In this work, the authors consider a homogeneous system of linear equations of the form $A_\alpha^{\otimes N} \mathbf{x} = 0$ 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 $N$. The authors 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.
On Nonnegative Solution of Multi-Linear System with Strong $\mathcal{M}_z$-Tensors
