# Question about multiplying a rank $1$ density matrix with a positive semidefinite matrix

Let $$\rho$$ be a density matrix (positive semidefinite and trace $$1$$), with its rank being $$1$$ such that $$$$\rho = v v^{*},$$$$ where $$v$$ is a $$n \times 1$$ unit vector.

Let $$M$$ be a positive semidefinite matrix such that all its eigenvalues are between $$0$$ and $$1$$.

I am trying to see whether the following two inequalities are correct: $$$$\text{Tr}\left(M^{2} \rho\right) \leq \text{Tr}\left(M \rho\right).$$$$

$$$$|| M v|| \leq \text{Tr}\left(M \rho\right),$$$$ where $$||\cdot||$$ is the $$2$$-norm of a vector.

The statement is obviously true when $$M$$ is a projector. But what about more general $$M$$? Note that if the first inequality is true, it implies the second one.

Recall the eigendecomposition $$M= P\Lambda P'$$. Since $$\mathrm{trace}(M^2\rho) = \mathrm{trace}(v^*M^2v) = v^*M^2v$$ and $$\mathrm{trace}(M\rho) = v^*Mv,$$ we have that $$v^*(M - M^2)v = v^*(P\Lambda P' - P\Lambda^2 P')v = v^*P(\Lambda - \Lambda^2)P'v\geq 0$$ as the difference $$\Lambda-\Lambda^2$$ is positive semidefinite (because the eigenvalues are bounded by 0 from below and by 1 from above).
Note that your assumptions on $$v$$ are not necessary.