilities for possible measurement results in whatever basis you have chosen is interpretation dependent. Some interpretations treat the state as representing the physically real state of an individual quantum system; in such an interpretation, the physically real state before any measurement is made would indeed be $\Psi$, not $\psi_1$ or $\psi_2$. Our basis for saying that would be that we prepared the system in such a way that we know the state $\Psi$ is what came out of our preparation process.

But other interpretations do not treat the state as representing the physically real state of an individual quantum system. They might treat it as representing an abstract ensemble of systems all prepared by the same process, or as representing the preparation process itself. In such an interpretation, you can't say anything about the state of an individual quantum system.

This is consistent with what PeterDonis says:

No, it has nothing to do with what I said in what you quote, since, as noted above, the state you wrote down is not an entangled superposition.

it was mentioned that the term "superposition of states" is also incorrect. But is it incorrect only because the term "superposition" does not inherently provide for the notion of "entanglement"?

Yes. See above.

Also, what I want to ask after pointing out this difference between pure state and mixed state is: is the problem in pop science (and also in my previous response) the confusion between mixed state and superposition of states?

The problem with pop science is that it is pop science and is not a reliable source if you want to learn actual science.

Apart from that, you appear to be placing way, way too much weight on ordinary language descriptions. Physics is not done in ordinary language. It is done in math. You keep trying to make hairsplitting distinctions about ordinary language that is already known to be inadequate to describe the physics anyway. You would be far better served by forgetting all about ordinary language and learning how to describe the physics with math.

 

[HighPhy]

Second, because the state is not entangled, calling it a "superposition" is basis dependent. If you are representing the state as a superposition of $\psi_1$ and $\psi_2$, that implies that you intend to measure the system and that $\psi_1$ and $\psi_2$ are eigenstates representing possible measurement results. And if you are doing that, then the relative amplitudes for $\psi_1$ and $\psi_2$ give you the probability of getting the measurement results corresponding to $\psi_1$ and $\psi_2$. In the state you wrote down, these probabilities are 50% each. But you could choose to make some different measurement which would lead you to write down the state in a different basis and give different probabilities. You could even choose a measurement for which $\Psi$ itself is an eigenstate, in which case the probability would be 100% that you would get the corresponding measurement result.

Interpretations aside (which, by the way, are not to be neglected), I omitted much of the context in which I wanted to present my response. My fault. I'll try to rectify that.
I would like to know if my claims can be fixed by the following reasoning. Or alternatively, if there is still something wrong.

The point I omitted is that these statements are all made before I make any measurements. Yes?

It is true that if I measure the observable corresponding to $\psi$, then there is a $50 \%$ chance after collapse the system will end up in the state $|\psi_1\rangle$.

However, let's say I choose to measure a different observable. Let's say the observable is called $\phi$, and let's say that $\phi$ and $\psi$ are incompatible observables in the sense that as operators $[\hat{\psi},\hat{\phi}]\neq0$. (I realize I'm using $\psi$ in a sense you would never use it). The incompatibility means that $|\psi_1 \rangle$ is not just proportional to $|\phi_1\rangle$, it is a superposition of $|\phi_1\rangle$ and $|\phi_2\rangle$ (the two operators are not simultaneously diagonalized).

Then we want to re-express $|\Psi\rangle$ in the $\phi$ basis. Let's say that we find
$$
|\Psi\rangle = |\phi_1\rangle
$$

For example, this would happen if
$$
|\psi_1\rangle = \frac{1}{\sqrt{2}}(|\phi_1\rangle+|\phi_2\rangle)
$$
$$
|\psi_2\rangle = \frac{1}{\sqrt{2}}(|\phi_1\rangle-|\phi_2\rangle)
$$
Then I can ask for the probability of measuring $\phi$ and having the system collapse to the state $|\phi_1\rangle$, given that the state is $|\Psi\rangle$, it's $100\%$.
So I have predictions for the two experiments, one measuring $\psi$ and the other $\phi$, given knowledge that the state is $\Psi$.

