A Note on Parallel Distinguishability of two Quantum Operations

We consider a homogeneous system of linear equations of the form $A_α^{\otimes N} {\bf x} = 0$ arising from the distinguishability of two quantum operations by $N$ uses in parallel, where the coefficient matrix $A_α$ depends on a real parameter $α$. It was conjectured by Duan et al. that the system has a non-trivial nonnegative solution if and only if $α$ 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.

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