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One prerequisite for secure and reliable artificial intelligence services is tracing the copyright of backend deep neural networks. In the black-box scenario, the copyright of deep neural networks can be traced by their fingerprints, i.e., their outputs on a series of fingerprinting triggers. The performance of deep neural network fingerprints is usually evaluated in robustness, leaving the accuracy of copyright tracing among a large number of models with a limited number of triggers intractable. This fact challenges the application of deep neural network fingerprints as the cost of queries is becoming a bottleneck. This paper studies the performance of deep neural network fingerprints from an information theoretical perspective. With this new perspective, we demonstrate that copyright tracing can be more accurate and efficient by using triggers with the largest marginal mutual information. Extensive experiments demonstrate that our method can be seamlessly incorporated into any existing fingerprinting scheme to facilitate the copyright tracing of deep neural networks.