Density matrix and the importance of non orthogonality of states in its construction

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.

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

Screenshoot from the article : • Can you post the source? I'm not sure this statement makes sense to me, and I'd like to see it in the original context. – Jahan Claes Jan 15 '18 at 0:30
• @JahanClaes I edited ! – StarBucK Jan 15 '18 at 0:48
• I don't really know much about quantum optics, but I'm pretty sure that the expression they give for the density matrix is not the most general form of the density matrix. In other words, there are density matrices that can't be written like this. This is confirmed in section VII of their reference  if you want to read more. I don't have access to the textbook they reference in , but it's probably worth a look. – Jahan Claes Jan 15 '18 at 1:37
• Please see the Definition paragraph of the following en.wikipedia.org/wiki/… Wikipedia page, where the required representation formula is proved by actual construction. – David Bar Moshe Jan 15 '18 at 8:31
• @JahanClaes Ok I will read it – StarBucK Jan 15 '18 at 10:53