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

The Tsallis and R\'enyi entropies are important quantities in the information theory, statistics and related fields because the Tsallis entropy is an one parameter generalization of the Shannon entropy and the R\'enyi entropy includes 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 this paper, we show limiting behaviors of the Tsallis and R\'enyi entropies for discrete-time quantum walks on the line which are starting from the origin and defined by arbitrary coin and initial state. The results show that the Tsallis entropy behaves in polynomial order of time with the parameter dependent exponent while the R\'enyi entropy tends to infinity in logarithmic order of time independent of the choice of the parameter. Moreover, we show the difference between the R\'enyi entropy and the logarithmic function characterizes by the R\'enyi entropy of the limit distribution of the quantum walk. In addition, we show an example of asymptotic behavior of the conditional R\'enyi entropies of the quantum walk.