Can we say $\rho=V \rho V^*$, if $\rho$ is a positive semi-definite matrix (and $Tr(\rho)=1$) and $V$ is an isometry matrix ($V^*V=I$)?

Question

We have $\rho$ and $V$ which $\rho$ is a positive semi-definite matrix with $Tr(\rho)=1$ and $V$ is an isometry matrix $V^*V=I$.

Now, is it true to say:

$\rho=V \rho V^*$

I should mention that $\rho \in \mathfrak{D}(H^{A})$ and $V: H^{B} \rightarrow H^{C} $. $H^{A}$, $H^{B}$, and $H^{C}$ are three hilbert spaces.

I have no idea, This may be a stupid question.

Answer 1

No. Consider $\rho = \begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix}$ and $V = \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$. Then $$V\rho V^* = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}.$$