Does Schrodinger's Cat Paradox Suck?

[Q-reeus]

What I worry about is degeneracy-- it would seem that if the individual particles in some seemingly macroscopic paddle were in their ground states, the whole system still does not have a unique "ground state", as there can be any phase relationship among the parts, even before you account for the identical particle multiple wavefunctions. It would seem to generate a huge manifold of equal-energy "ground states" for the whole system, so if you started out with a mixture of those before you "froze" it, you'd still have a mixture after freezing it-- not a pure state. That might not affect its behavior as it evolves as in its frozen condition, but when you later interact it with something, or heat it up, one might imagine the mixed state would "pop out" again, like magnetic domains. For some kinds of interactions that might not matter, but for others it might, and it would still not be literally a superposition, it would be a mixture of ground states with great similarities.

Interesting. My crude analogy here would be 'glass' vs 'crystal' - the former has an intrinsic disorder entropy even at absolute zero, the latter in principle may not (not sure here about isotope mix or nuclear spin ordering). So I guess an ultra deep frozen cat would be more 'glass' than 'crystal' in the quantum states sense. At another level, I take it then a macroscopic super-current at relatively high temp (say in a so-called high temperature superconductor) could be described as in a coherent ground state but not a pure state?

...I would just say the classical notions of vibration or no vibration does not encompass the behavior of the paddle.

OK this is no doubt your pure CI coming out!

 

[Rap]

The state of the cat must be viewed as a substate of the whole system, it is a projection that does not obey the Schroedinger equation. That equation applies to the closed system on the Hilbert space, not open substates that are projections onto subspaces of the Hilbert space. The subspaces do not preserve the postulates of quantum mechanics (in particular, they evolve into mixed states under decoherence, not superposition states), and this is the source of a lot of misunderstanding about the cat paradox.

Indeed, that is perhaps the key difference between a micro system and a macro system, it is the meaning of the Heisenberg divide: a micro system, as a substate, can recover its status as a pure state by measuring it and isolating it-- even though it remains a substate of something larger, it can be treated as a pure state going forward (and exhibit interference and so on). But a macro system, once evolved into a mixed state via external interactions, can never recover its pure state status, it is forever a substate of something larger, and will never exhibit interference. It is just wrong to say that baseballs don't give two-slit patterns because their wavelengths are too small, they are simply not in pure states period.

Ok, I think I understand, and I see that I was wrong to say that what is observed in opening the box is a superposition of e.g. dead microstates. Let me rephrase and see if I get what you are saying: What the scientist observes when opening the box is not a well defined wave function that is some superposition of all the e.g. dead states, but rather an observation that can be attributed to some ill-defined wave function that lies somewhere in the space spanned by the e.g. dead microstates. Is that the "projected subspace" you refer to?

That is also what you said, focusing on the need for isolation, so what I'm saying can be cast as the remark that macro systems like cats can never be sufficiently isolated from their own histories to be treated as systems that obey the Schroedinger equation as the unitary evolution of a state vector in a Hilbert space-- it's just wrong quantum mechanics to describe them that way.

I'm not sure I understand this - What I read is that a macro system cannot be treated quantum mechanically, and I disagree with that, just as I would disagree that a classical macrosystem cannot be treated as an N-body problem, at least in principle.

 

[Ken G]

Interesting. My crude analogy here would be 'glass' vs 'crystal' - the former has an intrinsic disorder entropy even at absolute zero, the latter in principle may not (not sure here about isotope mix or nuclear spin ordering). So I guess an ultra deep frozen cat would be more 'glass' than 'crystal' in the quantum states sense.

That sounds like either a reasonable or an excellent analogy, I can't really tell which without much deeper analysis.

At another level, I take it then a macroscopic super-current at relatively high temp (say in a so-called high temperature superconductor) could be described as in a coherent ground state but not a pure state?

That's an interesting question, it might be a bit like the laser. A laser is highly coherent, but not exactly coherent, and I'm really not sure if it is more accurate to say we have a pure state that has a bandwidth, or if the bandwidth is also expressing a mixed character to the state. When doing laser spectroscopy, the bandwidth is treated as a mixed state of plane waves, but it's probably more coherent than that. But I doubt it is a pure state either, where every photon has the same definite wavefunction.

OK this is no doubt your pure CI coming out!

No doubt! But I would argue this goes beyond interpretation-- it is just correct quantum mechanics, and even MWI would say the same thing. None of the interpretatons know how to reconcile the language of the quantum mechanical wave function of the closed system with the language that applies to whatever piece we actually observe from our vantage point as part of the system. The only one that can navigate that is Bohmian, and it comes with all kinds of excess baggage-- and claims the paddle must be either vibrating or not vibrating, possibly sequentially one or the other, but never both.

 

[Ken G]

Let me rephrase and see if I get what you are saying: What the scientist observes when opening the box is not a well defined wave function that is some superposition of all the e.g. dead states, but rather an observation that can be attributed to some ill-defined wave function that lies somewhere in the space spanned by the e.g. dead microstates. Is that the "projected subspace" you refer to?

Exactly-- the wave function is ill defined because the scientist is now part of it, and cannot attribute their own involvement, so must instead treat a projection that will end up looking like a mixed state. So decoherence can happen in a closed system that does not include an observer, but it will still be unitary evolution-- the paradoxical elements only appear when we have a different flavor of decoherence, the kind that involves untraced noise modes within the observer themself, for that hopelessly derails the concept of a pure state wavefunction, or at least any chance of getting that language to connect with the answers a physicist actually wants in an experiment. That's classic Bohr-- physics is done by macro brains.

I'm not sure I understand this - What I read is that a macro system cannot be treated quantum mechanically, and I disagree with that, just as I would disagree that a classical macrosystem cannot be treated as an N-body problem, at least in principle.

The problem is worse than just complexity. I agree with you that in principle, formal Hilbert-space quantum mechanics should apply to any closed system, no matter how complex, if it can be shown that the system starts off in a pure state when it gets closed. But that's just what cats won't be-- you can put a cat in an isolation booth, but you can't isolate it from its own history, it is already in a mixed state because you can only put the cat in the booth, not everything the cat has ever interacted with. In particular, you aren't putting yourself in the booth with it, but at some point you would have had to interact with it to tell that it is a cat in the first place. This is different from putting a "spin up" particle in an isolation booth, because there really isn't anything else there to know about the particle that is relevant to its behavior. That right there is the crux of the Heisenberg divide, a concept that seems to get little play any more but still seems profoundly relevant to me.

 

[Rap]

Exactly-- the wave function is ill defined because the scientist is now part of it, and cannot attribute their own involvement, so must instead treat a problemjection that will end up looking like a mixed state. So decoherence can happen in a closed system that does not include an observer, but it will still be unitary evolution-- the paradoxical elements only appear when we have a different flavor of decoherence, the kind that involves untraced noise modes within the observer themself, for that hopelessly derails the concept of a pure state wavefunction, or at least any chance of getting that language to connect with the answers a physicist actually wants in an experiment. That's classic Bohr-- physics is done by macro brains.

But how can we ever observe a pure state? Every pure state is connected to a macroscopic measuring device, which a macroscopic scientists observes. The connection between the device and the scientist is classical, so the effect of the scientist on the device is negligible as far as its readings are concerned. We are not trying to determine the wave function of the device, only the system it is measuring. The interaction between the scientist and the box upon opening can be minimal, like some kind of heartbeat detector which turns on a laser beam which emits a few photons per second when the cat's heartbeat stops. Opening the box is equivalent to opening an aperture and detecting or not detecting a stream of photons.

The problem is worse than just complexity. I agree with you that in principle, formal Hilbert-space quantum mechanics should apply to any closed system, no matter how complex, if it can be shown that the system starts off in a pure state when it gets closed.

But it applies for a mixed state as well. A mixed state is just a set of pure wavefunctions spanning the space, each multiplied by a probability. Each wave function can then be propagated using e.g. the Schroedinger equation.

But that's just what cats won't be-- you can put a cat in an isolation booth, but you can't isolate it from its own history, it is already in a mixed state because you can only put the cat in the booth, not everything the cat has ever interacted with. In particular, you aren't putting yourself in the booth with it, but at some point you would have had to interact with it to tell that it is a cat in the first place. This is different from putting a "spin up" particle in an isolation booth, because there really isn't anything else there to know about the particle that is relevant to its behavior. That right there is the crux of the Heisenberg divide, a concept that seems to get little play any more but still seems profoundly relevant to me.

