From Certifying Rank $k$ Projectors To Non-Positivity, Entanglement, And Non-Hermitian Witnesses Through Traces Of Matrix Powers

We all know that a density matrix $ρ$ is pure if and only if $\mathrm{Tr} \,ρ^2 = 1$. But is this the only way to prove the purity of $ρ$ using the trace of its powers? Here I systematically study the necessary and sufficient conditions of the more general question of guaranteeing that all eigenvalu