About nature of superposition of states

For quantum mechanics, a certain property of a subatomic particle, e.g. the spin of an electron, which can be either up or down, is a "superposition of states," and one of the two conditions, e.g. the fact that it has spin up or down, doesn't manifest itself until the situation is experimentally observed.
In fact, from what I understand, it does not exist until you observe it. So what I would like to understand is a very precise thing: does the electron have a real spin or does it only have it when it is observed?

Does the spin of an electron have an a priori value (unknown to me until I observe it), or does it have a superposed state? As Einstein said to someone, "Do you really believe that the Moon only exists when you look at it?"

I understand that the observation itself changes the state of the electron, but could this mean that its spin goes from up to down as a result of the observation, or vice versa, or does it mean that it just doesn't have a definite spin?

The fact is controversial to me, but if reality were only true when observed there would be paradoxical, or even physically impossible, consequences.

HighPhy said:

does the electron have a real spin

Unentangled electron has a definite spin in some direction. Sometimes this direction is known, other times it is not known.

HighPhy said:

So what I would like to understand is a very precise thing: does the electron have a real spin or does it only have it when it is observed?

Does the spin of an electron have an a priori value (unknown to me until I observe it), or does it have a superposed state?

The electron has a superposed state as spin. The observation of it is like a breaking of a symmetry, and forces it into one of the possible observable states (at least approximatively).

HighPhy said:

Does the spin of an electron have an a priori value (unknown to me until I observe it), or does it have a superposed state? As Einstein said to someone, "Do you really believe that the Moon only exists when you look at it?"

The Moon is not an electron. Moreover, having classical properties is not a necessary condition for existence. If an object has QM properties it has just as much existence as something with classical properties.

Hill said:

Unentangled electron has a definite spin in some direction.

One has to be very careful with such statements.

Mathematically, the Hilbert space of a single spin-1/2 (or "qubit"), which is the Hilbert space of the spin degree of freedom of a single unentangled electron, is such that every state in it corresponds to a point on a 2-sphere called the "Bloch sphere", and each such point can be thought of as a vector pointing from the origin (the center of the sphere) to that point, which in turn can be thought of as a "spin axis" pointing in a definite direction (even if we don't know which direction it is, because we haven't measured the spin and we don't know how the electron was prepared).

However, this only works for a single spin-1/2. It does not work for multiple spin-1/2 particles, and it does not work for single particles with spins greater than 1/2. So it is not a good idea to base any general interpretation of what QM is saying on this particular fact. It is better to be cautious and say that even in the case of a single qubit, the spin is undetermined until it is measured.

PeterDonis said:

One has to be very careful with such statements.

Mathematically, the Hilbert space of a single spin-1/2 (or "qubit"), which is the Hilbert space of the spin degree of freedom of a single unentangled electron, is such that every state in it corresponds to a point on a 2-sphere called the "Bloch sphere", and each such point can be thought of as a vector pointing from the origin (the center of the sphere) to that point, which in turn can be thought of as a "spin axis" pointing in a definite direction (even if we don't know which direction it is, because we haven't measured the spin and we don't know how the electron was prepared).

However, this only works for a single spin-1/2. It does not work for multiple spin-1/2 particles, and it does not work for single particles with spins greater than 1/2. So it is not a good idea to base any general interpretation of what QM is saying on this particular fact. It is better to be cautious and say that even in the case of a single qubit, the spin is undetermined until it is measured.

Thank you. Noted.

gentzen said:

The electron has a superposed state as spin.

Could you dig deeper? Even providing bibliographic sources where the question is addressed in a linear way, or as you wish...

HighPhy said:

The fact is controversial to me, but if reality were only true when observed there would be paradoxical, or even physically impossible, consequences.

If you want to learn QM, you have to free your mind. The universe is classical neither in Newtonian spacetime, nor in the dynamic properties of particles. QM has been around for about 100 years and, contrary to your belief, physics has neither stopped nor imploded. In fact, much of modern electronics is based on non-classical electron tunneling.

