A Note on Parallel Distinguishability of two Quantum Operations

Abstract

We consider a homogeneous system of linear equations of the form $A_{\hat{I}}{\pm }^{\otimes N}\mathbf{x}=0$ arising from the distinguishability of two quantum operations by $N$ uses in parallel, where the coefficient matrix $A_{\hat{I}}\pm$ depends on a real parameter $\hat{I}\pm$. It was conjectured by Duan et al. that the system has a non-trivial nonnegative solution if and only if $\hat{I}\pm$ 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 $N\le 10$ by presenting explicit non-trivial nonnegative solutions for the linear system.

14 pages