I am trying to calculate the probability of a state (density matrix) being in a specific other state.
Lets say I have a 2-dimensional state with the states given by the orthonormal basis states $|0\rangle, |1\rangle$.
Say I have a density matrix $\hat{\rho} = |0\rangle\langle0|$
I want to calculate, what is the chance of $\hat{\rho}$ being in the superposition state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$, and $\hat{\sigma} = |\psi\rangle\langle\psi|$.
From what I've read, this is simply calculating the following:
$\operatorname{Tr}({\hat{\sigma}\hat{\rho}}) = \langle\psi|\hat{\rho}|\psi\rangle$
In my simple example, $\langle\psi|\hat{\rho}|\psi\rangle = |\alpha|^2$. This is not the probability of $\hat{\rho}$ being in the state $|\psi\rangle$ as that should be equal to $0$ as there is no $|1\rangle$ component in $\hat{\rho}$.
To me, this only makes sense when $|\psi\rangle$ is a basis state of $\hat{\rho}$. So what is it that I've just calculated and how would I obtain what I am looking for?