Let $\rho_0,\rho_1, \rho_2$ be Positive Semidefinite, Hermitian, and trace one complex matrices with $\rho_i^2 = \rho_i$. Let $V$ be Hermitian and $V^{2} = I $. I search for an upper bound of $$|tr(\rho_1 [\rho_0, V])tr(\rho_2 [\rho_0, V])|$$ ideally separating the trace of the commutator. As the commutator is not necessarily PSD, I cannot use submultiplicity. Using linearity seems not to help either.
Edit: the following part is redundant (see comment): Furthermore, I search for a non-zero lower bound for $$tr( \rho_1 \rho_0 \rho_2 \rho_0)$$ As the eigenvalues of $\rho_i$ are $0$ or $1$, I get only zero as a lower bound.