&=& {\ensuremath{|i\rangle}}{\ensuremath{\langlei|}}, \  \ i = 1,2,...,d, \\
Y_{ij} &=&  \frac{1}{\sqrt{2}}({\ensuremath{|i\rangle}}{\ensuremath{\langlej|}} + {\ensuremath{|j\rangle}}{\ensuremath{\langlei|}}), \ \ 1 \leq i < j \leq d, \\
Z_{kl} &=&  \frac{i}{\sqrt{2}}({\ensuremath{|k\rangle}}{\ensuremath{\langlel|}} - {\ensuremath{|l\rangle}}{\ensuremath{\langlek|}}), \ \ 1 \leq k < l \leq d.\end{aligned}$$

Let us now focus on the situation in which the Hilbert space is a tensor product $\mathcal{H}=\mathcal{H}_A \otimes \mathcal{H}_B$ of Hilbert spaces of two subsystems $A$ and $B$ with dimensions $d_A$ and $d_B$ respectively. We can consider the complete set of orthonormal observables in $A$ as $\{ A_k: k = 1,2...,d_A^2\}$ and in $B$ as $\{ B_k: k = 1,2...,d_B^2 \}$, and construct a set of $d^2_A + d^2_B$ observables as $\{ M_k \} =\{ A_k \otimes I, I \otimes B_k\}$. Although this set is not complete, it can be utilized to define a very useful form of covariant matrices, known as the block covariant matrices. The block covariant matrix for a given bipartite state $\rho_{AB}$ and orthonormal observables $\{M_k\}$ is defined as follows.

**Def. 6** *Let $\rho_{AB}$ be a quantum state of a bipartite system, and let $\{ M_k \} =\{ A_k \otimes I, I \otimes B_k\}$ be a set of orthonormal observables as stated above. Then, the block covariance matrix $\gamma 