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 $\rho$ is pure if and only if $\mathrm{T}\mathrm{r}{\rho }^2=1$. But is this the only way to prove the purity of $\rho$ using the trace of its powers? Here I systematically study the necessary and sufficient conditions of the more general question of guaranteeing that all eigenvalues of a Hermitian matrix belong to a specified set through its trace of powers. More importantly, these characterization results are automatically witnesses certifying a quantum state as entangled, a Hermitian operator has least one negative eigenvalue and a linear operator as non-Hermitian. I demonstrate the effectiveness of these witnesses, analyze their performance, and study their strength, weakness together with resource requirement through numerical simulation as well as analytical work. I also discuss briefly the effects of numerical stability, uncertainty in measurement and rounding errors on these problems.