A Note on Parallel Distinguishability of two Quantum Operations

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.