I don't understand "isolate it from its own history". In principle, I could close the box knowing the cat's pure wave function. Or, I can get more realistic, and say its in a mixed state - probabilities of live wave functions are, I don't know, equal a priori, probabilities of dead are zero. Then I propagate both using the Schroedinger equation. Just before I open the box, I have a superposition of live and dead in the first case, and a mixed state combination of live and dead in the second. Now I "open the box". Just as I could in principle specify the pure state when I closed the box, I can in principle determine the pure state when I open it. Yes, practically, that's not likely, but not in principle impossible. Or, if I started with a pure case, but open the box and just observe a dead cat, I can say that my observation yields a mixed state which contains none of the alive components of the pure state I calculated before opening. I would call that a "collapse" of the wave function, but I am not adamant about that terminology. If I start with a mixed case, I can propagate that, then open the box, and again the mixed state collapses to a mixed state of dead cat only. Again, the interaction between the scientist and the box upon opening can be minimal for the mixed case measurement.

 

[Ken G]

But how can we ever observe a pure state? Every pure state is connected to a macroscopic measuring device, which a macroscopic scientists observes.

We don't observe pure states. In fact, we don't observe states at all-- we observe outcomes of observations. States, be they pure or mixed, are theoretical devices we use to understand what we actually do observe. Now, if we observe that a particle has "spin up", then we are allowed to treat the particle as being in a pure spin state going forward, because there's nothing else there to know-- there isn't unknown information being glossed over (if spin is what we care about). This is in contrast to the state of ourselves and the observational device we used that gave spin up-- those contain all kinds of unknown information we are indeed glossing over. So there we have a case where the substate (the particle) is in a pure state going forward into a new isolated environment where we can treat it as a closed system, or an environment with known attributes (like fields) which alter the Hamiltonian. The point is, there we can use the postulates of quantum mechanics going forward (but not backward) in time. The larger system that includes us and our instrument, however, cannot be so treated-- it is not a pure state, owing to all that untracked information and relevant history.

The problem with the usual rather bogus way of introducing the cat paradox is that the alive or dead status of the cat is treated like the spin state of the particle, which is just wrong quantum mechanics, because of all that untreated information that goes into an alive or dead cat (not present in a spin state of a particle). So the question is not if the cat can be in a superposition state, we already know the cat cannot be in a superposition state. The question is, can the system that includes the cat be in a pure state, and if it is, how can there be an alive and a dead cat tangled up in that pure state somewhere? That's the right way to describe the paradox, and it suggests what I think is the right answer-- the Bohr answer, the concept of an alive or dead cat is not a description of a global quantum mechanical system, it is a description of a projection onto a subspace, and projections like that are routinely going to be mixed states.

So there's no paradox as to why the cat is in a mixed state of alive or dead, that's just correct quantum mechanics. The paradox doesn't appear until we open the box and become part of the closed quantum mechanical system. We cannot perceive the quantum mechanical system because we are just a part of it and that's not how our brains work anyway, our brains work by throwing out most of the information there and concentrating on a subspace (like one that distinguishes the cat from ourselves instead of the holistic treatment of the full wave function of the closed system). Then we get the real question here: how does the subspace we are concentrating on get actualized into a set with specific attributes,keeping a set that is consistent with our information and throwing away all that isn't? This has to be answered in the language of information processing, it has to be something about how we do physics itself, not something involving any particular law of physics like quantum mechanics, which is necessarily subject to how we do physics.

But it applies for a mixed state as well. A mixed state is just a set of pure wavefunctions spanning the space, each multiplied by a probability.

But that isn't a quantum mechanical wave function, it is a mixture of wave functions. A mixture of things that obey quantum mechanical formalism is not something that obeys quantum mechanical formalism, but we know how to do quantum mechanics on it (density matrices and so forth). In particular, it produces no cat paradox, the cat paradox requires us to have a pure state in which there is an alive or dead cat in there somewhere, and that requires the state of the whole system.

I don't understand "isolate it from its own history". In principle, I could close the box knowing the cat's pure wave function.

No, that is just exactly what you could never do, not even in principle. Because the only way to know that would be to do measurements on the cat, but that would involve a measuring device, so immediately the cat becomes a subspace of the thing that is a pure state. Unlike measuring the spin of a single particle, where there is no information being ignored, if you measure an "entire cat", there is vast amounts of information you could never get a handle on, like herding cats (literally). There's no measurement like that which even in principle could result in complete information about the cat's wave function that could be treated as a closed system going forward, too much of the data (all the phase coherences) that would need to be tracked is going to be entangled with the instruments doing the measuring, not to mention the brain processing that information.

This is the key point-- the information that goes into determining a wave function is not in the entity being observed, it is in the environment doing the observing and processing that information. Physics is done by physicists, even if we can imagine the action of hypothetical physicists not actually present in the environment. If that environment does include a real brain, it might be able to treat the entity as having a pure-state wavefunction (as for the spin of a particle), but it could never be empowered to treat a cat in a pure-state wavefunction, there would always have to be too much overlooked information (indeed, judicious overlooking of information is more or less the foundational principle of physics). It is only ever the whole system including the observer that could be treated as a pure state, and only if it started out in a pure state, which brings in the issue of history.

Just before I open the box, I have a superposition of live and dead in the first case, and a mixed state combination of live and dead in the second.

But that makes all the difference. If you have a mixed combination, you have no paradox-- you have a purely classical situation, like a coin that is flipped and covered.

Now I "open the box". Just as I could in principle specify the pure state when I closed the box, I can in principle determine the pure state when I open it.

I dispute that, but even if it were possible, you would never get a pure state that is a superposition of an alive cat and a dead cat that way. The very definition of what an alive cat is requires that certain types of information about the cat be processed by a brain (even a hypothetical one) capable of making that determination, but that processing will require coupling the cat to the brain, bringing in all the untracked information in the brain. Again, both physics and language itself are examples of judicious overlooking of unwanted information, completely anathema to a concept of a pure state wavefunction. Quantum mechanics invokes the concept of a pure state expressly for the purposes of later dispelling it, there is no such thing as quantum mechanics involving only pure states. The way I like to put that is, if an electron could think, it wouldn't use quantum mechanics. I believe that is very consistent with Bohr's approach to the role of the mind of the physicist.

Or, if I started with a pure case, but open the box and just observe a dead cat, I can say that my observation yields a mixed state which contains none of the alive components of the pure state I calculated before opening.

Yes, certainly, we invoke mixed states like that all the time, even in classical physics. There isn't any quantum mechanics at the mixed-state level, the quantum mechanics is all what is happening at the pure-state level, and the cat is not described as a pure state in that example. Indeed, a mixture of suitably detailed pure states is exactly the same thing as a classical description. So when what we mean by "an alive cat" has much more to do with the nature of that mixture, than the nature of the pure states that go into it, then we say we have a classical treatment of a cat, not a quantum mechanical treatment.

I would call that a "collapse" of the wave function, but I am not adamant about that terminology.

To say a wave function is collapsed, you must have a wave function in the first place. A mixture is not a wave function, it is a mixture of wave functions. Classically, it is the mixture that matters, not the quantum mechanics of the wave functions-- the evolution of a mixture is a classical evolution, what the individual wave functions are doing gets lost (like a thermodynamic treatment of an ideal gas where we are not a whit for what any given particle is actually doing, only the generic possibilities for what they are allowed to do). When a cat is a super-complicated statistical average of a bunch of possible individual wavefunctions, then it is a classical object, not a quantum mechanical one.

If I start with a mixed case, I can propagate that, then open the box, and again the mixed state collapses to a mixed state of dead cat only. Again, the interaction between the scientist and the box upon opening can be minimal for the mixed case measurement.

Absolutely true-- but only because you are talking about a classical situation through and through.

 

[Dadface]

Nearly eighty years on and this famous experiment is still being discussed.Fair enough but in discussions of this type certain relevant features of the experimental design and set up seem to be overlooked.These features are in connection with the fact that the experiment is a thought experiment,not a real experiment.Thought experiments can be very useful but they must be used with great care and with a consideration of any limitations the experiments may have.The relevant limitations of this experiment can be defined under two headings...Observations and Isolation:

OBSERVATIONS

Schrodinger designed his experiment in such a way that once the box was closed no further relevant observations could be made,not even in principle,until the box was opened again.It becomes pointless to use a theory to make predictions about "happenings" within "hidden" regions(eg the closed box) where,because of the design of the experiment,no evidence can be gained to verify,or otherwise,those predictions.

ISOLATION

A major requirement of the experiment is that whilst the box is closed its contents and all of the surroundings are isolated from each other.If isolation here means "total isolation" then there can be no linkages at all between the contents and all parts of the rest of the universe including no interaction by means of gravitational forces.If it is total isolation that is required then the experiment fails straight away,even as a thought experiment.
Perhaps partial isolation is all that is required and if so before proceeding we would need a definition of "partial isolation".Any takers?