Moreover, QM was initially forced on the scientific community by experimental results. You need to take up your grievances with nature. The irony is that if your wish were granted and you were transported to a classical universe, there would be no chemistry, no stars and no life! You have the QM nature of the universe to thank for your own existence as a complex system!

HighPhy said:

For quantum mechanics, a certain property of a subatomic particle, e.g. the spin of an electron, which can be either up or down, is a "superposition of states," and one of the two conditions, e.g. the fact that it has spin up or down, doesn't manifest itself until the situation is experimentally observed.
In fact, from what I understand, it does not exist until you observe it. So what I would like to understand is a very precise thing: does the electron have a real spin or does it only have it when it is observed?
...

I understand that the observation itself changes the state of the electron, but could this mean that its spin goes from up to down as a result of the observation, or vice versa, or does it mean that it just doesn't have a definite spin?

The fact is controversial to me, but if reality were only true when observed there would be paradoxical, or even physically impossible, consequences.

Without contradicting any of the above answers:

A Mach Zehnder interferometer (single photon at a time version) works as it does (all photons going to the same single output port of a beam splitter) precisely because it is in a superposition of states. In your parlance: the "up" outcome (going one path) and the "down" outcome (going the other path) interfere constructively/destructively where they recombine. You might reasonably conclude that both exist, but each only 50% exist. (No one knows exactly, there are a number of ways to think of it.) If that were not the case, the photons would leave the two output ports of the beam splitter going equally (and randomly) out the two ports (rather than only one).

In actual experiments, you cannot say which path the photon took. So there is no "observation" - only a "paradoxical" result. That being, the photon takes both paths in some sense - and interferes with only itself.

PeroK said:

The Moon is not an electron. Moreover, having classical properties is not a necessary condition for existence. If an object has QM properties it has just as much existence as something with classical properties.

Just one final point. If an electron didn't exist until you looked for it, why would you find an electron? And, not something else? Likewise with the Moon. If the Moon does not exist and then you look at a place in space, why would you find the Moon and not a comet or an asteroid or an electron? What is it about the act of measuring that conjures something that did not exist previously exist?

What QM says is that the electron's dynamical properties do not have well-defined values until you measure them. The electron exists and its dynamic properties are defined by a wave-function. But, it's only when you measure the dynamic properties that they take a well-defined value.

It's even more clear with the Moon. In principle, none of the Moon's atoms has a precisely well-defined position until you measure them. But, the effect of that microscopic uncertainty on an object the size of the Moon is immeasurably small. The probability of finding the Moon precisely where we expect it is so close to certainty that emphasising the minuscule uncertainty of where we might find it is misguided, IMO. QM is compatible with macroscopic objects having apparently well-defined position and momentum simultaneously and moving deterministically along a classical trajectory.

If you apply the uncertainty principle to the Moon then you can see how irrelevant it is.

Just to sharpen up some of the statements. Every electron has the same spin angular momentum: ##\frac{\sqrt{3}\hbar}{2}##. This is true whether the electron is measured or not. When measuring the component of spin along a particular direction, I can only measure ##\pm \frac{\hbar}{2}##, with some probability of one or the other.

So a statement like "measuring the direction of the spin" does not have the same meaning as it does classically, and even if one has a pure state in the spin-up state, that does not imply it is in a pure state in the other two orthogonal directions.

HighPhy said:

Does the spin of an electron have an a priori value (unknown to me until I observe it), or does it have a superposed state?

As @Vanadium 50 has pointed above, the magnitude of the spin is definite, it is the direction of the spin (up or down on a given axis) that is in general a superposition. There are statistical differences between “neither up nor down but a superposition of up/down” and “either up or down, but I won’t know which until I look”, these can be tested experimentally, the experiments have been done, and the they have decidedly settled the question: no theory that says there is a definite value can be consistent with the experimental results. (For more on this you can Google for “Bell’s theorem”or read some of our many threads on it.) However…

As Einstein said to someone, "Do you really believe that the Moon only exists when you look at it?"

