A note on parallel distinguishability of two quantum operations

In this work, the authors consider a homogeneous system of linear equations of the form A(alpha)(circle times N)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 <= 10 by presenting explicit non-trivial nonnegative solutions for the linear system.