Physical interpretation of quantum superposition

[csmcmillion]

I understand the basics of the mathematical descriptions of systems in superposition. But what I'm not clear about is whether such a system is actually in more than one state at the same time. Is this a matter of QM interpretation or is it simply moot b/c we can make no measurements that can contradict the statement?

 

[homeomorphic]

Matter of interpretation.

The idea of it being a moot point IS an interpretation.

I don't know if super-imposed states are "real" or not. Maybe it doesn't matter, but, on the other hand, maybe attempting to answer the question will lead to progress. So, I don't think it's exactly a moot point. Maybe it is, maybe not. Even if not, maybe it's still a good idea to try to figure it out, because if nothing else, maybe we'll come to understand it better.

Schrödinger's cat illustrates the absurdity of things actually being in such a super-imposed state, at least on a large scale. From what I understand, maybe the resolution is that decoherence makes such super-imposed alive/dead states extremely unlikely/unstable on large scales like that of a cat. However, on a small scale, things like electrons are pretty much Schrödinger's cats.

 

[fleem]

Before wondering whether one "understands" something, one must have a rigorous definition of "understand". I think most people feel they understand something when they have an analogy that they are comfortable with. For example, a child may say he understands how electrons orbit the atom after being told its like planets orbiting a star (even though it isn't really like that--but that beside the point here). This also implies the analogy required for "understanding" should be more fundamental, so reductionism is also typically included in the definition of "understand". We do not have a classical analogy for QM because QM is not classical. We also do not have a more fundamental rule that yields superposition along with some other behavior we thought was unrelated. QM is certainly more fundamental than classical mechanics (because classical mechanics is a statistical analysis of many QM events), so it is a mistake to try to find a classical analogy to QM things--it will only lead to wrong assumptions about QM. We simply observe experiments and learn how to predict. That's actually what we did when we learned the classical mechanics rule of F=MA. Only after we find a more fundamental rule set (a set that describes superposition along with other things we first thought were unrelated) might we say we "understand" superposition. The concepts we learned from studying the classical world include things like continuums (like classical pictures of time and space), real numbers, mutually exclusive states, etc.. We should be leary of presuming these things must also be axioms of QM.

 

[martinbn]

It is good to keep in mind that whether a state is in superposition or not is a meaningless question. It depends on a choice of basis.

 

[granpa]

https://www.physicsforums.com/showthread.php?t=306959

when an electron in an atom transitions from one shell to another does it do it instantly or is there a time when it is partially in one and partially in the other. in other words in a superposition of both states.

In the framework of time-dependent perturbation theory, yes, the state will be a superposition of states, and there is a probability for it to decay to the ground state each unit time.

 

[homeomorphic]

It is good to keep in mind that whether a state is in superposition or not is a meaningless question. It depends on a choice of basis.

It's not meaningless once you have chosen a basis, though. For example, the dead and alive states of the cat. Dead state makes sense. Alive state makes sense. Superposition of them? Doesn't make sense.

Changing basis to make dead plus alive in the basis just dodges the question.

 

[Ken G]

It is good to keep in mind that whether a state is in superposition or not is a meaningless question. It depends on a choice of basis.

Mathematically, that's true, but observationally, it isn't always. A classic example of what you are talking about, from the physical perspective, is the polarizaton of linearly polarized light. If we have light that is linearly polarized in the "up-down" direction, then that's a pure state, and we know what an up-down polarization measurement will yield every time. But if we choose a basis of polarization at 45 degrees to up-down, we would say we have a superposition state-- even though it's exactly the same state. So in that case, the interpretation of a superposition state is clear-- it's just stemming from our choice of basis.

But this interpretation is not always so clear cut. Sometimes when we talk about a superposition state, from a physical standpoint, we might be talking about a state that is pure, mathematically, but that does not correspond to an eigenstate of any observation that we can do (say a superposition of two energy levels in a hydrogen atom). So even though we may be able to construct an operator that the pure state is an eigenstate of, if it doesn't correspond to anything we know how to observe, we're not sure what that superposition really means. The empirical foundations of science get a bit iffy when we cannot connect a state to any definite observable outcome. I think that's what the OP is really asking-- what does a state mean, physically and/or empirically, when it does not come with any definite observable outcome. This is a very good question-- it is the heart of "quantum weirdness."