Although subatomic particles do not have definite attributes unless we look, and although the moon is made up these particles, large collections of these particles (like the moon, or Schrodinger’s cat in a box, or just about any classical object) do have definite attributes and exist even when we aren’t looking. This is because of a phenomenon called “quantum decoherence “, which was incompletely understood when the moon question was first posed. David Lindley’s book “Where did the weirdness go?” is a good layman-friendly explanation.

HighPhy said:

Could you dig deeper?

How much deeper? Others in this thread already digged somewhat deeper, for example:

Nugatory said:

it is the direction of the spin (up or down on a given axis) that is in general a superposition. There are statistical differences between “neither up nor down but a superposition of up/down” and “either up or down, but I won’t know which until I look”, these can be tested experimentally, the experiments have been done, and the they have decidedly settled the question: no theory that says there is a definite value can be consistent with the experimental results.

And in a certain sense, already the very first answer in this thread digged deeper:

Hill said:

Unentangled electron has a definite spin in some direction. Sometimes this direction is known, other times it is not known.

I skipped over the "unentangled" part (i.e. my answer assumes that the electron is in a pure state, for mixed states the situation is "murkier"). On the other hand, Hill skipped over other details when he talked of "direction", for which he got criticised immediately. Purely real valued superpositions of up- and down-spin correspond to a "direction", which you could "theoretically" measure by a Stern-Gerlach type experiment (the fact that a single electron is negatively charged makes the experiment somewhat "unpractical"). General complex valued superpositions of up- and down-spin lead to the situation described in PeterDonis' critique of Hill's answer. But it is worse than merely details, because the ability to measure those more general states implies the ability to experimentally distinguish between i and -i. Which is actually possible, at least in the case of light, where there are materials that react differently to circularly polarized light than to anticircularly polarized light.

HighPhy said:

Even providing bibliographic sources where the question is addressed in a linear way, or as you wish...

Scott Aaronson gives a simplified high-level explanation in his "comment on a much-discussed new preprint by Matthew Pusey, Jonathan Barrett, and Terry Rudolph (henceforth PBR)" from 2011 under the headline The quantum state cannot be interpreted as something other than a quantum state:

Scott Aaronson said:

In quantum mechanics, mixed states—the most general type of state—have exactly the same observer-relative property. That isn’t surprising, since mixed states include classical probability distributions as a special case. As I understand it, it’s this property of mixed states, more than anything else, that’s encouraged many people (especially in and around the Perimeter Institute) to chant slogans like “quantum states are states of knowledge, not states of nature.”

By contrast, pure states—states with perfect quantum coherence—seem intuitively much more “objective.” Concretely, suppose I describe a physical system using a pure state |ψ>, and you describe the same system using a different pure state |φ>≠|ψ>. Then it seems obvious that at least one of us has to be flat-out wrong, our confidence misplaced! In other words, at least one of us should’ve assigned a mixed state rather than a pure state. The PBR result basically formalizes and confirms that intuition.

In the special case that |ψ> and |φ> are orthogonal, the conclusion is obvious: we can just measure the system in a basis containing |ψ> and |φ>. If we see outcome |ψ> then you’re “unmasked as irrational”, while if we see outcome |φ>, then I’m unmasked as irrational.

But in the end, I have no idea of the level of your existing knowledge. Do you know the difference between pure and mixed states? Do you see why assuming a pure state of the electron is nearly the same thing as assuming an unentangled electron (but still a slightly stronger assumption)? Do you know what is meant by a Stern-Gerlach type experiment? Or what is anticircular polarization, and what it has to do with distinguishing between i and -i?

gentzen said:

Do you know the difference between pure and mixed states?

Yes. It could be said that it has a certain connection with "Schroedinger's cat", couldn't it?

gentzen said:

Do you see why assuming a pure state of the electron is nearly the same thing as assuming an unentangled electron (but still a slightly stronger assumption)?

