\otimes {\ensuremath{|\phi_k\rangle}}{\ensuremath{\langle\phi_k|}}).$$ Then using the properties of covariance matrices, we get $$\begin{aligned}
\gamma(\rho_{AB},\{M_k\})  &=& \gamma(\sum_k p_k ({\ensuremath{|\psi_k\rangle}}{\ensuremath{\langle\psi_k|}} \otimes {\ensuremath{|\phi_k\rangle}}{\ensuremath{\langle\phi_k|}}), \{M_{k'}\}) \nonumber \\
&\geq&  \sum_k p_k \gamma({\ensuremath{|\psi_k\rangle}}{\ensuremath{\langle\psi_k|}} \otimes {\ensuremath{|\phi_k\rangle}}{\ensuremath{\langle\phi_k|}}, \{M_{k'}\}) \nonumber \\
&=& \sum_k p_k \{\gamma({\ensuremath{|\psi_k\rangle}}{\ensuremath{\langle\psi_k|}}, \{A_{k'}\}) \oplus \gamma({\ensuremath{|\phi_k\rangle}}{\ensuremath{\langle\phi_k|}}, \{B_{k''}\})\} \nonumber \\
&=& \kappa_A \oplus \kappa_B,\end{aligned}$$ where we have used concavity property of the covariance matrices in the second line, and in the third line, we have used the fact that the block covariant matrix of a product state takes the block diagonal form. This theorem can also be proven for the symmetric covariant matrices in the same manner. Clearly, if there exist no such $\kappa_A$ and $\kappa_B$, the state $\rho_{AB}$ must be entangled.$\square$

Clearly, it is not evident from Theorem 7 that the covariance matrix criterion leads to an efficient and physically plausible operational indication for separability. The main problem is to identify possible $\kappa_A$ and $\kappa_B$, as this requires an optimization over all pure state decompositions of $\rho_{AB}$. Therefore, we now focus on the cases, where the above criterion can be used efficiently to identify entangled states, by stating several corollaries of the above theorem.

**Corollary 1**\
*Let $
\gamma = 
\begin{pmatrix}
\mathcal{A} && \mathcal{X} \\
\mathcal{X}^{T} && \mathcal{B}
