# Whether the following two inequalities are correct: Tr(M^2 rho) <= Tr(M rho). norm(Mv) <= Tr(M rho), where norm(⋅) is the 2-norm of a vector.

Let $\rho$ be a density matrix (positive semidefinite and trace 1), with its rank being 1 such that
$\rho =v{v}^{\ast },$
where v is a $n×1$ 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:
$\text{Tr}\left({M}^{2}\rho \right)\le \text{Tr}\left(M\rho \right).$
$||Mv||\le \text{Tr}\left(M\rho \right),$
where $||\cdot ||$ is the 2-norm of a vector.
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Conor Daniel
Recall the eigendecomposition $M=P\mathrm{\Lambda }{P}^{\prime }$. Since
$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\left({M}^{2}\rho \right)=\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\left({v}^{\ast }{M}^{2}v\right)={v}^{\ast }{M}^{2}v$
and
$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\left(M\rho \right)={v}^{\ast }Mv,$
we have that
${v}^{\ast }\left(M-{M}^{2}\right)v={v}^{\ast }\left(P\mathrm{\Lambda }{P}^{\prime }-P{\mathrm{\Lambda }}^{2}{P}^{\prime }\right)v={v}^{\ast }P\left(\mathrm{\Lambda }-{\mathrm{\Lambda }}^{2}\right){P}^{\prime }v\ge 0$
as the difference $\mathrm{\Lambda }-{\mathrm{\Lambda }}^{2}$ is positive semidefinite (because the eigenvalues are bounded by 0 from below and by 1 from above).