Can you expand on this? Isn't entanglement usually defined only for pure states? What I know is that pure states can be entangled... So I guess I have no knowledge about what you are saying.

gentzen said:

Do you know what is meant by a Stern-Gerlach type experiment? Or what is anticircular polarization, and what it has to do with distinguishing between i and -i?

These are arguments I have heard about but never studied.

gentzen said:

Purely real valued superpositions of up- and down-spin correspond to a "direction"

So do complex superpositions of up and down spin. For a spin-1/2, as I said, every spin state corresponds to a "direction" via the Bloch sphere representation. But this only works for a single spin-1/2 degree of freedom. It doesn't work for multiple spin-1/2s or for higher spins.

gentzen said:

the ability to measure those more general states implies the ability to experimentally distinguish between i and -i.

Not at all. Measurement results are still real valued; all of the operators involved are Hermitian. The fact that complex numbers happen to appear in our mathematical representation of states does not mean they must appear in measurement results.

gentzen said:

assuming a pure state of the electron is nearly the same thing as assuming an unentangled electron (but still a slightly stronger assumption)?

I'm not sure what you mean by this or how you are getting it from Aaronson's article.

HighPhy said:

Do you know the difference between pure and mixed states?

Yes. It could be said that it has a certain connection with "Schroedinger's cat", couldn't it?

Kinda sorta yes... But that response leaves me a bit skeptical that you do in fact know the difference.

There's a fairly approachable overview: https://web.pa.msu.edu/people/mmoore/Lect34_DensityOperator.pdf
Maybe too much for an I level thread is https://pages.uoregon.edu/svanenk/solutions/Mixed_states.pdf

If you find either of those to be more than you are up for, Giancarlo Ghiardi's book "Sneaking a look at God's cards" is way more user-friendly and will give you enough background to be able to at least follow the discussion.

gentzen said:

assuming a pure state of the electron is nearly the same thing as assuming an unentangled electron (but still a slightly stronger assumption)?

PeterDonis said:

I'm not sure what you mean by this or how you are getting it from Aaronson's article.

If the electron is entangled with another particle, then the state of the electron alone can no longer be described by a pure state, but you need its reduced density matrix. (Let me ignore the question of whether the entanglement between the electron's momentum and spin state arising in a Stern-Gerlach magnet also forces a reduced density matrix corresponding to a mixed state. My guess is that it does.)
But it is not exactly the same thing, because the electron could also be in a mixed state for other reasons. For example, it could have been prepared in a thermal state.

gentzen said:

If the electron is entangled with another particle, then the state of the electron alone can no longer be described by a pure state

Ah, I see. The entangled state itself is pure, but it's not a pure state of the single electron by itself.

OK. I want to see if I am clear about the physical situation.

Instead of there being a vector that identifies the spin of the electron, there is a probability distribution that describes the probability of each outcome among all possible outcomes.

When you observe the spin, this probability distribution collapses to a defined state, and then your measurement changes that probability distribution. Depending on the case, it can collapse it to a very simple one - probability 1 for a certain value and 0 for all others.

That its spin changes from up to down as a result of observation is not possible. If you have already identified that the spin is up, and therefore the probability distribution was already up with 100%, the measurement will give result "up", and the electron does not change state.

Is my description correct?

HighPhy said:

Instead of there being a vector that identifies the spin of the electron, there is a probability distribution that describes the probability of each outcome among all possible outcomes.

Yes.

HighPhy said:

When you observe the spin, this probability distribution collapses to a defined state,

If we restrict our attention to pure states, then the spin always has a defined state. A defined state is different from defined measurement outcomes.

HighPhy said:

and then your measurement changes that probability distribution. Depending on the case, it can collapse it to a very simple one - probability 1 for a certain value and 0 for all others.

Spin is a 3D vector quantity. In QM, only spin about one axis can be defined - spin about the other two axes remains undefined. If we measure about the z-axis, we get either ##\pm \frac \hbar 2## and the state collapses to z-spin-up or z-spin-down. Subsequent measurements of a free particle will always give the same outcome. But, measurements about the x or y-axis will give ##\pm \frac \hbar 2## with equal probability.

