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The Tsallis and R\'enyi entropies are important quantities in the informationtheory, statistics and related fields because the Tsallis entropy is an oneparameter generalization of the Shannon entropy and the R\'enyi entropyincludes several useful entropy measures such as the Shannon entropy,Min-entropy and so on, as special choices of its parameter. On the other hand,the discrete-time quantum walk plays important roles in various applications,for example, quantum speed-up algorithm and universal computation. In thispaper, we show limiting behaviors of the Tsallis and R\'enyi entropies fordiscrete-time quantum walks on the line which are starting from the origin anddefined by arbitrary coin and initial state. The results show that the Tsallisentropy behaves in polynomial order of time with the parameter dependentexponent while the R\'enyi entropy tends to infinity in logarithmic order oftime independent of the choice of the parameter. Moreover, we show thedifference between the R\'enyi entropy and the logarithmic functioncharacterizes by the R\'enyi entropy of the limit distribution of the quantumwalk. In addition, we show an example of asymptotic behavior of the conditionalR\'enyi entropies of the quantum walk.

Asymptotic analysis of the one-dimensional quantum walks by the Tsallis and R\'enyi entropies