I'm reading Higher-order interference in quantum physics by Rozema et al. Their Figure 1 is a simple setup

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There are two paths. The upper path $|u\rangle$ is open and the lower path $|d\rangle$ is blocked. U is a unitary interaction between the two modes after the blocker, and there are N input states $\rho$. Each $\rho$ is spanned by $|u\rangle$ and $|d\rangle$. Here's what they say the intensities are:

$$I_{00}=NP_{00}=N\langle u|U\rho U^\dagger|u\rangle$$

is intensity when both paths are open.

$$I_{01}=N\rho_{uu}\langle u|U|u\rangle \langle u|U^\dagger|u\rangle$$

with $\rho_{uu}=\langle u|\rho|u\rangle$, is intensity when upper path is open.

$$I_{10}=N\rho_{dd}\langle u|U|d\rangle \langle d|U^\dagger|u\rangle$$ is intensity when lower path is open.

I think I understand $I_{00}$. We are just projecting the state after the unitary (i.e. $U\rho U^\dagger$) onto the upper path. I'm not sure why the other two are written the way they are. I know that this has a rather simple explanation, yet I cannot seem to find it. Please let me know why we can write intensities this way.


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