Moreover, if the particle is in a magnetic field, then the state will naturally evolve from the initial state or z-spin-up or z-spin-down. Look up Larmor Precession.

HighPhy said:

That its spin changes from up to down as a result of observation is not possible. If you have already identified that the spin is up, and therefore the probability distribution was already up with 100%, the measurement will give result "up", and the electron does not change state.

In principle, yes, if you keep measuring about the same axis, you get the same outcome.

HighPhy said:

Is my description correct?

More or less, but this is why we develop a mathematical formalism. You could ask a thousand different hypothetical questions and piece together a thousand specific answers. The ultimate answer is to learn the formalism of spin states. Then you can answer any question for yourself.

gentzen said:

Purely real valued superpositions of up- and down-spin correspond to a "direction"

PeterDonis said:

So do complex superpositions of up and down spin. For a spin-1/2, as I said, every spin state corresponds to a "direction" via the Bloch sphere representation. But this only works for a single spin-1/2 degree of freedom. It doesn't work for multiple spin-1/2s or for higher spins.

Well, I was thinking mostly in terms of optics and polarization. A more correct translation of that situation to an electron is that the spin states perpendicular to the direction of propagation are much easier to measure directly (by Stern-Gerlach type experiments) than the ones parallel to the direction of propagation. (I didn't answer earlier, because I first wanted to check the details for SG and electron spin in those QM books where I own dead-tree editions. Which I did now. In Griffiths, the electron propagates in y-direction, so at least I was lucky enough that my statement was not completely wrong for the setup from that specific book.)

gentzen said:

the ability to measure those more general states implies the ability to experimentally distinguish between i and -i.

PeterDonis said:

Not at all. Measurement results are still real valued; all of the operators involved are Hermitian. The fact that complex numbers happen to appear in our mathematical representation of states does not mean they must appear in measurement results.

Because i occurs in the nonrelativistic Schrödinger equation, there is a real physical difference between i and -i. But at least in optics, they still correspond to the direction that is more tricky to measure. I once wrote/fixed the polarization code of a lithography simulator, and strugged to build a simple test case for circular vs anticircular polarization. When a Maxwell-Solver was used to compute the mask transmission, then differences between the two were visible. But for the faster Kirchhoff approximation to the mask transmission, I initially struggled. I now tried to remember the trick. Once I remembered it again, it was obvious in hindsight: I had only used real valued mask transmissions, because most masks in actual use are designed like that. The trick was to also use complex valued mask transmissions, like one corresponding to a quarter wavelength phase shift.

My optics analogies were wrong, but the distinctions they suggested still remain somewhat true for electrons: Even so it seems easy to change the direction of propagation of a "particle" from y-direction to x-direction, it is only "theoretically easy" to do so without changing the spin in case the "particle" is not electrically neutral. But in that case, the Stern-Gerlach type experiment itself becomes difficult.

Let me be clear that my optical analogies had been more wrong than I was aware of. And because they were wrong, my post that you corrected was certainly confusing, both for experts and novices.

gentzen said:

Even so it seems easy to change the direction of propagation of a "particle" from y-direction to x-direction, it is only "theoretically easy" to do so without changing the spin in case the "particle" is not electrically neutral

But maybe one could use Lamor precession to rotate the spin of the "particle" instead of the direction of propagation. At least it seems possible "theoretically".

HighPhy said:

Instead of there being a vector that identifies the spin of the electron, there is a probability distribution that describes the probability of each outcome among all possible outcomes.

For a spin-1/2, these are the same thing: the spin state is mathematically equivalent to a vector describing the spin axis pointing in a specific direction.

Also, for a given spin measurement there are only two possible outcomes. If you measure the spin in the same direction as the vector describing the spin axis, only one of those two outcomes will be obtained, with 100% probability. Otherwise there will be a nonzero probability for both outcomes.

HighPhy said:

When you observe the spin, this probability distribution collapses to a defined state, and then your measurement changes that probability distribution.