 

[Ken G]

Actually, neither of those seem like problems to me. I agree with you that we cannot talk about happenings in an unobserved region, but we can talk about how we are treating the state of an unobserved system, and that is all that is needed here. The purpose of language about a state is simply to be able to make the proper predictions of the happenings that are in fact observable given the setup that defines the state. The paradox centers on what treatment we should give to the cat before the box is opened-- it is clear enough how to treat it after it is opened. I claim the real problem with the way the paradox is normally stated is that it is wrong quantum mechanics-- the cat is never in a superposition state, because it is a projected substate of that system, and that generally yields a mixed state not a superposition state. However, that does not make the paradox go away-- the whole system can presumably still be treated as a pure state, even if the whole system must include the entire history of the universe. The key point is, such a system includes the physicist, so it is inescapable that physics is something that is done by physicists, not something that happens to nature.

As for gravity, do you really think the experiment comes out differently, or the quantum mechanical treatment used by the physicist must differ, if done in deep space?

 

[Dadface]

Theories are informed by observations,they must conform to observations and if new predictions are made then for these to gain any credence it must be possible ,even if just in principle,to gain the necessary observations to back these predictions up.Schrodinger designed,perhaps unwittingly,an experiment which disallowed the gaining of such observations.
Deep space.Where's that?Isn't this stretching the thought experiment even further by taking everything to an imaginary place where,I'm guessing, it's assumed that gravitational interactions are vanishingly small.If so in the context of the isolation that is needed for the experiment,when does vanishingly small become zero?

 

[Ken G]

Theories are informed by observations,they must conform to observations and if new predictions are made then for these to gain any credence it must be possible ,even if just in principle,to gain the necessary observations to back these predictions up.Schrodinger designed,perhaps unwittingly,an experiment which disallowed the gaining of such observations.

I agree with your perspective on the interaction between theory and observations, but I don't see how your conclusion follows. It seems to me Schroedinger is asking a simple question-- how should our theory treat the state of the system in that thought experiment to be consistent with all the observations we could make? In particular, the theory says that we can correctly predict observations by treating all pure states as if they evolved according to the Schroedinger equation, which is a unitary evolution that only maps pure states into pure states.

The paradox is the fact that experimental outcomes never represent a mapping of a pure state into a pure state, because the outcome of the experiment never constrains the state of the macro apparatus sufficiently to be able to describe it as a pure state. (Here Schroedinger may err to claim that the cat is ever in a superposition state, but the paradox need not be expressed in that particularly pictorial way.) Hence there is a fundamental disconnect with the postulates of quantum mechanics and the way we do physics. Bohr resolves that simply by saying those are the limitations of the postulates of quantum mechanics, there is no need to take them any more seriously than that. I think he's right.

Deep space.Where's that?Isn't this stretching the thought experiment even further by taking everything to an imaginary place where,I'm guessing, it's assumed that gravitational interactions are vanishingly small.

That is the purpose of thought experiments.

If so in the context of the isolation that is needed for the experiment,when does vanishingly small become zero?

No, vanishingly small means the same thing it always does in physics-- too small to affect the observable outcome of the experiment sufficiently to concern the goals of the physicist.

 

[Dadface]

I am referring to the fact that no observations can be made whilst the box is closed.We can make observations before the box is closed and after it is opened and to explain how the event proceeded during the closed time needs nothing more than a bit of simple forensic investigation,general everyday experience,and common sense.Quantum theory may be able to predict the outcome but it can also be interpreted by some as making weird unproveable predictions such as a simultaneously dead and alive cat.Although it was designed with QM in mind,QM is not needed to explain this particular experiment.
The point I am making about thought experiments is that any limitations they have must be considered.Let me rephrase my question with yet another thought experiment.Assume that the box is at a place where the contents could not be considered as isolated because of their appreciable gravitational interactions with the surroundings.Now let the box go on a journey along a path of reducing gravitational interaction.Where along the path does the interaction fall to such a small value that the box contents become isolated enough to meet the criteria needed by Schroedingers experiment?In other words what exactly is the isolation needed and when can small be considered as zero?
We could go to the limit and take the box to an imaginary place where there are no interactions at all resulting in total isolation but such a place would be separated from the rest of the observeable universe.Yet another thought experiment breakdown.

 

[Ken G]

I am referring to the fact that no observations can be made whilst the box is closed.We can make observations before the box is closed and after it is opened and to explain how the event proceeded during the closed time needs nothing more than a bit of simple forensic investigation,general everyday experience,and common sense.Quantum theory may be able to predict the outcome but it can also be interpreted by some as making weird unproveable predictions such as a simultaneously dead and alive cat.Although it was designed with QM in mind,QM is not needed to explain this particular experiment.

Yes, that is very much Bohr's view-- we use QM for certain things, but not for others, it is our tool not our master. But many theorists find the aesthetic beauty of QM to be compelling as a more fundamental theory that transcends the way we use it and becomes a kind of "actual law" that reality obeys. It sounds like you and I would both agree that this view is elevating physics to a kind of philosophical status that is not justified by the prescriptions of science, but that is in itself a very interesting debate. One thing we cannot ignore is that quantum mechanics obeys the Correspondence Principle-- it never makes a wrong prediction, even when applied on scales where it was not intended, so its problem is that it is unwieldy but not wrong on classical scales. But if it's not wrong on those scales, then if we can get a large enough box around a closed system, it has to evolve according to the Schroedinger equation-- the issue then becomes, what does that evolution mean if it is not what is used by the physicist to make predictions and understand outcomes? That question is what separates the Copenhagen interpretation from the others.

The point I am making about thought experiments is that any limitations they have must be considered.

I would frame that same concern as saying that a thought experiment may be hypothetical, but it must not be magical-- its hypothetical elements have to be included as part of an actual (albeit hypothetical) experiment. So the results have to be framed as experimental outcomes, but we can still ask what states of the system we would use along the way to reach those predictions, and we can ask if those states make sense with what we observe in everyday life. Some say that the cat really is in a superposition state, and we just don't notice it, and others say that what we notice is what physics is all about. I point out that many claims of the superposition state are actually wrong quantum mechanics, because they treat a substate rather than a closed state.

Let me rephrase my question with yet another thought experiment.Assume that the box is at a place where the contents could not be considered as isolated because of their appreciable gravitational interactions with the surroundings.Now let the box go on a journey along a path of reducing gravitational interaction.Where along the path does the interaction fall to such a small value that the box contents become isolated enough to meet the criteria needed by Schroedingers experiment?

If you want to study at what point a system is "closed enough" to be treated as a closed state, you simply carry out the experiment many times, opening it at various gravitational strengths, and see when you start to get the predictions you expect for a closed system. But none of that would get at the issue of the cat paradox, because that paradox already exists for a perfectly closed system-- because when you open it to learn something about the system, at that point it is not closed any more, regardless of whether it was before. So there's always something entering the problem that you are not treating in the "pure state" description of the system, the very way we do physics always assumes we will open the system at the end, and that's where Bohr says the literal interpretation of the postulates breaks down.

We could go to the limit and take the box to an imaginary place where there are no interactions at all resulting in total isolation but such a place would be separated from the rest of the observeable universe.Yet another thought experiment breakdown.

Perhaps we are not that far apart-- I say that experiment breaks down not because it is isolated from the rest of the observable universe, but because it has to return to the observable universe at the point where we use it to test our understanding of it. It is this return that ruins the thought experiment, not the problem with isolation during the unitary evolution of the system.

 

[yuiop]

Having read the last few posts in this thread, there is still some confusion about whether the cat is (1) in superposition of dead AND alive or (2) is definitely either alive or dead but not both, while the box is closed and the observer can make no observations of the state of the cat. Now persons taking either position (1) or (2) are making claims about the state of the cat with absolutely no evidence to support their position because by the definition of the thought experiment no observations can be made, only suppositions.

Now let us say you have introduced a student to quantum physics by way of Schroedinger's cat example and then go on to state that a quantum particle is also in a state of superposition before measurement. A student can then reasonably conclude that "superposition" is just physicists' "mumbo jumbo" for "we do not know what the state of the particle is, simply because we have not yet measured it". If there is a deeper meaning to superposition, then Schroedinger's cat is doing a very poor job of demonstrating what it is.

