I want to know how many ways the identity operator $I$ on a (finite) Hilbert space $\mathcal{H}$ can be written as sum of outer products of states like $|\psi_i\rangle\langle \psi_i|$.
For example, $$I=\sum_i |u_i\rangle\langle u_i|,$$ where $\{|u_i\rangle\}_i$ is any orthonormal basis of $\mathcal{H}$.
But also, if $\mu$ is the Haar measure over $\mathcal{H}$, then I believe that $$I=\int |x\rangle\langle x| d\mu(x) $$ is also correct?
My question is whether there are any other ways to write the identity operator as a sum of outer products. If $\mathcal{H}=\mathcal{H_1}\oplus \mathcal{H_2}$ we can write the identity in the first form on $\mathcal{H_1}$ and in the second on $\mathcal{H_2}$. But is there another way?
I tried to find another 'discrete' sum but one where the vectors weren't orthonormal, but everything I tried failed. After reading other stack exchange questions I found out that these things are called 'resolutions of the identity', but I couldn't work out the answer to the above question.
I found an article that discusses this (for the discrete case): "A complete classification of quantum ensembles having a given density matrix". There they show many different ways to write the identity matrix. For example, you can write it as the outer product of $\{1,0\}$, $\{1/2, \sqrt{3}/2\}$, $\{1/2, -\sqrt{3}/2\}$ (with a normalisation factor).