 

[spocchio]

Remember that a state in the superposition in a certain basis can be a well state in another basis.
I think the problem comes when you consider a preferred base for your visualization, that is the q basis.

for example you can say that the state
|\psi\rangle = |q_1\rangle
isn't in a superposition, but it isn't in a superposition only in the position basis, in fact if we impose:
|u_1\rangle = {{|q_1\rangle + |q_2\rangle}\over \sqrt{2}}
|u_2\rangle = {{|q_1\rangle - |q_2\rangle}\over \sqrt{2}}
we obtain
|\psi\rangle = {{|u_1\rangle - |u_2\rangle}\over \sqrt{2}}

the problem of superposition, in my opinion, occurs when you do a measurement of a certain operator and your state is not an eigen state of your operator. in fact, we know how a measurement breaks linearity but we don't know when the breaking of linearity occurs (we are not even really sure if the linearity is broken..see decoherence).

 

[martinbn]

Ken_G of course you are right, but I don't think that the OP was talking about pure vs mixed states.

 

[homeomorphic]

But this interpretation is not always so clear cut. Sometimes when we talk about a superposition state, from a physical standpoint, we might be talking about a state that is pure, mathematically, but that does not correspond to an eigenstate of any observation that we can do (say a superposition of two energy levels in a hydrogen atom). So even though we may be able to construct an operator that the pure state is an eigenstate of, if it doesn't correspond to anything we know how to observe, we're not sure what that superposition really means. The empirical foundations of science get a bit iffy when we cannot connect a state to any definite observable outcome. I think that's what the OP is really asking-- what does a state mean, physically and/or empirically, when it does not come with any definite observable outcome. This is a very good question-- it is the heart of "quantum weirdness."

Yeah, I think that's what I was trying to get at with the cat.

Maybe it's better to phrase the question as a question of whether the state vector is real, rather than superposition. I guess that's slightly more general, but along the same lines.

 

[Ken G]

Ken_G of course you are right, but I don't think that the OP was talking about pure vs mixed states.

I wasn't talking about mixed states either. The question is about pure states, but it is also about when is a pure state a superposition state in some fundamentally empirical kind of way. Formally, there is no distinction between a pure state and a superposition state, the former is the true mathematical description and the latter is basis dependent (which was your point). But there is something deeper in the OP question, which makes it a question about the connection between the mathematics of QM, and the empirical basis of physics itself. That is quite an uneasy relationship, and is at the heart of a lot of questions about quantum interpretations. So it can't be made to go away by recognizing that a superposition state can be turned into a basis state-- what it cannot always be turned into is an eigenstate of an observation that we know how to do. So another way to frame the question is, what is physics talking about when it talks about "the state of a system," when that state does not conform to any definite observable we can connect to that system. Note that the ensemble interpretation sees great importance in this issue, and uses it to deny that a wave function is a "state" of an individual system at all. DeBroglie-Bohm also denies that for related reasons (while taking a very different view of the situation), and Copenhagen denies that there is any meaning to the state of the system beyond what we can say about the system (which is its wave function). So my answer to the OP is "it is interpretation dependent."

 

[martinbn]

I wasn't talking about mixed states either. The question is about pure states, but it is also about when is a pure state a superposition state in some fundamentally empirical kind of way. Formally, there is no distinction between a pure state and a superposition state, the former is the true mathematical description and the latter is basis dependent (which was your point). But there is something deeper in the OP question, which makes it a question about the connection between the mathematics of QM, and the empirical basis of physics itself. That is quite an uneasy relationship, and is at the heart of a lot of questions about quantum interpretations. So it can't be made to go away by recognizing that a superposition state can be turned into a basis state-- what it cannot always be turned into is an eigenstate of an observation that we know how to do. So another way to frame the question is, what is physics talking about when it talks about "the state of a system," when that state does not conform to any definite observable we can connect to that system. Note that the ensemble interpretation sees great importance in this issue, and uses it to deny that a wave function is a "state" of an individual system at all. DeBroglie-Bohm also denies that for related reasons (while taking a very different view of the situation), and Copenhagen denies that there is any meaning to the state of the system beyond what we can say about the system (which is its wave function). So my answer to the OP is "it is interpretation dependent."

Ok, but then I understand the question differently than you. I think that superpositions worry the OP. For him a state |a> is something 'reasonable' while |a>+|b> is border line mystic. My comment was that they should all be viewed equally, because there is no absolute way of saying if a state is of the form |a>.

 

[f95toli]

Ok, but then I understand the question differently than you. I think that superpositions worry the OP. For him a state |a> is something 'reasonable' while |a>+|b> is border line mystic. My comment was that they should all be viewed equally, because there is no absolute way of saying if a state is of the form |a>.

