How to measure relative phases of quantum states

If I have a large number of identical systems in identical quantum state $\Psi$ and an observable $A$ whose eigenstates are $\alpha_n$:

$$\Psi = \sum_n c_n \alpha_n$$

I can get absolute values of $|c_n|^2$ by measuring $A$, but with which experiment I can get relative phases of $c_n$?

• arxiv.org/pdf/1410.0916v1.pdf – valerio Jun 9 '16 at 13:30
• @valerio92 could you maybe summarize the article? I'm not a professional physicist. – xaxa Jun 9 '16 at 16:17

The phase ambiguity is a bit worse than you think. There is a global phase ambiguity in $|\Psi⟩$, for sure, but if the state $$|\Psi⟩=\sum_n c_n |\alpha_n⟩ \tag 1$$ is all you have around, then there is also a phase ambiguity in the phase of each individual $c_n$. This is because if you change $|\alpha_n⟩$ to $|\alpha'_n⟩=e^{i\theta_n}|\alpha_n⟩$, the transformed state is also an eigenstate of $A$ with eigenstate $|\alpha_n⟩$, so the two have exactly equal standing. If your universe only ever contains $|\Psi⟩$ (and a measurement device for $A$), then you only ever care about the absolute values $|c_n|^2$ of the coefficients.
The phase of these coefficients comes into play if you have two or more states, i.e. if you introduce some second state $$|\Phi⟩=\sum_n b_n |\alpha_n⟩ \tag 2$$ into the mix. Here transforming the eigenstates by some phase, as before, will still change the phases of the individual coefficients, but it will leave their differences $\arg(c_n)-\arg(b_n)$ unchanged. These are the phase differences that can be detected by experiment.
So, to come to the question, how do you detect them? There are a bunch of ways, but they all involve interference in some way. As a simple example, take your two states $|\Phi⟩$ and $|\Psi⟩$, and a superposition of them, $$\frac{|\Phi⟩+|\Psi⟩}{\sqrt 2} = \sum_n \frac{c_n+b_n}{\sqrt 2} |\alpha_n⟩,\tag 3$$ and measure $A$. Then you will measure the eigenvalue $\alpha_n$ with a probability $$P_n=\frac12 |c_n+b_n|^2$$ which is sensitive to the relative phase of the two coefficients.
• That's a nice catch about phases of eigenstates... Now I'm a bit confused: to which one of them does a system collapse after measurement according to QM? Of a set of systems in identical state, will each one collapse to the same $e^{i\theta_n}\alpha_n$ or $\theta_n$ may be different? – xaxa Jun 9 '16 at 15:24