 

[Ken G]

I agree with much of what you are saying, it sounds like you are making a plea for the value of being able to describe states as theoretical entities without apology, even when no specific observable is specified. This is akin to claims that a wave function is an expression of "everything that could possibly be known" about a system, or maybe even what nature itself "knows" about that system. However, I would say there are equal pitfalls to adopting a stance that nature is a kind of information processor, when in fact it is physicists that are information processors, as there is in adopting language that forces us to treat all theoretical constructs as "mumbo jumbo."

Somehow there is a fine line to walk there, and that line is very much at the heart of the dispute between the Copenhagen interpretation and others. So in that sense, instilling a healthy respect for careful consideration of that "fine line" is very much the point of the cat paradox. To me, walking that line successfully allows us to postulate hypothetical observers (and their brains) without any actually being present, but we are not allowed to postulate hypothetical knowledge of a system without the means for that knowledge to be specified being (hypothetically) present. Thus no experimental outcome has any meaning until we allow that the measuring device (and the brain that interprets it) be present, because its presence is what gives meaning to the outcome in principle-- whether or not the device is actually there in practice.

 

[Rap]

(Responding to Rap: I don't understand "isolate it from its own history". In principle, I could close the box knowing the cat's pure wave function.)

No, that is just exactly what you could never do, not even in principle. Because the only way to know that would be to do measurements on the cat, but that would involve a measuring device, so immediately the cat becomes a subspace of the thing that is a pure state. Unlike measuring the spin of a single particle, where there is no information being ignored, if you measure an "entire cat", there is vast amounts of information you could never get a handle on, like herding cats (literally). There's no measurement like that which even in principle could result in complete information about the cat's wave function that could be treated as a closed system going forward, too much of the data (all the phase coherences) that would need to be tracked is going to be entangled with the instruments doing the measuring, not to mention the brain processing that information
...
If that environment does include a real brain, it might be able to treat the entity as having a pure-state wavefunction (as for the spin of a particle), but it could never be empowered to treat a cat in a pure-state wavefunction, there would always have to be too much overlooked information

So you are saying that even in priciple, we cannot assign a wave function to a macroscopic object because "too much of the data (all the phase coherences) that would need to be tracked is going to be entangled with the instruments doing the measuring". I do not understand this "too much data being entangled with the instrument" effect. I assume this effect is continuous, gets continually worse as the size of the system being investigated contains more and more particles. Can you describe this effect a bit more fully, perhaps with just a few particles which demonstrates the effect, even if the effect is miniscule?

(Responding to Rap: Just before I open the box, I have a superposition of live and dead in the first case, and a mixed state combination of live and dead in the second.)

But that makes all the difference. If you have a mixed combination, you have no paradox-- you have a purely classical situation, like a coin that is flipped and covered.

I would say there is no paradox even if you did have a pure wave function. What is the paradox that occurs in assuming a pure state is known when the box is closed? Is it just the effect you mentioned above - i.e. it cannot be done?

(Responding to Rap: Now I "open the box". Just as I could in principle specify the pure state when I closed the box, I can in principle determine the pure state when I open it.)

I dispute that, but even if it were possible, you would never get a pure state that is a superposition of an alive cat and a dead cat that way. The very definition of what an alive cat is requires that certain types of information about the cat be processed by a brain (even a hypothetical one) capable of making that determination, but that processing will require coupling the cat to the brain, bringing in all the untracked information in the brain. Again, both physics and language itself are examples of judicious overlooking of unwanted information, completely anathema to a concept of a pure state wavefunction. Quantum mechanics invokes the concept of a pure state expressly for the purposes of later dispelling it, there is no such thing as quantum mechanics involving only pure states. The way I like to put that is, if an electron could think, it wouldn't use quantum mechanics. I believe that is very consistent with Bohr's approach to the role of the mind of the physicist.

I'm sorry, I misspoke - if you assume that you have a pure (and of course non-superposed) state when the box is closed, then you can do QM calculations (e.g. Schroedinger's equation) and you will calculate, for some time later (e.g. just before the box is opened), a pure state which is a superposition. This state will tell you the probability of opening the box and seeing the cat dead or alive, pretty much, since there will be little chance that, upon opening the box, the interference of the environment will affect that outcome.

There isn't any quantum mechanics at the mixed-state level, the quantum mechanics is all what is happening at the pure-state level, and the cat is not described as a pure state in that example. Indeed, a mixture of suitably detailed pure states is exactly the same thing as a classical description. So when what we mean by "an alive cat" has much more to do with the nature of that mixture, than the nature of the pure states that go into it, then we say we have a classical treatment of a cat, not a quantum mechanical treatment.

I am not, and I wish to avoid, invoking the decoherence approximation here. A mixture of suitably detailed pure states is NOT EXACTLY the same as a classical description. There IS quantum mechanics at the mixed state level - Upon closing the box, having measured a mixed state, each pure state of the mixture is propagated forward by e.g. the Schroedinger equation. Each pure state propagates to a another pure state, yielding a fully defined propagated mixed state at a time just before the box is opened, which is then used to calculate the probability of finding the cat dead or alive when the box is opened.

Note - when I say "having measured a mixed state", it may be as little as saying "I see a live cat, geiger counter not clicked", and then assigning equal probability to every pure state of the mixture which conforms to this observation, zero to the rest, which is, of course, more than just a measurement.

 

[Rap]

Having read the last few posts in this thread, there is still some confusion about whether the cat is (1) in superposition of dead AND alive or (2) is definitely either alive or dead but not both, while the box is closed and the observer can make no observations of the state of the cat. Now persons taking either position (1) or (2) are making claims about the state of the cat with absolutely no evidence to support their position because by the definition of the thought experiment no observations can be made, only suppositions.

Now let us say you have introduced a student to quantum physics by way of Schroedinger's cat example and then go on to state that a quantum particle is also in a state of superposition before measurement. A student can then reasonably conclude that "superposition" is just physicists' "mumbo jumbo" for "we do not know what the state of the particle is, simply because we have not yet measured it". If there is a deeper meaning to superposition, then Schroedinger's cat is doing a very poor job of demonstrating what it is.

The short answer is yes, "superposition" is just physicists' "mumbo jumbo" for "we do not know what the state of the particle is, simply because we have not yet measured it" and there is no deeper meaning to superposition.

The problem is again, the use of the word "state". Let's define "state" as the wave function or, equivalently, the quantum state of the particle. I think you are using the word "state" at a particular time to mean the quantum state of a particle if it were measured at that time. Let's call that the "measure-state". (Note that a particle is not in a well defined measure-state unless you specify what measurement you would make - If you measure position, then you get a different measure-state than if you measure momentum. For the cat, this is not something to worry about.)

So I think your statement is: "superposition" is just physicists' "mumbo jumbo" for "we do not know what the measure-state of the particle is, simply because we have not yet measured it"

So yes, I agree completely, its obviously true. A superposition consists of a bunch of possible measure-states, each with their own amplitude. It is an indication of the fact that the scientist does not know which measure-state the particle is in. When you make a measurement, the state collapses to one of those measure-states, and then you know.

If you have a measure state, Schroedingers equation tells you how it changes as time goes on. As time goes on, it becomes a superposition. The measure state evolves into a superposition state, which is a bunch of possible measure states, which is physicist's "mumbo jumbo" for "we do not know which measure state it has evolved into".

To Copenhagen people, there is no deeper meaning to superposition.

NOTE - this only applies to quantum wave functions, in a very isolated system. If you flip a coin and cover it with your hand without looking at it, it is not in a superposed state because you cannot describe it with a wave function. A wave function analysis only works for an isolated system, and your hand, the coin, everything else is open to the universe so you can't do QM. Thats why the cat is inside a box. Its the same with classical mechanics - if you take 12 toothpicks and create a little box, and stand outside, you cannot calculate the thermodynamics or velocities or anything inside that box for the next day or two just knowing what goes on inside the box at time zero. You have to know what the rest of the universe is doing, what the weather will be in an hour or a day or two. Quantum mechanics is much more restrictive. You cannot even have a stray photon wander into your box without upsetting the QM calculations.

 

[Ken G]

So you are saying that even in priciple, we cannot assign a wave function to a macroscopic object because "too much of the data (all the phase coherences) that would need to be tracked is going to be entangled with the instruments doing the measuring". I do not understand this "too much data being entangled with the instrument" effect. I assume this effect is continuous, gets continually worse as the size of the system being investigated contains more and more particles. Can you describe this effect a bit more fully, perhaps with just a few particles which demonstrates the effect, even if the effect is miniscule?