I am not sure I agree completely. There are for example a few qubit implementations where the two states are very well "defined" experimentally meaning their "classical" states are quite "reasonable". A couple of examples would be a charge qubit where the states are given by zero or one electron on an metallic island, or a superconducting flux qubit where the two states correspond to current flowing clockwise or counter-clockwise in a a micrometer sized loop.
Both of these qubit implementations "work" in the classical case as well (which experimentally just means raising the temperature), but then the states will of course be EITHER one or zero electrons etc.

My point is that one or zero electrons are quite "natural" (and classical) states, whereas a superposition is not. Moreover, due the decoherence a qubit will always end up in one of these two states. Hence, I think one can safely argue that the "classical" states and the superposition are not on equal footing.

 

[Ken G]

Ok, but then I understand the question differently than you. I think that superpositions worry the OP. For him a state |a> is something 'reasonable' while |a>+|b> is border line mystic. My comment was that they should all be viewed equally, because there is no absolute way of saying if a state is of the form |a>.

Yes, but the reason such states are borderline mystic is not because of their mathematical relationship to a basis system, it is because of their empirical relationship to observations we can actually do. There is indeed an empirically distinguishable meaning to a state of form |a>, it is the certainty of measurement outcome A that corresponds to state |a>, which is used as the empirical confirmation that we do in fact have state |a>. How do we know when we have state |a>+|b> where there is no equivalent definite observable there? It becomes a different kind of empirical animal, even if its mathematical place in the Hilbert space is no different.

Thus, it is in that empirical sense, rather than Hilbert-space mathematical sense, that the question persists: what does it mean for a system to be in a state that does not correspond to a definite outcome of any measurement that we know how to do on that system? What does one even mean, empirically, by "the state" of the system in that situation? This question is very much at the heart of all the different quantum interpretations.

 

[Ken G]

My point is that one or zero electrons are quite "natural" (and classical) states, whereas a superposition is not. Moreover, due the decoherence a qubit will always end up in one of these two states. Hence, I think one can safely argue that the "classical" states and the superposition are not on equal footing.

That is similar to what I am saying as well. Note that what we mean by a definite outcome to an observation is where decoherence comes in-- we intentionally choose measurement devices that we know act in such a way as to set up decoherence of the desired kind, decoherence that will separate the different possible eigenvalues of that particular observation. So we can say that there are types of decoherence that we know how to produce, the types for which |a> is an eigenstate that will definitely give A, and there are types of decoherence that we don't know how to produce, for which |a>+|b> might be an eigenstate but we don't really know what that means because we've never set up that kind of decoherence.

I presume QM formalists view it as an uninteresting detail if we don't know how to set up the decoherence for which |a>+|b> is an eigenstate, it doesn't in any way affect the place of |a>+|b> in the mathematical structure of QM. But those who are more worried about the connection between QM formalism and what is empirically testable, see a more fundamental difference between states like |a> and states like |a>+|b>, and adopt the ensemble interpretation (or some other Copenhagen variant) as a result.

 

[martinbn]

Here is quote from Dirac's book.

"The general principle of superposition of quantum mechanics
applies to the states, with either of the above meanings, of anyone
dynamical system. It requires us to assume that between these
states there exist peculiar relationships such that whenever the
system is definitely in one state we can consider it as being partly
in each of two or more other states. The original state must be
regarded as the result of a kind of superposition of the two or more
new states, in a way that cannot be conceived on classical ideas. Any
state may be considered as the result of a superposition of two or
more other states, and indeed in an infinite number of ways. Con-
Conversely any two or more states may be superposed to give a new
state. The procedure of expressing a state as the result of super-
superposition of a number of other states is a mathematical procedure
that is always permissible, independent of any reference to physical
conditions, like the procedure of resolving a wave into Fourier com-
components. Whether it is useful in any particular case, though, depends
on the special physical conditions of the problem under consideration."

 

[Ken G]

Right, there is no question that formal quantum mechanics views superposition states as pure states in a different basis, so makes no formal distinction. Yet when we frame the issue empirically, rather than mathematically, it is Dirac's last sentence that is the jumping off point for what I believe is being asked in this thread. When it is not at all useful to think of a state as a superposition, then to what extent shall we say that the state "really is" in such a superposition? This is asking for the meaning of the wave function, which is a matter of interpretation-- it's the cat paradox. The quantum mechanics that Dirac is talking about does not tell us what the wave function means, it only tells us its mathematical status in a predictive theory.