In a paper I am reading, they are studying states of an harmonic oscillator.
They say that because the coherent states $|\alpha\rangle$ are not orthogonal, we can expand an arbitrary density matrix as a linear combinaison of coherents states :
$$\rho = \int d\alpha ~ w(\alpha) |\alpha\rangle \langle\alpha|$$
I don't understand the non orthogonality argument ? Does it really matter ?
Remark : I am not used at all with the density matrices, I saw it some time ago and I never really used it in practice.
[Answer to the comments] :
I have read the definition part of the page https://en.wikipedia.org/wiki/Glauber%E2%80%93Sudarshan_P_representation
And I don't get some things.
First, they say that :
$$\rho=\int d\alpha P(\alpha)|\alpha\rangle \langle \alpha |$$
is diagonal in the coherent states basis. But I don't totally get it. It would mean that :
$$ \rho | \beta \rangle = \lambda | \beta \rangle$$
But, as the coherent states are not orthogonal it is not obvious for me. Indeed we have :
$$ \rho | \beta \rangle = \int d\alpha P(\alpha) \langle \alpha | \beta \rangle | \alpha \rangle$$
Why would the quantity $P(\alpha) \langle \alpha | \beta \rangle | \alpha$ be proportional to a Dirac ?
What's more, you said (in the comment) that they prove this formula but for me they start from this point they don't prove we can write it this way.
 : Extra question : how do we know if we have a good basis to write the density matrix ? Is the condition just that we have to have vectors that can span all our Hilbert space when we write $\sum_i |\psi_i\rangle \langle \psi_i|$ or there are extra (or fewer) conditions ?
The context : we have a state that has an harmonic oscillator hamiltonian (so we are in the Fock space of the harmonic oscillator).
Link to the article : https://arxiv.org/abs/1206.3405