  We consider a homogeneous system of linear equations of the form
$A_\alpha^{\otimes N} {\bf x} = 0$ arising from the distinguishability of two
quantum operations by $N$ uses in parallel, where the coefficient matrix
$A_\alpha$ depends on a real parameter $\alpha$. It was conjectured by Duan et
al. that the system has a non-trivial nonnegative solution if and only if
$\alpha$ lies in a certain interval $R_N$ depending on $N$. We affirm the
necessity part of the conjecture and establish the sufficiency of the
conjecture for $N\leq 10$ by presenting explicit non-trivial nonnegative
solutions for the linear system.
