Let $n$ be a positive integer and let $P_1,\dots,P_n\in M_n(\mathbb{C})$ be a collection of positive semidefinite $n\times n$ matrices satisfying the following properties:
- $\operatorname{Tr}P_i=1$ for each $i\in\{1,\dots,n\}$
- $P_iP_j = 0$ whenever $i\neq j$.
It is necessarily the case that there is an orthonormal basis $u_1,\dots,u_n$ such that $P_i = u_i u_i^*$ for each $i\in\{1,\dots,n\}$?
My thoughts:
- The matrices $P_1,\dots,P_n$ are normal and commuting, so they are simultaneously diagonalizable. In particular, we might as well assume without loss of generality that the are diagonal.
Well, you've gotten to the point where they are diagonal. Note that a product of diagonal matrices is $0$ iff they do not share non-zero entries, i.e. for any $i \neq j$ and $1 \leq k \leq n$ we get $(P_i)_{k,k} \neq 0 \Rightarrow (P_j)_{k,k} = 0$. Use the pigeon hole principle to conclude.