What I'm saying is that you don't get pure states naturally, they happen only in very controlled environments, and even then only in the mind of the physicist. The classic way to get a pure state is to pass a single particle through a Stern-Gerlach arrangement so that it has spin up or spin down, but of course that is not actually the state of the particle-- the particle has a zillion indistinguishable partners all over the universe and there's no such thing as a single particle wave function for real. However, we can isolate the particle to the point where the "true" wavefunction isn't needed, and we get instead the concept of a "single particle wavefunction." It's a good concept because it gives us correct predictions, so we get away with it even though it is not the quantum mechanically correct state of the particle in the universe. Or put differently, it is only the correct wavefunction if we allow that quantum mechanics is a tool used by physicists, not a true description of nature. So let's adopt that stance for now-- quantum mechanics is a set of rules that a neural processor uses to make correct predictions of macroscopic measurements.

So in this way of thinking, we do get pure states of single particles, but only just after we do a measurement. As you say, this then evolves into a superposition if all that happens is the particle is acted on by potentials-- but that does not include doing measurements on the particle. So the pure state only exists as a kind of interloper between two measurements, and neither measurement is itself describable as a pure state of anything. Now we can ask the question of whether a cat could ever be in a pure state, even in principle (it obviously could't ever be in practice). Could we imagine measuring every particle in a cat, and piecing together all those individual particle wave functions? No, because a cat is comprised of lots of identical particles with exchange energies and so on, so we need to measure multiple-particle wave functions to get the right coherences. How do we do that? Worse, we have to find ways to measure each part of the cat in such a way that it does not mess up other measurements. To avoid completely obliterating the cat with devastating energies, the measurements have to take a finite time, so there will be uncertainty as to exactly what time the measurements applied to. So if we get electron A had spin up and electron B had spin down at some time t, it could really have been spin up at time t+dt. and the other spin down at time t-dt, so how do we know that the measurement on the one electron didn't mess up the other one during that intervening time? I would say it is not even possible in principle to put a cat into a pure state.

Now, this is not just an issue of the number of particles in a cat-- we could imagine sending a beam of particles through a Stern-Gerlach and separating any number of them into pure-state "spin up" and "spin down", and if they were identical particles we could imagine the appropriate Slater determinent to get the full wavefunction. But that's just a beam of particles that are basically independent entities, it's not doing anything, it's not being a cat. And it can't be alive or dead. So the whole crux of the paradox is that we don't think cats can be alive and dead at the same time, but that's expressly because they are complex systems of interacting particles, not a beam of sterile interactions. So the fact that cats are complex enough to be alive or dead is exactly why they cannot be in pure states, and that's just what our intuition says about them.

The situation is even worse for the concept of a superposition state of alive/dead cat. To the universe, a cat is just a collection of particles and fields, there is no need for the universe to decide if a cat is alive or dead or even a cat. That's all going on in the mind of the physicist, it's a result of a certain type of information processing, and to judge if a cat is alive or dead requires coupling to that information processor. So now we are not only trying to specify the state of every particle in a cat, we are trying to also decide if it is a cat, and if it is alive or dead, so we also have to specify the state of every particle in the brain that is making that determination-- or at least include how the brain makes that decision. So now we have couplings to noise modes we are not including in our description of the pure state of the alive/dead cat, just to say that it is indeed a cat in the first place, and if the state we are treating it as will really test out correctly. So we have yet another reason why a cat cannot be in a pure state-- the meaning of "a cat" necessitates that it be in a mixed state, as a substate of the cat+brain that allows us to say that it is a cat. If we try to say the brain is also in a pure state, we need another brain to give that meaning, so we have the Wigner's friend problem. For these reasons, I conclude it is impossible even in principle for a cat to be in a pure state, it would just be wrong quantum mechanics.

As for how that state of affairs gets built up in a sequence of ever more complex systems, I would say the key is when the entanglements between the system and the brain doing the quantum mechanics on that system become important. For a single spin up particle, the way the brain gets entangled with that spin up state doesn't matter, because when you project onto the outcome of experiments on that particle, the brain entanglements project out-- everything going on in the brain that associates with "spin up" is separate from everything going on that would have been associated with "spin down" had that been the outcome. But that's not true when you entangle a brain with a cat-- there could be a lot going on in the brain that connects to either a dead cat or an alive cat, because the entanglements are to the individual particles in the cat, not the whole cat as if it was a single particle. So aliveness is not like spin-- it is not an attribute of a particle, it is an outcome of mental processing that mixes all kinds of different behaviors of the individual particles in the cat. So when you couple the brain to the cat, then project onto just the cat, you always end up with a mixed state, never a pure one-- even in principle.

I would say there is no paradox even if you did have a pure wave function. What is the paradox that occurs in assuming a pure state is known when the box is closed? Is it just the effect you mentioned above - i.e. it cannot be done?

The paradox is that if a cat can be in a pure state, we get a disconnect with our intuition that says the pure state could never involve both an alive and dead cat. But the unitary evolution of a pure state could easily lead to a superposition state in regard to aliveness, and that's what seems impossible to our intuition. We know it is always sufficient to treat a cat as being in a mixed state of alive or dead, so if quantum mechanics says it can be a superposition, we wonder why we never needed to think of it that way. I think the resolution of that paradox, which is the usual way the cat paradox is expressed but I don't see as the real paradox here, is that a cat is never in any kind of pure state, let alone a superposition of dead and alive. But that still leaves the real paradox here-- even if the substate that is the cat is in a mixed state, how do we actualize one or the other if the pure state of the larger closed system that includes us is in a pure state that has both alive and dead cats embedded into it as projections? In other words, the core paradox of quantum mechanics is not how you get mixed states, that's easy, you get mixed states when you couple to macro systems and then project onto substates that yield particular outcomes. The core paradox is how do you collapse the mixed state into a single outcome, not how you collapse the superposition into a mixed state.

There IS quantum mechanics at the mixed state level - Upon closing the box, having measured a mixed state, each pure state of the mixture is propagated forward by e.g. the Schroedinger equation. Each pure state propagates to a another pure state, yielding a fully defined propagated mixed state at a time just before the box is opened, which is then used to calculate the probability of finding the cat dead or alive when the box is opened.

That is all true, but you have to ask, when is the propagation of the pure states the key physics there, and when is the statistical behavior of the mixture what matters. When we derive ideal gas laws, we don't need to propagate all the individual particle wavefunctions-- we know the mixture allows us to average over the detailed behavior, and the relevant physics comes after the averaging, not before it. We can essentially replace all the detailed quantum mechanics with a simple assumption of ergodicity, and pow, we get statistical mechanics and the behaviors of gases. Same for cats, I would say, although Penrose thinks the extent to which they are conscious requires some survival of the quantum realm. I'm not convinced that is true.

Note - when I say "having measured a mixed state", it may be as little as saying "I see a live cat, geiger counter not clicked", and then assigning equal probability to every pure state of the mixture which conforms to this observation, zero to the rest, which is, of course, more than just a measurement.

A lot more, yes. And that's just the problem-- when you say you see those things, all kinds of information processing is going on that says as much about your mind as it says about cat electron wavefunctions. So an "alive cat" is not a set of equally probable cat-electron wavefunctions consistent with aliveness, it is a much richer system that includes your brain, and projections onto the cat are not mixtures of pure-state cats, because there is no such thing as a pure-state cat. A cat is a fundamentally different construct than a wavefunction.

 

[Rap]

Now we can ask the question of whether a cat could ever be in a pure state, even in principle (it obviously could't ever be in practice). Could we imagine measuring every particle in a cat, and piecing together all those individual particle wave functions? No, because a cat is comprised of lots of identical particles with exchange energies and so on, so we need to measure multiple-particle wave functions to get the right coherences. How do we do that? Worse, we have to find ways to measure each part of the cat in such a way that it does not mess up other measurements. To avoid completely obliterating the cat with devastating energies, the measurements have to take a finite time, so there will be uncertainty as to exactly what time the measurements applied to. So if we get electron A had spin up and electron B had spin down at some time t, it could really have been spin up at time t+dt. and the other spin down at time t-dt, so how do we know that the measurement on the one electron didn't mess up the other one during that intervening time? I would say it is not even possible in principle to put a cat into a pure state.

I agree that you could not measure the pure wave function of a cat without destroying the cat, but this would also be the case if the contents of the box were classical particles. We could not measure the position and momentum of every classical particle in a cat without destroying the cat, so I don't see this as a particularly quantum problem. In the classical case, we could say that IF we knew the position and momentum of every particle in the box just after we closed it, and we had a large enough computer, we could calculate with practical certainty the position and momentum of every particle at some time later, just before we open the box, and, with a large enough computer, we could also calculate with practical certainty whether that arrangement corresponded to a live cat or a dead cat, which is what we would observe when we opened the box. Similarly, in the quantum case, we could say that IF we knew the pure wave function of the contents of the box just after we closed it, and we had a large enough computer, we could calculate with practical certainty the wave function at some time later, just before we open the box, and we could also calculate from that wave function the probability that the cat would be seen as alive or dead when we open the box. Of course, in the quantum case, we would have a choice of initial wave functions, but I think we could choose one in which, say, the uncertainties in position and momentum of the center of mass of the box and contents were "microscopic in magnitude". We would also face a similar choice of how to interpret the final wave function, but again, we could choose a set of "eigencats" whose uncertainties were microscopic, i.e. corresponding to the fact that upon opening the box, the "measurement" that occurs is not a destructive collapse of the wave function to a particular eigencat, but rather to a mixture.

