# A Note on Parallel Distinguishability of two Quantum Operations

## TL;DR

This paper investigates the conditions under which two quantum operations can be distinguished in parallel, confirming a conjecture for up to ten uses by explicitly constructing solutions to related linear systems.

## Contribution

It proves the necessity and sufficiency of a conjecture on the distinguishability of quantum operations for up to ten uses, providing explicit solutions.

## Key findings

- Confirmed the necessity part of the conjecture.
- Established sufficiency for N ≤ 10 with explicit solutions.
- Enhanced understanding of quantum operation distinguishability.

## Abstract

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.