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

We convert a conjectured inequality from quantum information theory, due to He and Vidal, into a block matrix inequality and prove a special case. Given $n$ matrices $A_i$, $i=1,\dots ,n$, of the same size, let $Z_1$ and $Z_2$ be the block matrices $Z_1:=(A_j{A}_i^{\ast }{)}_{i,j=1}^n$ and $Z_2:=(A_j^{\ast }{A}_i{)}_{i,j=1}^n$. Then the conjectured inequality is $${(\mathrm{\mid }\mathrm{\mid }{Z}_1\mathrm{\mid }{\mathrm{\mid }}_1-\text{trace}{Z}_1)}^2+{(\mathrm{\mid }\mathrm{\mid }{Z}_2\mathrm{\mid }{\mathrm{\mid }}_1-\text{trace}{Z}_2)}^2\le {\left(\sum _{i\mathrm{\ne }j}\mathrm{\mid }\mathrm{\mid }{A}_i\mathrm{\mid }{\mathrm{\mid }}_2\mathrm{\mid }\mathrm{\mid }{A}_j\mathrm{\mid }{\mathrm{\mid }}_2\right)}^2\mathrm{.}$$ We prove this inequality for the already challenging case $n=2$ with $A_1$ equal to the identity matrix.