As long as you are aware that "collapse" is just a mathematical process (since we are talking about basic QM without adopting any particular interpretation), yes.

HighPhy said:

Depending on the case, it can collapse it to a very simple one - probability 1 for a certain value and 0 for all others.

For spin measurements, this will always be the case, since spin is a discrete observable.

PeterDonis said:

For a spin-1/2, these are the same thing: the spin state is mathematically equivalent to a vector describing the spin axis pointing in a specific direction.

That's something of a mathematical coincidence - that the set of two dimensional complex unit vectors can be mapped to a 2D sphere in ##\mathbb R^3##. The vector in this case does not represent spin (angular momentum) about that axis. It only means that we know the component of spin about that axis. The other components of the spin are undetermined.

PeroK said:

The vector in this case does not represent spin (angular momentum) about that axis.

It does not represent classical spin about that axis, yes. But it is (for that particular case only, of a single spin-1/2 particle) a description of the spin state in terms of a vector in ordinary 3-space, which has a concrete physical meaning as the direction in which you can measure the spin and get a 100% determined result.

HighPhy said:

Yes. It could be said that it has a certain connection with "Schroedinger's cat", couldn't it?

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

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

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

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

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

HighPhy said:

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

The important point is that an electron is not a cat. And "alive" and "dead" are not fundamental quantum properties like spin. In the sense that, within reason, simply observing a cat doesn't cause the cat to assume a state of alive and dead.

In any case, there are plenty of threads dealing with this cat.

If you want to learn QM, then you have to forget the cat and learn the mathematical fomalism of QM as it applies to an electron. The reason that the same formalism does not applies to cats is something for another day. If you want to know about cats, then drop physics and study veterinary medicine!

HighPhy said:

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

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

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

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

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

Regarding Schrödinger’s cat, there is no paradox to explain. As Carl Friedrich von Weizsäcker writes in his book “The Structure of Physics” (the book is a newly arranged and revised English version of the book "Aufbau der Physik" by Carl Friedrich von Weizsäcker):

“In an article from 1935 (see Jammer 1974, pp. 215—218) he [Schrödinger] treats with irony the Copenhagen point of view by means a thought experiment. Let a living cat be locked up in a box and with it a deadly poison which can be released by a single radioactive atom inside the box. After one half-life of the atom the probability is ##1/2## for the cat being still alive, and ##1/2## for being dead. Schrödinger describes the ##\psi##-function of the system at this time with the words: ‘The half-alive and the half-dead cat are smeared out over the entire box.’

The answer is trivial: the ##\psi##-function is the list of all possible predictions. A probability ##1/2## for the two alternative possibilities (here: "living or dead") means that the two incompatible situations must now be considered equally possible at the instant of time meant by the predic-tion. There is no trace of a paradox.”

HighPhy said:

Apologies to everyone for insisting on this thread. I would like to shed some light on this point.
...
How can the phrase "both dead and alive" be a synonym for "a superposition of two states, dead or alive"? Is the inherent formulation of this thought experiment correct? And what role does superposition play?

Thanks, now I see where you see the connection between mixed states and Schroedinger's cat.

After I learned about partial coherence in optics and how to handle it (as part of my job), I started to suspect that this might be the key for me personally to make sense of all those "the ψ-function represents knowledge" explanations of the Copenhagen interpretation. Which was not wrong, but in the end it was also my job that later forced me to really learn QM. It is a monumental time investment and constant source of frustration, because there is just so much stuff to learn.

PeroK said:

The important point is that an electron is not a cat. And "alive" and "dead" are not fundamental quantum properties like spin. In the sense that, within reason, simply observing a cat doesn't cause the cat to assume a state of alive and dead.

I reformulate. Many sources I read said that the superposition means that the cat is both alive "and" dead. But AFAIK, the addition in the wave function doesn't mean "and". The way I learned it, it means a sort of "or", so the superposition should say that the cat is dead or alive, with the appropriate probabilities. Is this correct? I'd like to figure out if and where I'm going wrong.