But now let's say that there's a $50\%$ chance that the system is in the pure state $|\psi_1\rangle$, and a $50\%$ chance the system is in the pure state $|\psi_2\rangle$. Not a superposition, a genuine uncertainty as to what the state of the system is. If the state is $|\psi_1 \rangle$, then there is a $50\%$ chance that measuring $\phi$ will collapse the system into the state $|\phi_1\rangle$. Meanwhile, if the state is $|\psi_2\rangle$, I get a $50\%$ chance of finding the system in $|\phi_1\rangle$ after measuring. So the probability of measuring the system in the state $|\phi_1\rangle$ after measuring $\phi$, is
$$(50\% \ \mathrm{being \ in} \ \psi_1)(50\% \ \mathrm{measuring} \ \phi_1) + (50\% \ \mathrm{being \ in} \ \psi_2)(50\% \ \mathrm{measuring} \ \phi_1)=50\%$$
This is different than the pure state case.

 

[PeterDonis]

The point I omitted is that these statements are all made before I make any measurements. Yes?

If that is what you want the context of your statements to be, then my remarks about what things are interpretation dependent applies. And any discussion of things that are interpretation dependent belongs in a separate thread in the interpretations subforum, not in this thread.

I have predictions for the two experiments, one measuring $\psi$ and the other $\phi$, given knowledge that the state is $\Psi$.

Yes, all that looks fine.

But now let's say that there's a $50\%$ chance that the system is in the pure state $|\psi_1\rangle$, and a $50\%$ chance the system is in the pure state $|\psi_2\rangle$.

But let's say it with math: you have a mixed state which is 50% $\psi_1$ and 50% $\psi_2$. That means you have the following density matrix, written in the $\psi_1$, $\psi_2$ basis:

$$
\rho = \begin{pmatrix}
\frac{1}{2} & 0 \\
0 & \frac{1}{2}
\end{pmatrix}
$$

So the probability of measuring the system in the state $|\phi_1\rangle$ after measuring $\phi$, is
$$(50\% \ \mathrm{being \ in} \ \psi_1)(50\% \ \mathrm{measuring} \ \phi_1) + (50\% \ \mathrm{being \ in} \ \psi_2)(50\% \ \mathrm{measuring} \ \phi_1)=50\%$$

And another way of seeing that is to transform the above density matrix to the $\phi_1$, $\phi_2$ basis, and find that it looks the same.

This is different than the pure state case.

Yes, because you have a mixed state: as I said before, you don't have full knowledge of how the system is prepared, so you can't make the same predictions you would make if you did know that.

 

[PeterDonis]

How can the existence of an interpretation enable that you're correct in speaking that way, unless it was true, that what makes you incorrect in speaking that way, is some other kind of interpretation?

I'm not sure what you're asking here, but discussion of interpretations is off topic in this thread; it belongs in a separate thread in the interpretations subforum.

 

[PeterDonis]

I'm not sure what you're asking here, but discussion of interpretations is off topic in this thread; it belongs in a separate thread in the interpretations subforum.

And this subthread has now been moved to a separate thread in the interpretations subforum:

https://www.physicsforums.com/threads/bohmian-interpretation-of-schrodingers-cat.1060982/

 

[selfsimilar]

Apart from that, you appear to be placing way, way too much weight on ordinary language descriptions. Physics is not done in ordinary language. It is done in math. You keep trying to make hairsplitting distinctions about ordinary language that is already known to be inadequate to describe the physics anyway. You would be far better served by forgetting all about ordinary language and learning how to describe the physics with math.

You are saying shutup and calculate. Does that mean that the "and,or" is not a question in QM( math) in the same sense that we do not ask what are virtual particle that are represented by math in QFT.

 

[PeterDonis]

You are saying shutup and calculate.

In this forum, yes, because in this forum we just discuss the basic math of QM, without adopting any particular interpretation.

Does that mean that the "and,or" is not a question in QM( math) in the same sense that we do not ask what are virtual particle that are represented by math in QFT.

I don't understand what you mean by this.

 

[selfsimilar]

I don't understand what you mean by this.

I mean the math is silent on how to interpret the probabilities as "and" or "or".

 

[PeterDonis]

I mean the math is silent on how to interpret the probabilities as "and" or "or".

The math is not silent on what probabilities mean operationally: you can test them experimentally.

The math is "silent" on "and" or "or" because those words are not math.

 

[RUTA]

Apologies to everyone for insisting on this thread. I would like to shed some light on this point.

Most widely circulated scientific articles present Schroedinger's Cat paradox in the following way:

"Suppose a perfectly closed lead box, that is, in such a way that you cannot understand in any way what is inside it.
The cat until the act of observation is both alive and dead, when you check collapses the wave function and the cat is either alive or dead."

