# Non-technical description of "superposition"

1. Jan 2, 2016

### Negatratoron

Context of this post (skim or skip, but read the last paragraph):

I like to teach things to people, especially those who believe that they won't understand. For those hard cases you almost have to trick them into learning something. That is, just start building up concepts in simple terms, and only at the very end reveal what it is you're talking about. Because if you say "Here, I'm gonna explain quantum superpositions to you" they think "I can't", turn their brains off, and go into hopeless-but-politely-attentive mode.

It feels like a dance, speaking only in ways that do not trigger them to stop thinking. Never giving them a place where they can say "yeah but that's beyond my ability to comprehend". And walking away smiling, as they struggle with the cognitive dissonance they get from understanding something they wanted to think they could not. But that's a rant for another time.

Anyway, my worst fear is always that I give someone a misconception. Misconceptions are extremely troublesome. Having one is like having a proof of "false" banging around in your head, going around proving other falses. And the more damage has been done, the more reluctant one is to change their mind. So I take misconceptions very seriously.

I am not a physicist. This post is to clear up my own misconceptions about superpositions so I can help other people clear up theirs. Please read over it and let me know whether I'm on the right track by describing superpositions this way.

The Post:

I used to have the misconception (a common one, I think) that "superposition" means a particle is literally in two places at once. The truth is way more mundane than that.

When someone says "A particle's location is in superposition", they just mean "I have not recently measured the location of the particle, but here's my guess as to where it might be".

For instance, consider an electron in a superposition of two locations. We needn't think of it as being in two places at once - we simply don't know which location it is at. Its electric field is also in superposition. When we compute how this electric field affects another particle, we do not "sum" or "average" over the superposition of electric fields and apply the result to the other particle. Instead, the other particle is pulled into superposition too. It could be affected by the electric field one of the two ways (not by both). We simply don't know which it was, until we measure an appropriate observable.

The superposition of a system is relative to a given observer. It is based on what information that observer has about the system.

The uncertainty principle tells us that an observer can never have perfect information about a system. So during measurement, the system's superposition "collapses" to some extent (meaning the observer now knows some of the specifics of the system). Then the superposition "grows" again, as the variables the observer was unable to measure do affect the evolution of the system.

Schrodinger's cat is in a superposition of being alive and dead. It is in no way both alive alive and dead simultaneously. Nor does it have a 50% chance of being alive and a 50% chance of being dead. It has a 100% chance of being one and a 0% chance of being the other, our observer just doesn't know which is which. (oh how the media likes to sensationalize Schrodinger's cat)

Really, the superposition corresponds to our observer's confidence in each alternative. With a 50-50 superposition, our observer has no idea whether the cat is alive or dead. When they look in the box, they become highly confident in one alternative over the other, and the superposition is said to collapse.

If our observer's partner looks in the box, then THEY are in superposition of having seen a living cat and having seen a dead cat. When our observer observes their partner, perhaps a look of relief on their face conveys some information and the superposition collapses a little bit. But it collapses more fully when our observer looks in the box themselves.

A superposition is just the (possibly infinite, of course) set of states a system might be in, along with the confidence level for each alternative. Different observers might even consider the same system to be in different superpositions, depending on what information they have about it.

That's all.

Thanks for reading (or at least scrolling) to the bottom of this long post. Happy new year all.

2. Jan 2, 2016

### Staff: Mentor

You are right about it not being in two places at once, but it is terribly misleading to say instead that "we simply don't know which location it is at".
That's not what the standard mathematical formalism of QM says, and the experimental confirmation of violations of Bell's inequality shows that no explanation based on "it's something, but we don't know what" is workable in the way that you want it to be (Bohmian interpretations notwithstanding - they're workable, but not as a way of explaining superposition to non-specialists).

You are right that Schrodinger's cat is either dead or or alive; it's like a tossed coin that is either heads or tails whether we look or not, but we won't know which until we look. However, this isn't going to help you explain superposition because the first sentence above is wrong - Schrodinger's cat is not in a superposition of being alive and being dead.

3. Jan 2, 2016

### andrewkirk

@Negatratoron What you are describing sounds to me like a version of Everett's many-worlds hypothesis. It's important to understand that that hypothesis is an interpretation of quantum mechanics (QM), its statements are not those of QM itself, but rather a speculation about what QM may 'mean', in layperson's terms.
There are other interpretations, such as Copenhagen and Bohm-de Broglie, which are equally consistent with QM, but are not consistent with what you wrote above.
If you are explaining to a lay person, I would recommend making the following points:

1. QM itself is just a bunch of definitions and equations that a lay person would not understand without first doing some advanced maths.

2. The equations do not say anything about 'the nature of reality' but rather they enable accurate predictions of the probabilities of certain experimental observations.

3. A statement involving words like 'superposition', 'exists', 'collapses' or 'observer' will in most cases be an interpretation of QM, not QM itself.

4. There are multiple different interpretations of QM, which are all consistent with QM itself, but not consistent with one another. There are currently no experiments that can be done to decide between the different interpretations.

With that preamble, you could legitimately present the above, but in the context that it is an interpretation, not a presentation of QM, and that it is only one of a number of possible interpretations.

4. Jan 2, 2016

### Staff: Mentor

You are way over complicating it.

Its actually quite simple. All superposition is, is the statement pure states form a vector space. That's it, that's all.

