The Tsallis and Renyi 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 Renyi entropy includes several useful entr
opy 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 Renyi 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 Renyi entropy tends to infinity in logarithmic order of time independent of the choice of the parameter. Moreover, we show the difference between the Renyi entropy and the logarithmic function characterizes by the Renyi entropy of the limit distribution of the quantum walk. In addition, we show an example of asymptotic behavior of the conditional Renyi entropies of the quantum walk.
In this paper, we consider continuous-time quantum walks (CTQWs) on finite graphs determined by the Laplacian matrices. By introducing fully interconnected graph decomposition of given graphs, we show a decomposition method for the Laplacian matrices
. Using the decomposition method, we show several conditions for graph structure which return probability of CTQW tends to 1 while the number of vertices tends to infinity.
For a spatial characteristic, there exist commonly fat-tail frequency distributions of fragment-size and -mass of glass, areas enclosed by city roads, and pore size/volume in random packings. In order to give a new analytical approach for the distrib
utions, we consider a simple model which constructs a fractal-like hierarchical network based on random divisions of rectangles. The stochastic process makes a Markov chain and corresponds to directional random walks with splitting into four particles. We derive a combinatorial analytical form and its continuous approximation for the distribution of rectangle areas, and numerically show a good fitting with the actual distribution in the averaging behavior of the divisions.