This seems to me a misrepresentation of this thought experiment, typical of bad pop-science. Otherwise I am the one who has missed the point.

How can the phrase "both dead and alive" be a synonym for "a superposition of two states, dead or alive"? Is the inherent formulation of this thought experiment correct? And what role does superposition play?

Let me answer this as a quantum information theorist might. The difference between a classical bit of information (cbit) and a quantum bit of information (qubit) is that you can get from a pure state to another pure state continuously through other pure states for a qubit, while you are only passing through mixed states between pure states for the cbit.

For example, suppose your cbit is a box and a measurement of the box (opening it) reveals one of two outcomes: a ball (yes) or no ball (no). The probability space has two axes, one represents "yes" and the other "no". Those are pure states, i.e., they represent actual measurement outcomes of a single trial of the experiment. Any state between those pure states, e.g., 80% yes, 20% no, does not represent the outcome of some new measurement, it represents a distribution of the yes/no outcomes of the original measurement, i.e., it's a mixed state. But, if the ball-box combo was a qubit, then that 80-20 state would have to correspond to the outcome of some other measurement with 100% probability.

For example, the x-spin state $$\frac{|\text{x+}\rangle + |\text{x-}\rangle}{\sqrt{2}}$$ means you will get 50% "up" results and 50% "down" results when you make an x-spin measurement of electrons in this state. If the x direction is horizontal, then your Stern-Gerlach magnets are oriented horizontally and electrons in this state are deflected in equal degree to the right and left in equal numbers. Since this is a qubit, your electron state must also be a pure state for some measurement corresponding to an outcome with 100% probability. What is that measurement and its outcome in this case? A z-spin "up" state works. In other words, you pass electrons through vertically oriented (z direction) SG magnets and those that are deflected up (towards North magnetic pole) are selected for future measurement. Those electrons, if subjected to a z-spin measurement, will produce the outcome z-spin "up" with 100% certainty, but if subjected to an x-spin measurement will produce "up-down" (physically, right-left) outcomes in 50-50 fashion, i.e., $$|\text{z+}\rangle = \frac{|\text{x+}\rangle + |\text{x-}\rangle}{\sqrt{2}}$$. Now you understand the physical difference between a cbit and a qubit.

The problem with the way most people present Schroedinger's Cat is that they only talk about a measurement with outcomes of Live Cat (LC) and Dead Cat (DC). With that information alone, we could have a cbit. The problem Schroedinger was pointing out is that quantum mechanics is supposedly applicable to anything. Therefore, it should be possible to render the Cat-Box system a qubit rather than a cbit in which case the state $$\frac{|\text{LC}\rangle + |\text{DC}\rangle}{\sqrt{2}}$$ must represent the outcome of some measurement with 100% certainty. What is that measurement? And, what does its outcome mean physically? Can you turn the Cat-Box system into a qubit simply by adding a quantum trigger mechanism for the deadly gas? Does that help answer the questions we need answered to understand the Cat-Box system as a qubit? We could answer those questions for the spin of an electron, but his point was we have no answers for the Cat-Box system. So, is quantum mechanics really applicable to any thing?

We have a book forthcoming with Oxford UP in June 2024 titled "Einstein's Entanglement: Bell Inequalities, Relativity, and the Qubit" in which we propose a principle justification for the existence of qubits (whence the Hilbert space of quantum mechanics). It was written so that someone with knowledge of intro physics alone could understand it. Since you're a physics student, you might be interested in that.

 

[HighPhy]

We have a book forthcoming with Oxford UP in June 2024 titled "Einstein's Entanglement: Bell Inequalities, Relativity, and the Qubit" in which we propose a principle justification for the existence of qubits (whence the Hilbert space of quantum mechanics). It was written so that someone with knowledge of intro physics alone could understand it. Since you're a physics student, you might be interested in that.

I'm looking forward to it!

 

[selfsimilar]

Let me answer this as a quantum information theorist might.

Can quantum information theorist say something different than the original theory? AFAIK the theory says that you cannot know the exact value of the observable before measurement, even worse you cannot know things like "and", "or" question which this thread is about, you can only know the value after measurement. So I am really surprised why there are endless threads about that. The theory just has nothing to say about that (before measurement) by the postulates, that is that. Everything else is a made up idea trying to reach a similar structure to quantum mechanics by providing a "mechanism" before measurement henceforth called interpretation.