I'm trying to fulfil all the details in a proof of a result in quantum computing. Suppose that I have two linearly independent commuting projectors $P_1$ and $P_2$ defined on a vector space of dimension $2^n$ such that
- $P_1^2 = P_1$ and $P_2^2 = P_2$ having only eigenvalues equal to $0$ or $+1$ -- by definition of a projector
- by construction the traces $Tr(P_1) = Tr(P_2) = 2^{n-1}$ -- thus both projectors have $2^{n-1}$ linearly independent eigenvectors associated to the eigenvalue $+1$, and $2^{n-1}$ linearly independent eigenvectors associated to the eigenvalue $0$.
So, the eigenspace $E_{+1}(P_1)$ has dimension $2^{n-1}$. Now, if I apply the second projector on that subspace, I would like to conclude that the subspace would be split into two subspaces of dimension $2^{n-2}$. Is the fact both projectors commute (i.e., they are orthogonal) sufficient? Or do I miss another important property on the defined operators?
Ultimately, I would apply this reasoning $n-k$ times with $k$ distinct mutually commuting projectors!