So the fact that cats are complex enough to be alive or dead is exactly why they cannot be in pure states, and that's just what our intuition says about them.

This is again the difference between "we cannot measure the microstate of a cat without destroying it" which does not imply "we cannot assume a pure wave function for a cat".

The situation is even worse for the concept of a superposition state of alive/dead cat. To the universe, a cat is just a collection of particles and fields, there is no need for the universe to decide if a cat is alive or dead or even a cat. That's all going on in the mind of the physicist, it's a result of a certain type of information processing, and to judge if a cat is alive or dead requires coupling to that information processor. So now we are not only trying to specify the state of every particle in a cat, we are trying to also decide if it is a cat, and if it is alive or dead, so we also have to specify the state of every particle in the brain that is making that determination-- or at least include how the brain makes that decision. So now we have couplings to noise modes we are not including in our description of the pure state of the alive/dead cat, just to say that it is indeed a cat in the first place, and if the state we are treating it as will really test out correctly. So we have yet another reason why a cat cannot be in a pure state-- the meaning of "a cat" necessitates that it be in a mixed state, as a substate of the cat+brain that allows us to say that it is a cat. If we try to say the brain is also in a pure state, we need another brain to give that meaning, so we have the Wigner's friend problem. For these reasons, I conclude it is impossible even in principle for a cat to be in a pure state, it would just be wrong quantum mechanics.
...
As for how that state of affairs gets built up in a sequence of ever more complex systems, I would say the key is when the entanglements between the system and the brain doing the quantum mechanics on that system become important.

I don't see how the brain of the physicist interacts with the process. The question of whether the arrangement of particles in the classical case or the eigencats in the quantum case correspond to dead or alive cats is a computational problem. The brain of the physicist does not interact before the box is opened, and by that time the calculation is finished. Ok, the step from the final wave function to the prediction of what the scientist will observe upon opening the box is fuzzy in my mind, maybe this is where the crux of the problem is. But my point is that it is not in the steps leading up to that point.

The paradox is that if a cat can be in a pure state, we get a disconnect with our intuition that says the pure state could never involve both an alive and dead cat. But the unitary evolution of a pure state could easily lead to a superposition state in regard to aliveness, and that's what seems impossible to our intuition.

It is not unintuitive if you accept the wave function as an encoding of our (quantum) uncertainties about the situation. In the pure wave function scenario, I see the uncertainty in alive/dead as essentially stemming from the Heisenberg uncertainty. If we measure the position of a particle to a high degree of accuracy, then that wave function will be a superposition of precise momentum states. This does not mean that the particle possesses all these momenta at once (not very intuitive), it simply means we are uncertain about the momentum we would measure if we made a momentum measurement just after the position measurement (quite intuitive).

 

[yoda jedi]

Having read the last few posts in this thread, there is still some confusion about whether the cat is (1) in superposition of dead AND alive or (2) is definitely either alive or dead but not both, while the box is closed and the observer can make no observations of the state of the cat. Now persons taking either position (1) or (2) are making claims about the state of the cat with absolutely no evidence to support their position because by the definition of the thought experiment no observations can be made, only suppositions.

not with nonlinear quantum mechanics, there is a self induced collapse.

 

[Ken G]

Similarly, in the quantum case, we could say that IF we knew the pure wave function of the contents of the box just after we closed it, and we had a large enough computer, we could calculate with practical certainty the wave function at some time later, just before we open the box, and we could also calculate from that wave function the probability that the cat would be seen as alive or dead when we open the box.

We can all agree any real physicist will be forced to treat that cat as a mixed state, but the issue of the paradox is whether or not that is also the reality of the situation, or if there is some deeper pure state there that the physicist is involved in but cannot make use of. I think we both agree that in any correct application of quantum mechanics, the cat by itself is going to have to be treated as a mixed state, as it will always be a substate of a larger system based on the history of how it got there and the impossibility of separating it from that history without destroying it. I agree the same thing would be true of a purely classical cat for different reasons-- owing to the butterfly effect, a deterministic treatment of a classical cat would also have to destroy the cat to achieve the impossibly precise initial conditions needed to predict the future evolution as anything but a (classical) mixed state.

The bottom line is inescapable: a cat is a mixed state, quantum mechanically or classically. What survives of the quantum mechanical cat paradox, though, is if we expand the system to include everything that could possibly influence that (even its whole past light cone if necessary), and treat that as a pure state, where in that pure state is there a dead cat and an alive cat? In other words, the "right" paradox there was never how does a pure state become a mixed state, because the cat is always a mixed state, there's no "becoming" involved. The right paradox is, how does a mixed state get actualized as a single experimental outcome? Classically, we have no issue with that, we just say that the mixed state was a kind of mistake, the reality itself was always one or the other. Such a cavalier interpretation of what reality is does not conform to the postulates of quantum mechanics, unless one takes the Copenhagen approach of treating those postulates as tools rather than as an actual description of reality.

This is again the difference between "we cannot measure the microstate of a cat without destroying it" which does not imply "we cannot assume a pure wave function for a cat".

The reason we cannot assume a pure wavefunction for a cat is that we have no way to establish such a thing. Reality by itself is never going to do it-- quite the opposite, reality is going to couple and entangle that cat all over the show, forcing the cat to live in a subspace of whatever might possibly be construed as a pure wavefunction there. In my view, pure states just don't work that way, we must look to smaller and smaller systems, and closely observed ones, to find the meaning of a pure state-- not look to larger and larger ones, taking us ever farther from the actual demonstrated usefulness of the concept in some vain hope that it is easier to herd a trillion cats than ten of them (and what I'm counting here is modes, not particles-- a huge number of particles all in the same state is just one mode, like a Bose-Einstein condensate).

I don't see how the brain of the physicist interacts with the process. The question of whether the arrangement of particles in the classical case or the eigencats in the quantum case correspond to dead or alive cats is a computational problem.

Not so-- saying that a cat is alive or dead is not something that can be determined from what the electrons are doing, the electrons are just obeying the laws of physics if they are in a cat or in a rock. There has to be a brain somewhere to process that information and make the judgement that the cat is alive or dead. We might use the cat's own brain for that, and "ask the cat" if you will, but you certainly will never get a superposition that way-- if you ask the cat, you'll only get the answer of one cat.

The brain of the physicist does not interact before the box is opened, and by that time the calculation is finished. Ok, the step from the final wave function to the prediction of what the scientist will observe upon opening the box is fuzzy in my mind, maybe this is where the crux of the problem is. But my point is that it is not in the steps leading up to that point.

You're saying there's a concept of a wave function as a function of time, which is later used by the physicist to make a prediction or understand an outcome. I'm saying that wave function has no independent meaning, its whole purpose is to be used by the physicist to make a prediction or understand something. And I'm disputing that there ever was a pure state wavefunction there in the first place, there never was anything there to evolve in the way you describe. Instead, there as a vast mixture of wavefunctions, whose ultimate behavior is both statistical and classical. The classical part appears when we are forced to take a statistical average over the quantum mechanical elements, destroying all the phase coherences that were the hallmark of quantum mechanics in the first place. This does not mean that a quantum mechanical process can't happen in a cat, it just means that a cat cannot be a quantum mechanical pure state.

It is not unintuitive if you accept the wave function as an encoding of our (quantum) uncertainties about the situation. In the pure wave function scenario, I see the uncertainty in alive/dead as essentially stemming from the Heisenberg uncertainty. If we measure the position of a particle to a high degree of accuracy, then that wave function will be a superposition of precise momentum states. This does not mean that the particle possesses all these momenta at once (not very intuitive), it simply means we are uncertain about the momentum we would measure if we made a momentum measurement just after the position measurement (quite intuitive).

I agree with that description of a single particle, I'm saying that it doesn't apply to a cat. It applies to the parts of a cat, but those parts come together in a way that we will always have to statistically average with random phases, because we can never connect them to some "mother" pure state.