I've also read that " Schroedinger was wrong because quantum mechanics does imply that such superpositions are totally allowed and must be allowed and this fact can be experimentally verified – not really with cats but with objects of a characteristic size that has been increasing". And that "macroscopic objects have already been put to similar "general superposition states" and from a scientifically valid viewpoint, the thought experiment shows that superpositions are indeed always allowed – it is a postulate of quantum mechanics – even if such states are counterintuitive." Is this point of view correct?

This is the source of my confusion. Many sources say that "the cat is in a state that is a superposition of our "life states" alive and dead" means "the cat is both alive and dead" until we open the box.

Other sources say that Schrodinger's cat is not both dead and alive any more than an electron simultaneously exists at every point in space, also because a system cannot be in multiple states at once. So Schrodinger's cat would be always in a single state: there would be an equal probability of us "measuring" the cat to be either alive or dead once we open the box. Therefore, "the cat is in a state that is a superposition of our "life states" alive and dead" would not mean that "the cat is both alive and dead" until we open the box, because is in a single state, and this state is described as a superposition of life states.

I'm not speculating on any theory, I am just very confused. I'm describing some of the interpretations I read because I would like to see more clearly about which one is the most appropriate and reliable.

All this I say despite the fact that I am still aware that the cat represents a classical system. Therefore it does not behave like an electron. My question was not addressing so much Schroedinger's purpose (criticizing the Copenhagen interpretation) and the absurdity to which he wants to lead us. It was directed at the interpretation of "both alive and dead" and "superposition."

PeroK said:

The reason that the same formalism does not applies to cats

If we have a pure spin-1/2 state ##\vert \hat n\rangle##, then we can always find some linear combination of spin operators ##\sigma_{\hat n}## with ##\vert \hat n\rangle## as an eigenvector. Thus, it makes perfect sense to think of ##\vert \hat n\rangle## as a single state, which can be expanded in a basis of eigenstates of ##\sigma_z## so that
$$
\vert \hat n\rangle = \cos\left(\textstyle\frac{\theta}{2}\right)\vert +\rangle_z + e^{i\phi}\sin\left(\textstyle\frac{\theta}{2}\right)
\vert -\rangle_z\, , \tag{1}
$$
for some ##\theta## and ##\phi##. Whether one chooses to describe ##(1)## as a state that is spin-up and spin-down (with suitable probabilities) until one makes a measurement with ##\sigma_z##, or as a single quantum state expanded on two basis states is a matter of semantics: both description will lead to the same results. If we measure ##\sigma_z##: some of the time the outcome will be spin-up, some of the time the outcome will be spin-down. Moreover, if we measure in the direction ##\hat n##, there will be a single outcome.

Of course, things are different for a cat. There is no "cat" operator ##\sigma_{\hbox{cat}}## with eigenstate
$$
\vert\hbox{cat}\rangle= \cos\left(\textstyle\frac{\theta}{2}\right)\vert \hbox{dead}\rangle + e^{i\phi}\sin\left(\textstyle\frac{\theta}{2}\right)
\vert \hbox{alive}\rangle\, . \tag{2}
$$
The sense of the superposition ##(2)## as a single quantum state eigenstate of a non-existent ##\sigma_{\hbox{cat}}## operator, and thus analog of ##\vert \hat n\rangle## is rather abstract, but maybe the sense of the superposition of alive and dead cat could be clear as a generalization of the right hand side of ##(1)## (??).

Does this description fit?

@HighPhy it would be better if we understood where you are learning QM? Are you a university student?

PeroK said:

@HighPhy Are you a university student?

Yes.

HighPhy said:

Yes.

Then, forget Schrodinger's cat and learn the QM you need for your course. IMO, you need to immerse yourself in QM and forget the macroscopic world for the time being. The objective is to retrain your understanding of nature to include non-classical concepts. The more you focus on the macroscopic world, the less you focus on QM.

I would treat all popular sources as a distraction.