 Single-copy Werner filtering normalization bound

Write $W_F=(I\otimes I-w{\sum }_j{\sigma }_j\otimes {\sigma }_j)/4$, with $w=(4F-1)/3\in (1/3,1]$. For positive matrices $X=A^{†}A=a_{0}I+a\cdot \sigma$ and $Y=B^{†}B=b_{0}I+b\cdot \sigma$, the success probability is $p=a_0{b}_0-wa\cdot b$. It is at least $a_0{b}_0-|a||b|\ge \sqrt{(a_0^2-|a{|}^2)(b_0^2-|b{|}^2)}=|\det A\det B|$. Squaring the middle comparison leaves $(a_0|b|-b_0|a|{)}^2\ge 0$.