= \gamma(\rho_{AB},\{M_k\})$ is given in terms of its matrix elements as $\gamma_{i,j} = \langle M_i M_j \rangle - \langle M_i \rangle \langle M_j \rangle$, and has the block structure $$\gamma = 
\begin{pmatrix}
\mathcal{A} && \mathcal{X} \\
\mathcal{X}^{T} && \mathcal{B}
\end{pmatrix},
\label{eq:bcmc}$$ where $\mathcal{A} = \gamma(\rho_A,\{A_k\})$ and $\mathcal{B} = \gamma(\rho_B,\{B_k\})$ are the covariant matrices of the reduced subsystems, and $\mathcal{X}_{i,j} = \langle A_i \otimes B_j \rangle - \langle A_i \rangle \langle B_j \rangle$.*

Similarly, we can define the symmetric version of the block covariance matrix by replacing $\mathcal{A}$ and $\mathcal{B}$ with their symmetrized counterparts, while keeping $\mathcal{X}$ unchanged. Clearly, if $\rho_{AB} = \rho_A \otimes \rho_B$ is a product state, then its block covariant matrix reduces to the block diagonal form, $\gamma(\rho_{AB}, \{M_k\}) = \mathcal{A} \oplus \mathcal{B}$, as $\mathcal{X}_{i,j}$ become zero $\forall i,j$.

If $\rho$ is a pure state on a $d$-dimensional Hilbert space, then the corresponding covariance matrix $\gamma$ satisfies following properties:

1.  The rank of $\gamma$ is equal to $d-1$.

2.  The nonzero eigenvalues of $\gamma$ are equal to 1, hence $\mbox{tr}(\gamma)=d-1$.

3.  $\gamma^2 = \gamma$.

The corresponding symmetric covariance matrix $\gamma^S$ satisfies the following:

1.  The rank of $\gamma^S$ is equal to $2(d-1)$.

2.  The nonzero eigenvalues of $\gamma^S$ 