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Quantum networks have the potential to transmit multimedia data with high security and efficiency. However, ensuring high-fidelity transmission links remains a significant challenge. This study proposes a novel framework to enhance quantum network performance via link selection and transport strategy. Specifically, we formalize the quantum fidelity estimation and link selection as a best-arm identification problem, leverage median elimination to estimate fidelity and select the quantum link for each multimedia chunk transmission. To optimize the transmission of multimedia chunks in a quantum network, we can employ the scheduling strategy to maximize the cumulative benefit of chunk transmissions while considering the fidelity of the links and the overall network utilization. Through extensive experiments, our proposal demonstrates significant advantages. Compared to the randomized method, Minerva reduces bounce number and execution time by 12% ∼ 28% and 8% ∼ 32%, respectively, while improving average fidelity by 15%. Compared with the uniformly distributed method, our approach decreases bounce number by 24% ∼ 30% and execution time by 8% ∼ 32% and enhances average fidelity by 11% ∼ 21%.