If you are trying to explain it to someone that doesn't know what a vector space is - don't. It will simply confuse them as I know only too well in trying to explain it on this forum.

If you want to explain QM IMHO the following is the best approach:
http://www.scottaaronson.com/democritus/lec9.html

Thanks
Bill

Last edited: Jan 3, 2016
5. Jan 3, 2016

### A. Neumaier

Not quite. Pure states form the unit sphere in a complex Hilbert space. When forming superpositions of states you need to respect the normalization constraint.

6. Jan 3, 2016

### A. Neumaier

Saying that a quantum system is in a superposition presupposes in the simplest but typical case that you have a particular set of measurements in mind. Being in a superposition of two states, each of which is a pure state persistent under repeated measurement means that you can predict the outcome of the measurement only in a probabilistic way, with probabilities given by Born's rule.

This is different from a mixture of the pure states, where you can also predict the outcome of the measurement only in a probabilistic way, but with probabilities given by classical probability theory.

The difference becomes visible when you manipulate the system before measurement, e.g., in a Stern-Gerlach experiment by changing the external magnetic field.

7. Jan 4, 2016

### Khashishi

The problem with the OP presentation is that a superposition state doesn't come from the ignorance of the observer. Rather, if the observer knows that a system is in a particular superposition state, they know all there is to know about the system. When the observer doesn't know what state the system is in, they assign it a mixed state. A superposition state is not the same as a mixed state.

The particle state $\psi_1 = \frac{1}{\sqrt{2}} \left(\left|up\right> + \left|down\right>\right)$
is different than the state $\psi_2 = \frac{1}{\sqrt{2}} \left(\left|up\right> - \left|down\right>\right)$
although both states have a 50% probability to be measured to be up. Both of these states contain more information than the mixture of 50% up 50% down.

A superposition indicates that the state is known but is incompatible with some measurement basis.

8. Jan 4, 2016

### Negatratoron

Interesting; I'm gonna need to read up on Bell's inequality.

Point 3 is very interesting - I didn't know "superposition" and "observer" would be considered part of interpretations of QM, rather than QM itself.

(I think I consider interpretations to be meta-physics. And you know what they call meta-physics that matters? Physics. :) And, RE: Nugatory, I realize my saying "it's something, but we don't know what" is also meta-physics. According to google, it's called counterfactual definiteness, and I should not assume it.)

(I actually just bought the book Quantum Computing since Democritus; it's very enjoyable. My other ulterior motive for learning about QM is to gain a handle on what quantum computing might be able to do for us).

So a pure state $| \psi \rangle$ is a unit vector in a hilbert space $\mathcal{H}$. A pure state is in a superposition with respect to an observable $O$ iff it does not lie on an eigenspace of that observable.

The possible outcomes of measurement are the eigenvalues $\lambda_n$ of $O$. The probability that the outcome will be $\lambda_k$ is $\langle \psi | P_k | \psi \rangle$ where $P_k$ is the projection onto the eigenspace corresponding to $\lambda_k$.

After measurement, $| \psi \rangle$ is set equal to its projection onto the appropriate eigenspace (and re-normalized, of course). Then $\langle \psi | P_k | \psi \rangle = \langle \psi | \psi \rangle = 1$ so repeated measurements will yield the same value.

Alright, so a superposition state just means the observer knows that the outcome of a measurement will be nondeterministic. A mixed state means the observer knows the state could be one of many alternatives, with classical probabilities assigned to each one.

So schrodinger's cat is in a mixed state, not a superposition.

Say we have a stream of silver atoms flying along our Y axis. Their spin, measured along any axis, will take on one of two values.

We set up a magnetic field along the Z axis, so that depending on the particles' spins, they will end up in one of two locations. We collect the particles from one location and pass them through another magnetic field along the X axis, so they again end up in one of two locations. We collect the particles from one location and pass them through another magnetic field along the Z axis.

If their spin along each axis were merely a mixed state, we would expect them to finally land in only location. Instead, they land in two again. This is because our measurement of their spin along the X axis changed their state, putting their spin along the Z axis back into a superposition.

Question: If, in the middle apparatus, we do not block one of the streams, but let them both through, then will they wind up in only one location at the very end? (I've illustrated this question by modifying a picture I got off Wikipedia)

9. Jan 4, 2016

### andrewkirk

So do I. But there's nothing wrong with metaphysics, as long as one doesn't confuse it with physics.
I don't call it that. For instance I make the metaphysical assumption of non-solipsism - that there are more conscious processes than the one I experience. That assumption matters to me a great deal, and I get the impression that it also matters to many other people to believe they are not alone in the world. But I would not call it Physics. I subscribe to Popper's approach that science consists only of what is empirically falsifiable.

10. Jan 5, 2016

### A. Neumaier

Not quite. This is the way mixed states are usually introduced, but this is valid only in special circumstances where a mixed state is created as a mixture of known pure states. But once the mixture is assembled one cannot tell at all of which pure states it is composed, since there are infinitely many essentially different ways to dissect a mixure into pure states. None of these has a physical meaning - i.e., it is impossible to devise an experiment that would tell which pure states the mixture is composed of.

11. Jan 5, 2016

### A. Neumaier

The figure is ambiguous as the magnetic field deflects the beams, and the effect of the field on the beam depend on the relative angle. But if you do the same experiment with light and polarization filters, there is no ambiguity and the answer is yes.