 

[Rap]

We can all agree any real physicist will be forced to treat that cat as a mixed state, but the issue of the paradox is whether or not that is also the reality of the situation, or if there is some deeper pure state there that the physicist is involved in but cannot make use of. I think we both agree that in any correct application of quantum mechanics, the cat by itself is going to have to be treated as a mixed state, as it will always be a substate of a larger system based on the history of how it got there and the impossibility of separating it from that history without destroying it. I agree the same thing would be true of a purely classical cat for different reasons-- owing to the butterfly effect, a deterministic treatment of a classical cat would also have to destroy the cat to achieve the impossibly precise initial conditions needed to predict the future evolution as anything but a (classical) mixed state.

This is a point that has to be resolved first, because the rest of your argument, if I understand correctly, is based on the idea that we cannot consider a cat in a pure state when the box is closed. I agree that a wave function cannot be assigned to a given box and contents without destroying it, just as in the classical case you cannot measure the momentum and position of every particle in it without destroying it.

Just to be clear, do you agree that we could postulate a pure wave function representing a hypothetical cat etc., just as we can postulate a set of classical particles with specific position and momenta corresponding to a hypothetical cat, etc.? I understand that this will not apply to a real situation. Also that, given this situation, we could propagate forward with practical certainty, given a large enough computer. Also that, given sufficient knowledge of what microstates constitute a cat, dead or alive, a calculation could be made as to what the final state, classical or quantum, represents, classical being a certainty as to what the physicist observes when opening the box, quantum being not so much.

Note that the butterfly effect says that your errors will increase exponentially as you propagate, not that they will become infinite, so this is not a valid objection to either case. The number of significant digits you must carry in your computer will increase exponentially as the time interval you consider increases linearly. A finite time interval will yield a finite number of significant digits needed, thus a finite computer. Sure it might be 10^10000 times the size of the universe, but this is a practical limitation, not a limitation in principle.

 

[Ken G]

Just to be clear, do you agree that we could postulate a pure wave function representing a hypothetical cat etc., just as we can postulate a set of classical particles with specific position and momenta corresponding to a hypothetical cat, etc.?

It's not clear that is possible, but it doesn't seem to matter terribly, so I'll work with the assumption that in principle there is such a thing as a "pure state cat wavefunction", it just never appears in nature. The reason I am skeptical there is any such thing is that the process of making a cat is so fundamentally connected with classical phenomena, I'm not sure it would even be possible to identify a theoretical pure-state cat wavefunction. But I don't think it will matter if this is possible or not, what will matter is that it never happens in nature. Note also that we must include in the pure state enough surrounding air to keep the cat alive as long as needed.

Also that, given this situation, we could propagate forward with practical certainty, given a large enough computer. Also that, given sufficient knowledge of what microstates constitute a cat, dead or alive, a calculation could be made as to what the final state, classical or quantum, represents, classical being a certainty as to what the physicist observes when opening the box, quantum being not so much.

Yes, if a pure state is possible, we can imagine isolating it, and it will evolve deterministically into a new pure state, according to the correspondence principle.

Note that the butterfly effect says that your errors will increase exponentially as you propagate, not that they will become infinite, so this is not a valid objection to either case.

It just means that in the classical case, you'd again have to disintegrate the cat to measure its parts closely enough to predict its future for any significant amount of time, just as you would in the quantum case. The limitation is again that it takes energy to make precise measurements, as per entropic requirements even in classical physics.

 

[Rap]

It's not clear that is possible, but it doesn't seem to matter terribly, so I'll work with the assumption that in principle there is such a thing as a "pure state cat wavefunction", it just never appears in nature.

Ok, I just wanted to make sure I wasn't missing some point you were making. Next, can we describe a real cat in a real box as a mixture of a set of these hypothetical wave functions, each hypothetical wave function being consistent with whatever rather non-destructive measurements we make when we close the box, each hypothetical wave function being assigned equal weight in the mixture? Would that be an acceptable way of characterizing the situation when the box was closed (again, in principle only)?

 

[Ken G]

Ok, I just wanted to make sure I wasn't missing some point you were making. Next, can we describe a real cat in a real box as a mixture of a set of these hypothetical wave functions, each hypothetical wave function being consistent with whatever rather non-destructive measurements we make when we close the box, each hypothetical wave function being assigned equal weight in the mixture? Would that be an acceptable way of characterizing the situation when the box was closed (again, in principle only)?

If such a pure-state cat is possible, there's no need for us to even mix them together in the original state, we may as well just take the pure cat. No real cat can be described as a mixture of pure-state cats, the mixing happens throughout the cat, based on the history of all the cat subsystems, it's not just a global mixture of whole cats. But if it is possible in principle to have a pure state cat, then we may as well do the gedanken with a pure state alive cat in the box with a pure state undecayed nucleus and a pure state hypodermic needle attached to the undecayed nucleus, and go from there.

 

[Rap]

If such a pure-state cat is possible, there's no need for us to even mix them together in the original state, we may as well just take the pure cat. No real cat can be described as a mixture of pure-state cats, the mixing happens throughout the cat, based on the history of all the cat subsystems, it's not just a global mixture of whole cats.

It is possible in principle to postulate a pure state cat, but it is not possible to measure the pure state of a cat without destroying it. This is analogous to classical mechanics where it is possible to postulate the microstate of a gas in an isolated container at equilibrium, giving all the positions and momenta of the particles in the isolated container, but it is not possible to measure the actual microstate without destroying it. Nevertheless, we can assign equal probabilities to every microstate which yields the same temperature and pressure that we measure before isolating the container (i.e. closing the box). We can then propagate forward and prove that the temperature and pressure will very likely be practically the same when we check it at some time later. Statistical mechanics is just a way of carrying out that propagation approximately without having to track 10^23 particle positions and momenta of an enormous number of possible microstates.

But if it is possible in principle to have a pure state cat, then we may as well do the gedanken with a pure state alive cat in the box with a pure state undecayed nucleus and a pure state hypodermic needle attached to the undecayed nucleus, and go from there.

I am fine with such a gedanken experiment, the pure state at some time later consists of a superposition of live and dead cat wave functions, and these simply allow me to calculate the probability of finding a live or dead cat when the box is opened. To inquire into what is "really" going on inside the box before it is opened is not something that can be measured, therefore it is not a proper scientific question.

But then there is always the nagging problem that you pointed out, that it cannot be used in a real situation. When someone says it can never happen in practice, and if I'm such a Copenhagen sympathizer, how do I respond?

Its possible to have a hypothetical pure state cat, but one cannot take a real situation and assign a pure state to it, because, as you say, it will be destroyed by the measurement. I'm just trying to use these hypothetical pure states to analyze a real situation, analogous to the way hypothetical microstates could be used in classical mechanics and are used in statistical mechanics to analyse macroscopic (i.e. thermodynamic, measurable) states.

 

[Ken G]

I am fine with such a gedanken experiment, the pure state at some time later consists of a superposition of live and dead cat wave functions, and these simply allow me to calculate the probability of finding a live or dead cat when the box is opened.

But that's what I'm saying isn't true-- even if we start with pure states for each component of the system, when we couple them, the only pure state is now a combined system. The cat is now a substate of that system, and substates don't evolve according to the Shroedinger equation, so they don't evolve unitarily and they don't become superposition states. There is really no such thing as the state of a part of a system, but we as physicists can make correct predictions by using the concept of a mixed state to treat such substates, or in some special circumstances, we have enough information to treat a substate as a pure or superposition state. That ability is quickly lost for the cat in the box, even if it starts out in an impossible-to-know pure state.

In classical language, a mixed state is "one or the other we just don't know which, but nature does", but in quantum mechanical language, that is not actually the state of the cat, it is only the state we are choosing to use to treat the cat because it is going to make correct predictions. If one sympathizes with Bohr (as I do), one says that there is no such thing as "the real state" of anything in physics, there is only how we the physicist are choosing to treat the situation, so for me a mixed state is a fine treatment of that cat-- the cat is either alive or dead we just don't know which. If one takes a literal belief in quantum mechanical wavefunctions, then the cat simply has no real state at all, it is part of a larger system, period.

So my point is, whether we start with a putative (but impossible) pure-state cat, or if we adopt a mixture of pure states with some statistical distribution, doesn't matter for the cat paradox-- because correct quantum mechanics says that once we couple that cat to the mechanism that can kill it, there is no longer any such thing as the state of the cat in quantum mechanics. There is only a projection of the full state onto the cat degree of freedom, but that isn't a quantum mechanical state, it is a classical treatment of a quantum mechanical state. It makes no difference to the quantum mechanics if we now assert that the cat "really is" alive or dead and we have no way of knowing which, or if we assert that we have chosen to treat it that way in our mathematics-- the correct quantum mechanics is completely moot on the point, there is no cat-state wavefunction so there is no superposition of alive or dead.

