If we can prove that superposition exists, then why can't we measure it?

Question

If we experimentally proved that superposition exists, why can't we test if anything is in a superposition state? This seems like a contradiction because one would think that, if superposition exists, we should we able to find something in that state.

For example, as a proof of existence of the superposition, the double slit experiment is often given. But if double slit experiment shows that particles are in superposition, then it could show when superposition collapse, but this is seemingly not the case.

What is the explanation for this?

Answer 1

We do measure superpositions all the time.

• We can measure if light is diagonally polarized with a polarizing filter, but diagonal polarization is an equal superposition of horizontal and vertical polarization.
• We can measure the momentum of a particle, but a definite momentum state is a superposition of infinitely many definite position states.
• We can measure if a spin 1/2 particle has its spin pointing to the right, but this state is an equal superposition of spin up and spin down.
• We can measure if a spot on the screen in the double slit experiment is dark. This means the amplitude there is an equal and opposite superposition of the amplitudes from each slit.

These are all superpositions, in every sense of the word, even if they don't all make it into pop-sci.

I think you're looking for a situation where your measuring apparatus both "clicks and doesn't click" at the same time, but this is the wrong thing to look for. Measurements have definite results.

Answer 2

Superposition does not "exist," as much as it is a mathematical tool that we use to describe the world that does exist. We can't measure it because its not something to be measured.

Superposition is a concept in mathematics. A function has the property of superposition iff:

$$f(x_1+x_2) = f(x_1) + f(x_2)$$ $$f(ax) = af(x)$$

Any system which satisfies this property is said to have superposition. That's all.

The quantum physics principle you are referring to is that the quantum states of the system have superposition. This lets me say things like "The spin of the electron is a superposition of spin up and spin down." What experiments like the double-slit experiment show is that intuitive explanations which assume the spin is either up or down fail to properly predict what we show experimentally. We see experimental evidence that suggests that theories which assume the spin is either up or down yield incorrect results, as does the assumption that its spin is the "average" of spin up and spin down. We then show that the behavior is well predicted by the quantum mechanics, and that behavior has the property of superposition.

You can actually "solve" quantum mechanics waveforms without using the principle of superposition. You don't have to use that property at all. However, it becomes intractable very quickly. The fact that the equations of quantum mechanics permit us to treat a state as a superposition of several states simplifies the mathematics for us dramatically.

Answer 3

Usually, when we express a quantum state as a superposition, we expand the state $|\psi\rangle$ in terms of a set of orthogonal basis states $|n\rangle$: $$ |\psi\rangle=\sum_n \alpha_n |n\rangle . $$ We can choose any set of orthogonal basis states. Different sets are related to each other via unitary transformations $$ |n\rangle=U|m\rangle . $$

Now the important thing to realize is that when we make measurements, our measurement setup dictates which set of basis states we should use. During the measurement process the quantum state (according to particular interpretations of quantum mechanics) collapses to one particular element of that set of basis states, which means that it is now not in a superposition of that set anymore, but consists of only one of the elements. However, we can now go and expand that state in a different set of basis states and again get a superposition. This can be confirmed by doing another measurements where the setup dictates a different set of basis states.