Let $\rho \in \mathfrak{L}(A)$ and $\sigma \in \mathfrak{L}(A)$. Now I want to know can we say that:
$\rho \geqslant 0 $ and $\sigma \geqslant 0 \quad \quad $ if and only if $\quad \quad Tr(\rho \sigma) \geqslant 0$
$\rho \geqslant 0 $ and $\sigma \geqslant 0 $ means that both of them are positive semi-definite operator and $Tr$ means trace. $A$ is a Hilbert space and $\mathfrak{L}(A)$ means all linear maps on $A$.
If the above theorem is true could you please help me how can I show it?
Best Answer
Implication
$$\quad \quad Tr(\rho \sigma) \ge 0 \ \implies \ \rho \succcurlyeq 0 \ \text{and} \ \sigma \succcurlyeq 0$$
cannot hold; here is a counterexample:
$$\rho=I \ \text{and} \ \sigma=\begin{pmatrix}1&2\\2&0\end{pmatrix}.$$
We have $Tr(\rho \sigma)=1 > 0$, but $\sigma$ not positive definite.
For the other implication, see here.