So that would seem like the end of the cat paradox, but I'm saying that's just the end of the wrong cat paradox, the one that uses wrong quantum mechanics. The right cat paradox survives-- it is the one that says whether we treat the cat as already alive or dead and we just don't know which, or if we say that is simply how we are conceptualizing the cat, there should be a pure-state wavefunction for the whole system. Where in that pure state is there information about what we will observe when we open the box? It is nothing but statistical information-- that's what Einstein objected to, god is rolling dice, but there's no dice anywhere in the physical setup.

That is what I think forces us to the Copenhagen interpretation: our approach to doing physics is necessarily limited, we cannot recover what reality is actually doing there, it simply doesn't make sense. We don't have a fundamental theory, we have an effective theory, because we do physics by coupling quantum mechanical systems to classical processors, and something is lost in translation. And I would further add that we should have always expected that-- there is almost no more purely mystical idea than the idea that we create fundamental theories rather than effective ones.

 

[Rap]

But that's what I'm saying isn't true-- even if we start with pure states for each component of the system, when we couple them, the only pure state is now a combined system. The cat is now a substate of that system, and substates don't evolve according to the Shroedinger equation, so they don't evolve unitarily and they don't become superposition states. There is really no such thing as the state of a part of a system, but we as physicists can make correct predictions by using the concept of a mixed state to treat such substates, or in some special circumstances, we have enough information to treat a substate as a pure or superposition state. That ability is quickly lost for the cat in the box, even if it starts out in an impossible-to-know pure state.

Hmm - well I have been sloppy about the distinction between the cat alone and the cat plus the box plus all the other stuff. I never meant to imply that we were dealing with anything but the pure wave function of the box and all its contents (i.e, the "system"). But I don't understand why propagating the system wave function forward results in a wave function in which the cat is ill-defined. I agree, it won't be perfectly defined, there are miniscule probabilities that the cat will evaporate, or something, but the overwhelming probability is that there will be a cat in there somewhere.

I wonder if you would be willing to consider a simpler system which may help me to more fully understand what you are saying. Consider a box with constant volume containing a stoichiometric mixture of hydrogen and oxygen at a particular temperature and pressure, and a tiny piece of radioactive material which emits, I don't know, an alpha particle every hour on average. Suppose this alpha particle greatly increases the probability that the mixture will explode and produce water vapor (maybe not true, but that's not the point). The gas is like the cat, alive when you have the stoichiometric mixture, dead when you have water vapor. The only other parts of the system are the box and the bit of radioactive material. We could come up with any number of impossible-to-know pure states that describe the system. I will modify your quote to read:

"There is really no such thing as the state of a part of a system, but we as physicists can make correct predictions by using the concept of a mixed state to treat such substates, or in some special circumstances, we have enough information to treat a substate as a pure or superposition state. That ability is quickly lost for the gas in the box, even if it starts out in an impossible-to-know pure state."

Is this still a valid statement?

The next question is - how would you treat this simple system quantum mechanically? Can we speak of a superposition of exploded/unexploded after a certain time?

I would say that when we close the box we would describe the system as a mixed state consisting of every pure wave function corresponding to a microstate of the unexploded gas at that particular pressure and temperature along with the box and the un-emitted radioactive material, multiplied by a probability that is equal for each pure wave function. (Unlike the classical case we have a choice of wave functions corresponding to the momentum/position tradeoff for a single particle). This mixed state can be propagated forward. I understand that treating the gas alone as a separate entity which evolves unitarily is not correct, but I am having trouble imagining the implications of this. It seems to me that practically every interpretation of the evolved wave function will involve exploded or unexploded gas and a tiny piece of radioactive material.

That is what I think forces us to the Copenhagen interpretation: our approach to doing physics is necessarily limited, we cannot recover what reality is actually doing there, it simply doesn't make sense. We don't have a fundamental theory, we have an effective theory, because we do physics by coupling quantum mechanical systems to classical processors, and something is lost in translation. And I would further add that we should have always expected that-- there is almost no more purely mystical idea than the idea that we create fundamental theories rather than effective ones.

On this, I totally agree.

 

[Ken G]

"There is really no such thing as the state of a part of a system, but we as physicists can make correct predictions by using the concept of a mixed state to treat such substates, or in some special circumstances, we have enough information to treat a substate as a pure or superposition state. That ability is quickly lost for the gas in the box, even if it starts out in an impossible-to-know pure state."

Is this still a valid statement?

Yes, and although I like your simplification, I would say let's go whole hog and simplify it even more. Let's take a two-slit experiment, where we put a left circular polarizer in one slit, and a right circular polarizer in the other. Then let's send photons that are linearly polarized (so in a superposition of left and right circular polarization) through the slits, and get a two-slit diffraction pattern. Now let's do it again, but first down-convert the photons into a photon pair, both linearly polarized like the original (I think that's what down conversion would do, but if not we entangle them in some other way). So all we did is take a single-particle superposition state (like our imagined pure state cat) and convert it to a two-particle superposition state (with highly entangled correlations embedded in it, like the cat plus the device that can kill it). Will the individual particle substates of that entangled pair act like a superposition state? In other words, if we send one of the photons in each pair through the slits, will it make a two-slit diffraction pattern?

As I understand these things, the answer is no-- we will no longer get a two-slit pattern after down-converting those photons, because now it is the whole two-photon system that is in a superposition, but the substate projections of such a tightly entangled pair encodes "which way" information, so will not yield a two-slit pattern any more-- instead, the single-piarticle treatment will need to be that of a mixed state of the two polarizations, to make the correct predictions. In the analogy, the same holds for the cat.

Now, you are probably thinking about quantum erasure, so you know that we can get a two-slit pattern in a particularly ingenious way out of my apparatus, but it requires that we do two more things than what I said so far-- we have to sort the pattern into two batches, correlated with two possible outcomes on the other members of the pairs, and we have to use outcomes on the other members that "erases" which-way information. But we still don't get a two-slit pattern for the whole set of original photons, only the two sorted batches, so we cannot say the first photons were in a superposition state. However, they weren't strictly in a mixed state either-- there is additional information there we could imagine accessing such that the mixed state isn't officially correct either, it's just the treatment we choose when we pretend we have a one-particle state rather than a whole system.

And above all, there's no way to correlate cat outcomes with apparatus outcomes, such that we could sort the cat outcomes into two sets that behaved like superpositions of alive and dead cats in each set, because we can't do anything clever with the apparatus to erase "alive/dead" information, since the apparatus is classical too. There would need to be a way to kill a cat with a pure-state wavefunction, rather than a classical system, and I don't think that would ever be possible-- that's one manifestation of the "Heisenberg gap" right there.

On this, I totally agree.

Then we are fellow Copenhagen sympathizers, there seem to be fewer of us all the time!

 

[Hurkyl]

In classical language, a mixed state is "one or the other we just don't know which, but nature does"

I should point out that, classically, this is ontological point of view is merely a simplification, rather than something demanded by the mathematics of classical statistical mechanics.

Classically, mixtures remains stable; they can be neither created not destroyed. Furthermore, the components do not influence each other in any way.

In other words, you cannot design an experiment that can distinguish between "nature does know which" and "nature doesn't know which".

Occam's razor applies, then, for the theory and philosophy of classical mechanics. However, Einstein's razor applies in the quantum case: "Make things as simple as possible, but not simpler."

There is only a projection of the full state onto the cat degree of freedom, but that isn't a quantum mechanical state, it is a classical treatment of a quantum mechanical state.

This doesn't really make sense, unless by "quantum mechanical state" you really mean to restrict your attention solely to the states that are pure.

 

[yuiop]

Yes, and although I like your simplification, I would say let's go whole hog and simplify it even more. Let's take a two-slit experiment, where we put a left circular polarizer in one slit, and a right circular polarizer in the other. Then let's send photons that are linearly polarized (so in a superposition of left and right circular polarization) through the slits, and get a two-slit diffraction pattern. Now let's do it again, but first down-convert the photons into a photon pair,...

If you place left and right circular polarisers in front of the left and right slits respectively, you will NOT get a two slit diffraction pattern. See this counterexample https://www.physicsforums.com/showpost.php?p=3118570&postcount=63 to a comment by DrC. The two polarisers give potential "which way" information which destroys the interference pattern.

You only recover an interference pattern in the case of entangled photons after you carry out coincidence counting or in the case of "un-entangled" photons by placing a linear polariser in the path before or after the two circular polarisers.

P.S. Dr.C has not yet responded to the counterexample in that post.