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We convert a conjectured inequality from quantum information theory, due toHe and Vidal, into a block matrix inequality and prove a special case. Given$n$ matrices $A_i$, $i=1,\ldots,n$, of the same size, let $Z_1$ and $Z_2$ bethe block matrices $Z_1:=(A_jA_i^*)_{i,j=1}^n$ and $Z_2:=(A_j^*A_i)_{i,j=1}^n$.Then the conjectured inequality is \[ \left(||Z_1||_1-\trace Z_1\right)^2 +\left(||Z_2||_1-\trace Z_2\right)^2 \le \left(\sum_{i\neq j} ||A_i||_2||A_j||_2\right)^2. \] We prove this inequality for the already challengingcase $n=2$ with $A_1$ equal to the identity matrix.

On a Block Matrix Inequality quantifying the Monogamy of the Negativity of Entanglement