\end{pmatrix}
$ be the block covariance matrix of a bipartite state $\rho_{AB}$. Then, if $\rho_{AB}$ is separable, we have* $$||\mathcal{X}||^2 \leq [1 - \mbox{tr}(\rho_A^2)][1 - \mbox{tr}(\rho_B^2)],$$ *where* $||A|| = \mbox{tr} \ \sqrt{A^{\dagger}A}$ *is the matrix trace norm.*

**Corollary 2**\
*Let $\rho_{AB}$ be a quantum state shared between two subsystems $A$ and $B$ with dimensions $d_A$ and $d_B$ respectively (with $d_A \leq d_B$), $
\gamma^S = 
\begin{pmatrix}
\mathcal{A} && \mathcal{X} \\
\mathcal{X}^{T} && \mathcal{B}
\end{pmatrix}
$ be the corresponding symmetric covariance matrix, and let $\mathcal{J} = \{j_1,...,j_{d_A^2}\} \left(\subset \{1,...,d_B^2\}\right)$ be a set of $d_A^2$ distinct indices. Then if $\rho_{AB}$ is separable, we have* $$2 \sum_{i=1}^{d_A^2} \sum_{j \in \mathcal{J}} |\mathcal{X}_{i,j}| \leq [1 - \mbox{tr}(\rho_A^2)] + [1 - \mbox{tr}(\rho_B^2)].$$

Now we will give another operational entanglement criterion based on the Schmidt decomposition on operator space, then try to relate covariance matrix criterion with the cross-norm or realignment criterion. A general bipartite quantum state $\rho_{AB}$ on a Hilbert space $\mathcal{H}_{A} \otimes \mathcal{H}_{B}$, where $d_{A}$ and $d_{B}$ are the dimensions of the Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$ respectively, can be written as $$\rho_{AB} = \sum_{k=1}^{d_A^2}\sum_{l=1}^{d_B^2} \xi_{kl} \tilde{G}^A_k \otimes \tilde{G}^B_l,
\label{eq:sd_svd}$$ where $\xi_{kl}$ are real quantities, and $\{ \tilde{G}^A_i \}$ and $\{ \tilde{G}^B_j \}$ are complete 